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| LastUpdate 11/29/2024 |
| "Grapes" and "Function View" are famous as excellent free software that draws very easily just by substituting formulas of functions. In high school mathematics classes , when teaching graphs of quadratic functions , students create an xy correspondence table , plot the points on the xy-coordinate plane , and draw a graph by connecting the points with a smooth curve. "Grapes" and "Function View"draw graphs simply by substituting function expressions. Therefore , there is little sense of creating an xy correspondence table , plotting the points , and connecting them with a smooth curve. However , Excel is the software that makes it possible on a computer to create an xy correspondence table , plot points , and draw a graph with a sense of connecting them with a smooth curve. Excel is a spreadsheet sofrware that we use on a daily basis. When I went to a public library in Higashi-Omi , I happened to come across a book titled "Easy Mathematics Learning with Excel (from Trigonometory to Calculus) , Co-authered by Yukihisa Takahashi and Ha Watanabe , Ohmusha Publishinng". I found a description that made me realize that how to draw a graph of a funcion using "Excel" and the way of thinking about numerical sequences is exactly the copy function of "Excel". In addition , I introduced what can be done using "Excel" in "Simulation" , "Prime and Perfect Numbers" , and "Polor Equations" in Table of Contents 12-14. Please try it. You should be able to apply it to the other things as well. By using "Grapes" , "Function View" , and "Excel" according to the senses , skills , and knowledge you want students to acqure , I believe that more effective lessons can be organized. |
| http://kn-makkun.com/MakkunWp/excels.html |
| No | Table of contents |
| １ | Quadatic function |
| ２ | Trigonometric function |
| ３ | Exponential function |
| ４ | Logarithmic function |
| ５ | Varius curves |
| ６ | Differentiation (Part 1) |
| ７ | Differentiation (Part 2) |
| ８ | Integral |
| ９ | Number sequence |
| １０ | Complex number |
| １１ | Application of square root |
| １２ | Simulation |
| １３ | Prime snd perfect numbers |
| １４ | Polar equation |
| １５ | Download |
| １６ | How to use the sample data |
| Back to the Top Page |
| To table of contents |
| 【１０１】 Draw a graph of the quadratic functions of y=x^2 , y=2x^2 , and y=3x^2. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -4 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B20 , right-click , and select Paste. Enter =B12^2 in half-width in cell C12. Enter =2*B12^2 in half-width in cell D12. Enter =3*B12^2 in half-width in cell E12. Drag the cell range C12:E12 , right-click , and select Copy. Drag the cell range C13:C20 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E20 in the correspondence table of (1) above. ② Left-click [Insert]→[Scatter plot]→[Smooth line and marker]. (3) Arrange the xy coordinate plane on which the graoh is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add grid lines ② Right-click on the numbers on the y-axis → Axis formating → Scale interval Fixed 5.0 → Close (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click [Insert]→[Scatter plot]→[Smooth line].  ＜Consideration 1＞ Compare the spread of graphs of y=x^2 , y=2x^2 , and y=3x^2. ＜Consideration 2＞ In general , how does the spread of the graph of y=ax^2 change depending on the value of a ? |
| 【１０２】 Draw a graph of the quadratic functions of y=2x^2 and y=-2x^2. | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【１０１】 how to enter the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D20 in the correspondence table of (1) above. ② Left click ［Insert］→［Scatter plot］→［Smooth line and Marker］ (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formating → Scale interval Fixed 5.0 → Close (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it . ② Left-click ［Insert］→［Scatter plot］→［Smooth line］  ＜Consideration 1＞ State the positional relationship between the graphs of y=2x^2 and y=-2x^2. ＜Consideration 2＞ In general , what is the positional relationship between the graphs of y=ax^2 and y=-ax^2 ? |
| 【１０３】 Draw a graph of the quadratic functions of y=2x^２ , y=2x^2+3 , and y=2x^2-3. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【１０１】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E20 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line and Marker］. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formating → Scale interval Fixed 5.0 → Close (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph-area to activate it. ② Left-click ［Insert］→［Scatter plot］→［Smooth line]  ＜Consideration 1＞ Compare the spread of the graphs of y=2x^2 , y=2x^2+3 , and y=2x^2-3. ＜Consideration 2＞ State the positional relationship of the graphs of y=2x^2 , y=2x^2+3 , and y=2x^2-3. ＜Consideration 3＞ In general , what is the positional relationship between graphs of y=ax^2 and y=ax^2+b ? |
| 【１０４】 Draw a graph of the quadratic functions of y=2x^2 , y=2(x+2)^2 , and y=2(x-2)^2. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【１０１】 how to enter the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E20 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line and Marker］ (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formating → Scale interval Fixed 5.0 → Close (4) Deleate points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］  ＜Consideration 1＞ Compare the spread of the graphs of y=2x^2 , y=2(x+2)^2 , and y=2(x-2)^2. ＜Consideration 2＞ State the positional relationship of the graphs of y=2x^2 , y=2(x+2)^2 , and y=2(x-2)^2. ＜Consideration 3＞ In general , what is the positional relationship between graphs of y=ax^2 and y=a(x-b)^2 ? |
| 【１０５】 Draw a graph of the quadratic functions of y=2x^2 and y=2(x-2)^2+3. | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the folllowing correspondence table on the worksheet. ＜Reference＞ Please refer to 【１０１】 how to enter the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D20 in the correspondence table of (1) above. ② Left -click ［Insert］→［Scatter plot］→［Smooth line and Marker]. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formating → Auxiliary scale interval Fixed 1.0 → Close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］  ＜Consideration 1＞ Compare the spread of the graphs of y=2x^2 and y=2(x-2)^2+3. ＜Consideration 2＞ State the positional relationship of the graphs of y=2x^2 and y=2(x-2)^2+3. ＜Consideration 3＞ In general , what is the positional relationship between the graphs of y=ax^2 and y=a(x-b)^2+c. |
| 【１０６】 Draw a graph of the quadratic functions of y=-x^2 and y=-x^2+6x-7. | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【１０１】 how to enter the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D20 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line and Marker］ (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formating → Auxiliary scale interval Fixed 1.0 → Close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］  ＜Consideration 1＞ Compare the spread of the graphs of y=-x^2 and y=-x^2+6x-7. ＜Consideration 2＞ State the positional relationship of the graphs of y=-x^2 and y=-x^2+6x-7. |
| To table of contents |
| 【２０１】 Draw a graph of the trigonometric functions of ｙ＝ｓｉｎｘ , ｙ＝ｓｉｎ２ｘ , and ｙ＝ｓｉｎ３ｘ. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B192 , right-click , and select Paste. Enter = SIN(RADIANS(B12)) in half-width in cell C12. Enter = SIN(RADIANS(2*B12)) in half-width in cell D12 Enter = SIN(RADIANS(3*B12)) in half-width in cell E12 Drag the cell range C12:E12 , right-click , and select Copy. Drag the cell range C13:C192 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E192 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration 1＞ Find the period of the graph of ｙ＝ｓｉｎｘ from the graph. ＜Consideration 2＞ Find the period of the graph of ｙ＝ｓｉｎ2x from the graph. ＜Consideration 3＞ Find the period of the graph of ｙ＝ｓｉｎ３ｘ from the graph. ＜Consideration 4＞ In general , what can be said about the period of the graph of y=sinax ? |
| 【２０２】 Draw graphs of the trigonometric functions y=sinx and y=sin(x/2). | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B372 , right-click , and select Paste. Enter =SIN(RADIANS(B12)) in half-width in cell C12. Enter =SIN(RADIANS(B12/2)) in half-width in cell D12. Drag the cell range C12:D12 , right-click , and select Copy. Drag the cell range C13:D372 , right-click , and select Paste (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D372 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration 1＞ Find the period of the graph of y=sin(x/2) from the graph ? ＜Consideration 2＞ In general , what can be said about the period of the graph of y=sin(x/a) ? |
| 【２０３】 Draw a graph of the trigonometric functions of y=sinx , y= 2sinx , and y=3sinx ? | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B192 , right-click , and elect Paste. Enter =SIN(RADIANS(B12)) in half-width in cell C12. Enter =2*SIN(RADIANS(B12)) in half-width in cell D12. Enter =3*SIN(RADIANS(B12)) in half-width in cell E12. Drag the cell range C12:E12 , right-click , and elect Copy. Drag the cell range C13:C192 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E192 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration 1＞ Find the period of the graph of y=2sinx from the graph. ＜Consideration 2＞ Find the period of the graph of y=3sinx from the graph. ＜Consideration 3＞ State the difference between graphs y=sinx , y=2sinx , and y=3sinx. ＜Consideration 4＞ In general , what can be said about the amplitude of the graph of y=asinx ? |
| 【２０４】 Draw a graph of the trigonometric function y=2sin(x-60°). | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above.》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B222 , right-click , and select Paste. Enter =2*SIN(RADIANS(B12)) in half-width in cell C12. Enter =2*SIN(RADIANS(B12-60)) in half-width in cell D12. Drag the cell range C12:D12 , right-click , and select Copy. Drag the cell range C13:C222 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D222 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration＞ Make sure that y=2sin(x-60°) graph is the y=2sinx graph translated +60° along the x-axis. |
| 【２０５】 Draw a graph of the trigonometric functions y=cosx , y=cos2x , and y=cos3x. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B192 , right-click , and select Paste. Enter =COS(RADIANS(B12)) in half-width in cell C12. Enter =COS(RADIANS(2*B12)) in half-width in cell D12. Enter =COS(RADIANS(3*B12)) in half-width in cell E12. Drag the cell range C12:E12 , right-click , and select Copy. Drag the cell range C13:C192 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E192 in the correspondence table of (1) above ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area by horizontally by dragging the right and lrft edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration 1＞ Find the period of the graph of y=cosx from the graph. ＜Consideration 2＞ Find the period of the graph of y=cos2x from the graph. ＜Consideration 3＞ Find the period of the graph of y=cos3x from the graph. ＜Consideration 4＞ In general , what can be said about the period of the graph of y=cosax ? |
| 【２０６】 Draw a graph of the trigonometric functions y=cosx and y=cos(x/2). | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Procedure＞ 《How to complete the worksheet above.》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B372 , right-click and select Paste. Enter =COS(RADIANS(B12)) in half-width in cell C12. Enter =COS(RADIANS(B12/2)) in half-width in cell D12. Drag the cell range C12:D12 , right-click and select Copy. Drag the cell range C13:C372 , right-click and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D372 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 30.00 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale intercal Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration 1＞ Find the period of the graph of y=cos(x/2) . ＜Consideration 2＞ In general, what can be said about the period of the graph of y=cos(x/a) ? |
| 【２０７】 Draw a graph of the trigonometric functions y=cosx , y=2cosx , and y=3cosx. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above.》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 , right-click , and select Copy. Drag the cell range B14:B192 , right-click , and select Paste. Enter =COS(RADIANS(B12)) in half-width in cell C12. Enter =2*COS(RADIANS(B12)) in half-width in cell D12. Enter =3*COS(RADIANS(B12)) in half-width in cell E12. Drag the cell range C12:E12 , right-click , and select Copy. Drag the cell range C13:C192 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E192 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliay gridlines  ＜Consideration 1＞ Find the period of the graph of y=2cosx from the graph. ＜Consideration 2＞ Find the period of the graph of y=3cosx from the graph ＜Consideration 3＞ State the difference between graph ｙ＝ｃｏｓｘ , graph ｙ＝２ｃｏｓｘ , and graph ｙ＝３ｃｏｓｘ. ＜Consideration 4＞ In general , what can be said about the amplitude of the graph of ｙ＝ａｃｏｓｘ ? |
| 【２０８】 Draw a graph of the trigonometric function y=2cos(x+60°). | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -60 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B222 , right-click , and select Paste. Enter =2*cos(RADIANS(B12)) in half-width in cell C12. Enter =2*cos(RADIANS(B12+60)) in half-width in cell D12. Drag the cell range C12:D12, right-click , and select Copy. Drag the cell range C13:C222 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D222 on the correspondence table of (1) above. ② Left -click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration＞ Make sure that the graph of y=2cos(x+60°) is the graph of y=2cosx that has been translated by -60 °in the x-axis direction. |
| 【２０９】 Draw a graph of the trigonometric function y=tanx. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -180 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B56 , right-click , and select Paste. Enter -88 in half-width in cell B58. Enter =B58+2 in half-width in cell B59. Right-click cell B59 and select Copy. Drag the cell range B60:B146 , right-click , and select Paste. Enter 92 in half-width in cell B148. Enter =B148+2 in half-width in cell B149. Right-click cell B149 and select Copy. Drag the cell range B150:B192 , right-click , and select Paste. Enter =TAN(RADIANS(B12)) in half-width in cell C12. Right-click cell C12 and select Copy. Drag the cell range C13:C192 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) avobe. ① Drag and select the cell range B11:C192 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration 1＞ Find the period of the graph of y=tanx from the graph. ＜Consideration 2＞ Make sure that the straight lines x=90°and x=-90°are asymptote. |
| 【２１０】 Draw a graph of the trigonometric function of y=tan2x. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the woksheet above》 Enter -90°in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click cell B13 and select Copy. Drag the cell range B14:B56 , right-click , and select Paste. Enter -44 in half-width in cell B58. Enter =B58+1 in half-width in cell B59. Right-click cell B59 and select Copy. Drag the cell range B60:B146 , right-click , and select Paste. Enter 46 in half-width in cell B148. Enter =B148+1 in half-width in cell B149. Right-click cell B149 and select Copy. Drag the cell range B150:B192 , right-click , and select Paste. Enter =TAN(RADIANS(2*B12)) in half-width in cell C12. Right-click cell C12 and select Copy. Drag the cell range C13:C192 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:C192 in the correspnndencetable of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval 15.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval 5.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration 1＞ Find the period of the graph of y=tan2x from the graph. ＜Consideration 2＞ Make sure that the straight lines x=45°and x=-45°are asymptote. ＜Consideration 3＞ In general , what can be said about the period of th graph of y=tanax ? |
| 【２１１】 Draw a graph of the trigonometric functions y=tanx and y=2tanx. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【２０９】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cellrange B11:D192 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area vertically and horizontally by dragging the top and bottom edges and the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the number on the x-axis → Axis formating → Scale interval Fixed 15.0 → Close → Right-click on the number on the x-axis → Add gridlines ② Right-click on the number on the x-axis → Axis formating → Auxiliary scale interval Fixed 5.0 → Close → Right-click on the number on the x-axis → Add auxiliary gridlines ③ Right-click on the number on the y-axis → Axis-formating → Auxiliary scale interval Fixed 1.0 → Close → Right-click on the number on the y-axis → Add auxiliary gridlines  ＜Consideration 1＞ Find the period of the graph of y=2tanx from the graph. ＜Consideration 2＞ State the positional relationship between the graphs of ｙ＝ｔａｎｘ and ｙ＝２ｔａｎｘ. ＜Consideration 3＞ In general , what is the positional relationship between the graphs of ｙ＝ｔａｎｘ and ｙ＝ａｔａｎｘ. |
| 【２１２】 Draw a graph of the trigonometric function y=2tan{(1/2))x-30°}. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【２０９】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:C372 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth lines］. ③ Expand the graph area vertically and horizontally by dragging the right and reft edges and the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines  ＜Consideration 1＞ Make sure that the straight lines x=240° and x=-120° are asymptote of the graph of y=2tan{(1/2)x-30°} ＜Consideration 2＞ Make sure that the graph of y=2tan(1/2)x is translated by +60 in the x-axis direction. |
| 【２１３】 When 0°≦ｘ≦360° Draw a graph of the functions y=cos2x+2sinx+2 and y=-2(sinx)^2+2sinx+3. |
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| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cell range B14:B192 , right-click , and select Paste. Enter 0 in half-width in cell D12. Enter =D12+2 in half-width in cell D13. Right-click on cell D13 and select Copy. Drag the cell range D14:D192 , right-click , and select Paste. Enter =COS(RADIANS(2*B12))+2*SIN(RADIANS(B12))+2 in half-width in cell C12. Enter =-2SIN(RADIANS(B12))^2+2*SIN(RADIANS(B12))+3 in half-width in cell E12. Right-click on cell D12 and select Copy. Drag the cell range C13:C192 , right-click , and select Paste. Right-click on cell E12 and select Copy. Drag the cell range E13:E192 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) avove. ① Drag and select the cell range B11:C192 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. ④ Drag and select the cell range D11:E192 in the correspondence table of (1) above. ⑤ Left-click ［Insert］→［Scatter plot］→［Smooth line］ ⑥ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines   ＜考察＞ Make sure that the graphs for y=cos2x+2sinx+2 and y=-2(sinx)^2+2sinx+3 are the same. |
| 【２１４】 When 0°≦ｘ≦360° Draw a graph of the functions of y=sinx+cosx and y=√2sin(x+45°). |
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| ＜Procedure＞ (1) Create the following correspondence tabe on the worksheet.
＜Reference＞《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click cell B13 amd select Copy. Drag the cell range B14:B192 , Right-click , and select Paste. Enter 0 in half-width in cell D12. Enter =D12+2 in half-width in cell D13. Right-click cell D13 amd select Copy. Drag the cell range D14:D192 , Right-click , and select Paste. Enter =SIN(RADIANS(B12))+COS(RADIANS(B12)) in half-width in cell C12. Enter =SQRT(2)*SIN(RADIANS(B12+45)) in half-width in cell E12. Right-click cell C12 amd select Copy. Drag the cell range C13:C192 , Right-click , and select Paste. Right-click cell E12 amd select Copy. Drag the cell range E13:E192 , Right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:C192 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］ ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. ④ Drag and select the cell range D11:E192 in the correspondence table of (1) above. ⑤ Left-click ［Insert］→［Scatter plot］→［Smooth line］ ⑥ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 30.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formating → Auxiliary scale interval Fixed 10.0 → Close → Right-click on the numbers on the x-axis → Add auxiliary gridlines   ＜Consideration＞ Make sure that the gtaphs for ｙ＝ｓｉｎｘ＋ｃｏｓｘ and y=√2sin(x+45°) are the same. |
| To table of contents |
| 【３０１】 Draw a graph of the exponential functions of y=2^x , y=3^x , and y=4^x. | |||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -2 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click B13 and select Copy. Drag the cell range B14:B16, right-click , and select Paste. Enter =POWER(2,B12) in half-width in cell C12. Enter =POWER(3,B12) in half-width in cell D12. Enter =POWER(4,B12) in half-width in cell E12. Drag the cell range C12:E12, right-click , and select Copy Drag the cell range C13:C16 , right-click , and select Paste (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E16 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth lines and Marker] ③ Expand the graph area vertically by dragging the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formating → Scale interval Fixed 1.0 → Close ③ Right-click on the numbers on the y-axis → Axis formating → Auxiliary scale interval Fixed 0.5 → Close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete the points(markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］  ＜Consideration 1＞ Find the coordinates of the point through which the graphs of y=2^x , y=3^x , and y=4^x commonly pass. ＜Consideration 2＞ Find the asymptote that the graphs of y=2^x , y=3^x , and y=4^x have in common. ＜Consideration 3＞ State the difference between graphs y=2^x , y=3^x , and y=4^x. |
| 【３０２】 Draw a graph of the exponential functions of y=(1/2)^x , y=(1/3)^x , and y=(1/4)^x. | |||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Refer to 【３０１】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E16 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line and Marker］. ③ Expand the graph area vertically by dragging the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formating → Scale interval Fixed 1.0 → Close ③ Right-click on the numbers on the y-axis → Axis formating → Auxiliary scale interval Fixed 0.5 → Close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delite the points(markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］.  ＜Consideration 1＞ Find the coordinates of the point through which the graphs of y=(1/2)^x , y=(1/3)^x , and y=(1/4)^x commonly pass. ＜Consideration 2＞ Find the asymptote that the graphs of y=(1/2)^x , y=(1/3)^x , and y=(1/4)^x have in common. ＜Consideration 3＞ State the difference between the graphs y=(1/2)^x , y=(1/3)^x , and y=(1/4)^x. |
| 【３０３】 Draw a graph of the exponential functions of y=2^x and y=(1/2)^x . | ||||||||||||||||||||||||||||||||||||
＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【３０１】 how to complete the worksheet
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D18 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line snd marker］. ③ Expand the graph area vertically by dragging the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formating → Scale interval Fixed 1.0 → Close ③ Right-click on the numbers on the y-axis → Axis formating → Auxiliary scale interval Fixed 0.5 → Close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete the points(markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］.  ＜Consideration 1＞ State the positional relationship of the graphs of y=2^x and y=(1/2)^x. ＜Consideration 2＞ In general , what is the positional relationship between graphs y=a^x and y=(1/a)^x ? |
| 【３０４】 Draw a graph of the exponential functions of y=2^x , y=2^(x-2) , and y=2^(x+2). | |||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【３０１】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E18 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line and Marker］. ③ Expand the graph area horizontally by dragging the right and left edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formating → Scale interval Fixed 1.0 → Close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formating → Scale interval Fixed 5.0 → Close ③ Right-click on the numbers on the y-axis → Axis formating → Auxiliary scale interval Fixed 1.0 → Close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete the points(markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］.  ＜Consideration 1＞ State the positional relationship of the graph of y=2^x , y=2^(x-2) , and y=2^(x+2). ＜Consideration 2＞ In general , what is the positional relationship between graphs y=a^x and y=a^(x-k). |
| 【３０５】 Solve the equation 4^x - 2^(x+2) = 32 using a graph | ||||||||||||||||||||||||||||||||||||||||||||
＜Procedure＞ Draw the graphs of y=4^x and y=2^(x+2) +32. (1) Create the following correspondence table on a worksheet. ＜Reference＞ Please refer to 【３０１】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D20 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line and Merker］ ③ Expand the graph area vertically by dragging the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 10.0 → close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete the points(markers) in the correspondence table from the graph. ① Left-click the graph area to activare. ② Left-click ［Insert］→［Scatter plot］→［Smooth line］  ＜Consideration＞ Solve the equation 4^x-2^(x+2)=32 from the intersection of the graphs y=4^x and y=2^(x+2)+32 |
| 【３０６】 Solve the inequality 9^x < 2･3^x + 3 using a graph. | ||||||||||||||||||||||||||||
| ＜Procedure＞ Draw the graphs y=9^x and y=2･3^x+3 (1) Create the following correspondence table on a worksheet. ＜Reference＞ Please refer to 【３０１】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D16 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line and Marker] ③ Expand the graph area vertially by dragging the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is draw. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formatting → Auxiliary Scale interval Fixed 1.0 → close → Right-click on the numbers on the y-axis → Auxliary add gridlines (4) Delete the points(markers) in the correspondence table from the graph. ① Left-click the graph area to activate it. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］  ＜Consideration＞ Solve the inequality 9^x < 2･3^x + 3 from the intersection of the graphs of y=9^x and y=2･3^x+3. |
| 【３０７】 Compare the size of 8^(1/4) , 16^(1/3) , 64^(1/5) using a graph. | ||||||||||||
＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter the information in cells B12 , B13 , and B14 as shown in the worksheet above. Enter =POWER(8,1/4) in Cell C12. Enter =POWER(16,1/3) in Cell C13. Enter =POWER(64,1/5) in Cell C14 (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B12:C14 in the correspondence table of (1) above. ② Left-click ［Insert］→［Vertical bar］→［2-D vertical bar cluster］  ＜Consideration＞ Compare the sizes of 8^(1/4) 16^(1/3) 64^(1/5) from the bar graphs of 8^(1/4) , 16^(1/3) , and 64^(1/5) |
| To table of contents |
| 【４０１】 Draw a graph of the logarithmic functions of y=log(2)x , y=log(3)x , y=log(4)x | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Drag the cell range B12:B20 and right-click , and select Cell formatting. Select a number from the classification and set the number of decimal places to 5. Enter 1/16 in half-width in cell B12 (Displayed as a decimal). Enter data in the cell range B13:B20 using half-width characters as shown in the worksheet above. Enter =LOG(B12,2) in half-width in cell C12. Enter =LOG(B12,3) in half-width in cell D12. Enter =LOG(B12,4) in half-width in cell E12. Drag the cell range C12:E12 , right-click , and select Copy. Drag the cell range C13:C20 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E20 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line and Marker］ ③ Expand the graph area vertically by dragging the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add auxiliary gridlines ③ Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 0.5 → close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete points(markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］  ＜Consideration 1＞ Find the coordinates of point which the graphs of y=log(2)x , y=log(3)x , and y=log(4)x commonly pass through. ＜Consideration 2＞ Find the common asymptote for the graphs of y=log(2)x , y=log(3)x , and y=log(4)x. ＜Consideration 3＞ State the difference between the graphs of y=log(2)x , y=log(3)x , and y=log(4)x. |
| 【４０２】 Draw a graph of the logarithmic functions of y=log(1/2)x , y=log(1/3)x , and y=log(1/4)x. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet. ＜Reference＞ Refer to 【４０１】 how to complete the wprksheet.
(2) Draw a graph from the correspondence table in (1) avobe. ① Drag and select the cell range B11:E20 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line and Marker］ ③ Expand the graph area vertically by dragging the top and bottom of graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add auxiliary gridlines ③ Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 0.5 → close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］  ＜Consideration 1＞ Find the coordinates of the point that the graphs of y=log(1/2)x , y=log(1/3)x and , y=log(1/4)x pass commonly through. ＜Consideration 2＞ Find the common asymptote for the graphs of y=log(1/2)x , y=log(1/3)x and , y=log(1/4)x. ＜Consideration 3＞ State the difference between the graphs of y=log(1/2)x , y=log(1/3)x and , y=log(1/4)x. |
| 【４０３】 Draw a graph of the logarithmic functions of y=log(2)x and y=log(1/2)x. | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following corresspondence table on the worksheet. ＜Referrence＞ Please refer to 【４０１】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D20 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Smooth line and Marker］ ③ Expand the graph area vertically by dragging the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add auxiliary gridlines ③ Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 0.5 → close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］  ＜Consideration 1＞ State the positional relationship of the graphs of y=log(2)x and y=log(1/2)x. ＜Consideration 2＞ In general , what can be said about the positional relationship between the graphs of y=log(2)x and y=log(1/2)x. |
| 【４０４】 Draw a graph of the logarithmic functions of y=log(2)x and y=log(2)(1/x) | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【４０１】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D20 in the correspndence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line and Markers］ ③ Expand the graph area vertically by dragging the top and bottom of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the x-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add auxiliary gridlines ③ Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 0.5 → close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］  ＜Consideration 1＞ State the positional relationship between the graphs of y=log(2)x and y=log(2)(1/x). ＜Consideration 2＞ In general , what can be said about the positional relationship between the graphs of y=log(a)x and y=log(a)(1/x). |
| 【４０５】 Solve equation log(10)x+log(10)(x-3)=1 using graphs. | ||||||||||||||||||||||||||||||||||||||||||||
＜Procedure＞  Draw graphs of y=log(10)x and y=-log(10)(x-3)+1. (1) Create the following correspondence table on the worksheet. ＜Reference＞ Please refer to 【401】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D20 in the correpondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line and marker］ ③ Expand the graph area vertically by dragging the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 0.1 → close → Right-click on the numbers on the y-axis → Add auxiliary gridlines (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it ② Left-click ［Incert］→［Scatter plot］→［Smooth line］  ＜Consideration＞ Solve equation log(10)x+log(10)(x-3)=1 from the intersection of the graphs of y=log(10)x and y=-log(10)(x-3)+1. |
| 【４０６】 Solve inequality log(3)(x-3)+log(3)(x-5)<1 using a graph. | ||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ log(3)(x-3) < -log(x-5)+1 Draw graphs of y=log(3)(x-3) and y=-log(3)(x-5)+1. (1) Create the following correspondence table on a worksheet. ＜Reference＞ Please refer to 【401】 how to complete the worksheet.
(2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D18 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line and Marker］ ③ Expand the graph area vertically by dragging the top and bottom edges of the graph area. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.00 → close → Right-click on the numbers on the x-axis → Add gridline ② Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 0.1 → close → Right-click on the numbers on the y-axis → Add auxiliary gridline (4) Delete points (markers) in the correspondence table from the graph. ① Left-click on the graph area to activate it. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］  ＜Consideration＞ Solve inequality log(3)(x-3)+log(3)(x-5) < 1 from the intersection of the graphs of y=log(3)(x-3) and y=-log(3)(x-5)+1 |
| 【４０７】 Compare the size of 1/2, -log(2)(1/3) , and log(1/2)7 using the graphs. | ||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete a worksheet above》 Enter data into cells B12 , B13 , and B14 as shown in the worksheet above. Drag the cell range C12:C14 , right-click , and select Cell Formatting. Select a number from the crassification and set the number of decimal places to 5. Enter 1/2 in half-width in cell C12. Enter =-1*LOG(1/3,2) in half-width in cell C13. Enter =LOG(7,1/2) in half-width in cell C14. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B12:C14 in the correspondence table of (1) above. ② Left-click ［Incert］→［Vertical bar］→［2-D vertical bar cluster］  ＜Consideration＞ From the bar graphs of 1/2 , -log(2)(1/3) , and log(1/2)7 , compare the sizes of 1/2 , -log(2)(1/2) , and log(1/2)7. |
| To table of contents |
| 【５０１】 Draw a circle x^2+y^2+6x-8y-11=0 | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ x^2+y^2+6x-8y-11=0 x^2+6x+9+y^2-8y+16=11+9+16 (x+3)^2+(y-4)^2=36 , center(-3,4 ) , radius 6 (y-4)^2=36-(x+3)^2 y-4=+√{36-(x+3)^2} , -√{36-(x+3)^2} y=4+√{36-(x+3)^2} , 4-√{36-(x+3)^2} (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -9 in half-width in cell B12. Enter =B12+0.1 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cell range B14:B132 , Right-click , and select Paste. Enter =4+SQRT(36-(B12+3)^2) in half-width in cell C12. Enter =4-SQRT(36-(B12+3)^2) in half-width in cell D12. Drag the cell range C12:D12 , Right-click , and select Copy. Drag the cell range C13:C132 , Right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D132 in the correspondence table of (1) above. ② ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area to stretch and adjust. (3) Arrange the xy plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add auxiliary gridlines ② Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close →Right-click on the numbers on the y-axis → Add auxiliary gridlines  ＜Consideration＞ Make sure that it is a circle with center(-3,4) and radius 6. |
| 【５０２】 Find the coordinates of the common point between circle x^2+y^2=5 and straight line y=-x+1 by drawing a graph. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
＜Procedure＞  (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -2.236 in half-width in cell B12. Enter -2.2 in half-width in cell B13. Enter =B13+0.1 in half-width in cell B14. Right-click on cell B14 and select Copy. Drag the cell range B15:B57 , right-click , and select Paste. Enter 2.236 in half-width in cell C12. Enter =SQRT(5-B12^2) in half-width in cell C12. Enter =-SQRT(5-B12^2) in half-width in cell D12. Enter =-B12+1 in half-width in cell E12. Drag the cell range C12:E12 , right-click , and select Copy. Drag the cell range C13:C58 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E58 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines  ＜Consideration＞ Find the coordinates of the common point from the graphs circle x^2+y^2=5 and straight line y=-x+1. |
| 【５０３】 Draw an ellipse whose sum of distances from two fixed points (3,0) and (-3,0) is 10. | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ Let x^2/a^2+y^2/b^2=1 be the ellipse to seek. ｃ＝３ Since ２ａ＝１０ , ａ＝５ Since c=√(a^2-b^2) , 3=√(5^2-b^2) ｂ＝４ Therefore , x^2/5^2 + y^2/4^2 = 1 , y=4√(1 - x^2/25) , y=-4√(1 - x^2/25) (1) Create the following correspondence table in a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -5 in half-width in cell B12. Enter =B12+0.1 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cell range B14:B112 , right-click , and select Paste. Enter =4*SQRT(1-B12^2/25) in half-width in cell C12. Enter =-4*SQRT(1-B12^2/25) in half-width in cell D12. Drag the cell range C12:D12 , right-click , and select Copy. Drag the cell range C13:C112 , right-click , and select Paste. (2) Drag a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D112 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the right and left edges of the graph areas holizontally to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add gridlines  ＜Consideration＞ Make sure that ellipse x^2/5^2 + y^2/4^2 = 1 is drawn. |
| 【５０４】 Draw a parabola whose distance from rhe fixed point (2,0) is equal to the distance from the straight line x=-2. | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ Let y^2=4px be the parabola to seek. Since p=2 , y^2=8x y=2√(2x) , y=-2√(2x) (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+0.1 in half-width in cell B12. Right-click on cell B13 and select Copy. Drag the cell range B14:B52 , right-click , and select Paste. Enter =2*SQRT(2*B12) in half-width in cell C12. Enter =-2*SQRT(2*B12) in half-width in cell D12. Drag the cell range C12:D12 , right-click , and select Copy. Drag the cell range C13:C52 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D52 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of graph area vertically to stretch and adjust. (3) Arrange the coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines  ＜Consideration＞ Make sure that parabola y^2=8x is drawn. |
| 【５０５】 Draw a hyperbola whose dtfference between distances from two fixed points (5,0) and (-5,0) is 6. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ Let x^2/a^2 - y^2/b^2 = 1 be the hyperbola to seek. Since c=5 , 2a=6 , c=√(a^2+b^2) , a=3 , b=4 Therefore x^2/3^2 - y^2/4^2 = 1 , y=4√(x^2/9 - 1) , y=-4√(x^2/9 - 1) (1) Create the following correspondence table in a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 10 in half-width in cell B12. Enter =B12-0.1 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cell range B14:B82 , right-click , and select Paste. Enter -3 in half-width in cell B84. Enter =B84-0.1 in half-width in cell B85. Right-click on cell B85 and select Copy. Drag the cell range B86:B154 , right-click , and select Paste. Enter =4*SQRT(B12^2/9-1) in half-width in cell C12. Enter =-4*SQRT(B12^2/9-1) in half-width in cell D12 Drag the cell range C12:D12 , right-click , and select Copy. Drag the cell range C13:C154 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. See 【504】 (3) Arrange the xy coordinate plane on whic the graph is drawn. ① Right-click on the numbers on x-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close → Right-click on the numbers on x-axis → Add auxiliary gridlines ② Right-click on the numbers on y-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close → Right-click on the numbers on y-axis → Add auxiliary gridlines  ＜Consideration＞ Make sure that hyperbola x^2/3^2 - y^2/4^2 = 1 is drawn. |
| To table of contents |
| 【６０１】 Draw graphs of function y=-2x^3+6x+1 and y'=-6x^2+6. | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above.》 Enter -3 in half-width in cell B12. Enter =B12+0.1 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cell range B14:B72 , right-click and , select Paste. Enter =-2*B12^3+6*B12+1 in half-width in cell C12. Enter =-6*B12^2+6 in half-width in cell D12. Drag the cell range C12:D12 , right-click , and select Copy. Drag the cell range C13:C72 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:D72 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］. ③ Drag the top and bottom area of the graph area vertically to strech and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add gridlines  ＜Consideration 1＞ When y'<0 , what does the graph of y=-2x^2+6x+1 look like ? ＜Consideration 2＞ When y'>0 , what does the graph of y=-2x^2+6x+1 look like ? ＜Consideration 3＞ When y'=0 , what does the graph of y=-2x^2+6x+1 look like ? |
| 【６０２】 Find the maximam and minimum values by drawing a graph of function y=2x^3-3x^2-12x-6 (-2≦x≦4) | |||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -2 in half-width in cell B12. Enter =B12+0.1 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cell range B14:B72 , right-click , and select Paste. Enter =2*B12^3-3*B12^2-12*B12-6 in half-width in cell C12. Right-click on cell C12 and select Copy. Drag the cell range C13:C72 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:C72 in the orrespondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area vertically to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close → Right-click on the numbers on the y-axis → Add auxiliary gridlines ③ Right-click on the numbers on the y-axis → Axis formatting → Scale interval Fixed 5.0 → close  ＜Consideration 1＞ Find the maximum value when -2≦x≦4 from the graph. ＜Consideration 2＞ Find the minimum value when -2≦x≦4 from the graph. |
| 【６０３】 Find the number of different real solutions for cubic equation x^3-3x^2-a=0 using a graph. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ x^3-3x^2=a Draw graphs of y=x^3-3x^2 and y=a. (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter a= in cell B81. Enter 1 in half-width in cell C81. Enter -2 in half-width in cell B12 Enter =B12+0.1 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cellrange B14:B72 , right-click , and select Paste. Enter =B12^3-3*B12^2 in half-width in cell C12. Enter =$C$8 in half-width in cell D12. Drag the cellrange C12:D12 , right-click , and select Copy. Drag the cellrange C13:C72 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) avove. ① Drag and select the cell range B11:D72 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area vertically to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 1.0 → close →Right-click on the numbers on the y-axis → Add auxiliary gridlines  ＜Consideration 1＞ When a=1 , a=-5 , find the number of different real solutions. ＜Consideration 2＞ When a=0 , a=-4 , find the number of different real solutions. ＜Consideration 3＞ When a=-2 , find the number of different real solutions. ＜Consideration 4＞ How does the number of different real solutions change depending on the value of the constant a ? |
| 【６０４】 When x≧0 , draw a graph to confirm that the inequality x^3-3x^2+4≧0. | |||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+0.1 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cell range B14:B44 , right-click , and select Paste. Enter =B12^3-3*B12^2+4 in half-width in cell C12. Right-click on cell C12 and select Copy. Drag the cell range C13:C44 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:C44 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area vertically to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines  ＜Consideration＞ When x≧0 , look at the graph and confirm that the inequality x^3-3x^2+4≧0 holds true. |
| 【６０５】 Draw a graph of the curve y=logx and the tangent and normal at point(1,0) on it. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
＜Procedure＞  Tangent ： ｙ－０＝１（ｘ－１） ∴ ｙ＝ｘ－１ Normal ： ｙ－０＝－１（ｘ－１） ∴ ｙ＝－ｘ＋１ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0.05 in half-width in cell B12. Enter =B12+0.05 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cell range B14:B111 and right-click , and select Paste. Enter =LN(B12) in half-width in cell C12. Enter =B12-1 in half-width in cell D12. Enter =-B12+1 in half-width in cell E12. Drag the cell range C12:E12 and right-click , and select Copy. Drag the cell range C13:C111 and right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E111 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Arrange the top and bottom of the graph area vertically to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines  ＜Consideration 1＞ Make sure that y=x-1 is tangent to point (1,0) on y=logx. ＜Consideration 2＞ Make sure that y=-x+1 is normal to point (1,0) on y=logx. |
| 【６０６】 Draw the graph of ellipse x^2/4 + y^2 = 1 and the tangent at the point (6/5,4/5) on it. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ Since y^2 = 1 - x^2/4 , y=±√(1-x^2/4) Since the tangent formula , x/4 × 6/5 + y × 4/5 = 1 Therefore , tangent : y=-(3/8)x + 5/4 Draw the graphs of y = √(1-x^2/4) , y = -√(1-x^2/4) , and y = -3/8 + 5/4 (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -2 in half-width in cell B12. Enter =B12+0.05 in half-width in cell B13. Right-click on cell B13 and select Copy. Drag the cell range B14:B32 , right-click , and select Paste. Enter =SQRT(1-B12^2/4) in half-width in cell C12. Enter =-SQRT(1-B12^2/4) in half-width in cell D12. Enter =-3*B12/8+5/4 in half-width in cell E12. Drag the cell range C12:E12 , right-click , and select Copy. Drag the cell range C13:C92 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:E92 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line] ③ Drag the left and right edges of graph area horizontally and the top and bottom edges of graph area vertically to strech and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Axis formatting → Auxiliary scale interval Fixed 0.2→ close → Right-click on the numbers on the x-axis → Add auxiliary gridlines ② Right-click on the numbers on the y-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 0.2 → close → Right-click on the numbers on the y-axis → Add auxiliary gridlines  ＜Consideration＞ Make sure that y=(-3/8)x+5/4 is the tangent to the point (6/5,4/5) on the ellipse x^2/4+y^2=1. |
| To table of contents |
| 【７０１】 Draw a graph of function y=x√(9-x^2). | |||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -3 in half-width in the cell B12. Enter =B12+0.1 in half-width in the cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B72 , right-click , and select Paste. Enter =B12*SQRT(9-B12^2) in half-width in the cell C12. Right-click on the cell C12 and select Copy. Drag the cell range C13:C72 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:C72 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area vertically to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 1.0 → close → Right-click on the numbers on the x-axis → Add gridlines ② Right-click on the numbers on the y-axis → Axis formatting → Auxiliary scale interval Fixed 0.5 → close → Right-click on the numbers on the y-axis → Add auxiliary gridlines  ＜Consideration 1＞ Find the local maximum from the graph. ＜Consideration 2＞ Find the local minimum from the graph. ＜Consideration 3＞ Find the maximum value from the graph. ＜Consideration 4＞ Find the minimum value from the graph. |
| 【７０２】 For 0≦x≦2π , draw the graph of the function y=x+2sinx. | |||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B192 , right-click , and select Paste. Enter =RADIANS(B12)+2*SIN(RADIANS(B12)) in half-width in cell C12. Right-click on the cell C12 and select Copy. Drag the cell range C13:C192 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:C192 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area vertically to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Axis formatting → Scale interval Fixed 60.0 → Close → Right-click on the numbers on the x-axis → Add gridlines  ＜Consideration 1＞ Use a graph to find the value of x that takes the local maximum value when 0≦x≦2π. Then caluculate the local maximum value. ＜Consideration 2＞ Use a graph to find the value of x that takes the local miniimum value when 0≦x≦2π. Then caluculate the local minimum value. ＜Consideration 3＞ Use a graph to find the value of x that takes the maximum value when 0≦x≦2π. Then caluculate the maximum value. ＜Consideration 4＞ Use a graph to find the value of x that takes the miniimum value when 0≦x≦2π. Then caluculate the minimum value. |
| 【７０３】 Draw a graph of the function y=x+√(4-x^2) | |||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -2 in half-width in cell B12. Enter =B12+0.1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B52 and right-click , and select Paste. Enter =B12+SQRT(4-B12^2) in half-width in cell C12. Right-click on the cell C12 and select Copy. Drag the cell range C13:C52 and right-click , and select Paste. Enter 2 in half-width in cell C52. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:C52 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area vertically to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines  ＜Consideration＞ Find the minimum value and the value of x at that time from the graph. |
| 【７０４】 When -1≦x≦1 , draw a graph of the function y=e^(2x)-2e^x+1 | |||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter -1 in half-width in cell B12. Enter =B12+0.05 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B52 , right-click , and select Paste. Enter =EXP(2*B12)-2*EXP(B12)+1 in half-width in cell C12. Right-click on the cell C12 and select Copy. Drag the cell range C13:C52 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B11:C52 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area vertically to stretch and adjust. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Riht-click on the numbers on the x-axis → Add gridlines  ＜Consideration 1＞ When -1≦x≦1 , find the maximum value and the value of x at the time using the graph. ＜Consideration 2＞ When -1≦x≦1 , find the minimum value and the value of x at the time using the graph. |
| To table of contents |
| 【８０１】 Find the area enclosed by the curve y=x^2 , the x-axis , and the straight line x=1 using the piecewise method. | ||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0.01 in halh-width in cell B12. Enter =B12+0.01 in halh-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B111 , right-click , and select Paste. Enter =B12^2 in halh-width in cell C12. Enter =0.01*C12 in halh-width in cell D12. Right-click the cell range C12:D12 , and select Copy. Drag the cell range C13:C111 , right-click , and select Paste. (2) Find the piecewisw from the correspondence table in (1) abobe. ① Find the sum of the cell range D12:D111 in the correspondence table in (1) above. Enter =SUM(D12:D111) in half-width in cell D112 .of the worksheet. ＜Consideration 1＞ Calcurate the following definite integral and find the area enclosed by the curve y=x^2 , the x-axis , and the straight line x=1.  ＜Consideration 2＞ Compare the value of the definite integral obtained in consideration 1 with the value of D112 , which is the sum of the cell range D12:D111 in the correspondence table of the worksheet. |
| 【８０２】 Draw a graph of cycloid ｘ＝ｔ－ｓｉｎｔ、ｙ＝１－ｃｏｓｔ (０≦ｔ≦２π). Also , find its length. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 0 in half-width in cell B12. Enter =B12+2 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B192 , right-click , and select Paste. Enter =RADIANS(B12)-SIN(RADIANS(B12)) in half-width in cell C12. Enter =1-COS(RADIANS(B12)) in half-width in cell D12 Drag the cell range C12:D12 , right-click , and select Copy. Drag the cell range C13:D192 , right-click , and select Paste. Enter =C13-C12 in half-width in cell E13. Enter =D13-D12 in half-width in cell F13. Enter =SQRT(E13^2+F13^2) in half-width in cell G13. Drag the cell range E13:G13 , right-click , and select Copy. Drag the cell range E14:E192 , right-click , and select Paste. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range C11:D192 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Smooth line］ ③ Drag the top and bottom edges of the graph area vertically to stretch and adjust. (3) Arrange the xy-coordinate plane on which the graph is drawn. ① Right-click on the numbers on the x-axis → Add gridlines (4) Find the length of the cycloid from the correspondence table in (1) above. ① Find the sum of the cell range G13:G192 in the correspondence table of (1) above. Enter =SUM(G13:G192) in half-width in cell G193 of the worksheet.  ＜Consideration 1＞ Calculate the following definite integral and find the length of the cycloid ｘ＝ｔ－ｓｉｎｔ、ｙ＝１－ｃｏｓｔ (０≦ｔ≦２π)  ＜Consideration 2＞ Compare the value of the definite integral obtained in Consideration 1 with the value of cell G193 , which is the sum of the cell range G13:G192 in the correspondence table of the worksheet. |
| To table of contents |
| 【９０１】 Find the general terms of the arithmetic progression where the 3rd term is 5 and the 12th term is 59. | |||||||||||||||||||||||||||||||||
| ＜Procedure＞ Let    If you solve ① and ② simultaneously , ａ＝－７、ｄ＝６ Therefore ,   (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B111 , right-click , and select Paste. Enter =6*B12-13 in half-width in cell C12. Right-click cell C12 , right-click , and select Copy. Drag the cell range C13:C111 , right-click , and select Paste. ＜Consideration 1＞ Solving for a(n)=6n-13=143 gives n=26. Check on the worksheet that 143 is the 26th term. ＜Consideration 2＞  Check on the worksheet that the 96th term is 563. |
| 【９０２】 Find the general terms of the geometric progression where the 3rd term is 12 and the 7th term is 192. | ||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ Let    Solving ① and ② simultaneously gives ａ＝３、ｒ＝±２ Therefore ,  (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter the information in the cell range B11:D111 as shown in the table above. Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B46 , right-click , and select Paste. Enter =3*2^(B12-1) in half-width in cell C12. Enter =3*(-2)^(B12-1) in half-width in cell D12. Drag the cell range C12:D12 , right-click , and select Copy. Drag the cell range C13:C46 , right-click , and select Paste. ＜Consideration 1＞   Check on the worksheet that the 8th term is 384 and -384. ＜Consideration 2＞ Solving a(n)=3*2^(n-1)=786432 gives n=19. Solving a(n)=3*(-2)^(n-1)=786432 gives n=19. Check on the worksheet that 786432 is the 19th term. |
| 【９０３】Find the sum from the first term to the nth term of an arithmetic progression where the first term is 50 and the commom difference is -6. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
＜Procedure＞  (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Drag the cell range B11:E11 , right-click and select Cell formatting. Select Tablar format → Classification string→ OK Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B61 , right-click , and select Paste. Enter =-6*B12+56 in half-width in cell C12. Right-click on the cell C12 and select Copy. Drag the cell range C13:C61 , right-click , and select Paste. Enter 50 in half-width in cell D12. Enter =D12+C13 in half-width in cell D13. Right-click on the cell D13 and select Copy. Drag the cell range D14:D61 , right-click , and select Paste. Enter =-3*B12^2+53*B12 in half-width in cell E12. Right-click on the cell E12 and select Copy. Drag the cell range E13:E61 , right-click , and select Paste. ＜Consideration 1＞ Make sure that the sum in column D of the worksheet matches -3n^2+53n in column E. ＜Consideration 2＞   Therefore , when n=9 , it is maximum. Check on the worksheet that when n=9 , it is the maximum. |
| 【９０４】Find the sum from the first term to the nth term of a geometric progression with an initial term of 1/2 and a common ratio of 2. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
＜Procedure＞  (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the woksheet above》 Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B46 , right-click , and select Paste. Enter =2^(B12-2) in half-width in cell C12. Right-click on the cell C12 and select Copy. Drag the cell range B13:C46 , right-click , and select Paste. Enter 0.5 in half-width in cell D12. Enter =D12+C13 in half-width in cell D13. Right-click on the cell D13 and select Copy. Drag the cell range D14:D46 , right-click , and select Paste. Enter =(2^B12-1)/2 in half-width in cell E12. Right-click on the cell E12 and select Copy Drag the cell range E13:E46 , right-click , and select Paste. ＜Consideration 1＞ Make sure that the sum in column D of the worksheet matches (2^n-1)/2 in column E. ＜Consideration 2＞  Solving this gives n=11. Check on the sheet that the fist term the sum exceedds 1000 is the sum up to the 11th term. |
| 【９０５】 Find the sum from 1^2 to n^2. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
＜Procedure＞   (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B46 ,right-click . and select Paste. Enter =B12^2 in half-width in cell C12. Right-click on the cell C12 and select Copy. Drag the cell range C13:C46 ,right-click . and select Paste. Enter 1 in half-width in cell D12. Enter =D12+C13 in half-width in cell D13. Right-click on the cell D13 and select Copy. Drag the cell range D14:D46 ,right-click . and select Paste. Enter =B12*(B12+1)*(2*B12+1)/6 in half-width in cell E12. Right-click on the cell E12 and select Copy. Drag the cell range E13:E46 ,right-click . and select Paste. ＜Consideration＞ Make sure that the sum in column D of the worksheet matches n(n+1)(2n+1)/6 in column E. |
| 【９０６】 Find the sum from 1^3 to n^3. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
＜Procedure＞  (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B46 , right-click , and select Paste. Enter =B12^3 in half-width in cell C12. Right-click on the cell C12 and select Copy. Drag the cell range C13:C46 , right-click , and select Paste. Enter 1 in half-width in cell D12. Enter =D12+C13 in half-width in cell D13. Right-click on the cell D13 and select Copy. Drag the cell range D14:D46 , right-click , and select Paste. Enter =(B12*(B12+1)/2)^2 in half-width in cell E12. Right-click on the cell E12 and select Copy. Drag the cell range E13:E46 , right-click , and select Paste. ＜Consideration＞ Make sure that the sum in column D of the worksheet matches {n(n+1)/2}^2 in column E. |
| 【９０７】Find the general term of the following sequence. ３，４，６，１０，１８，３４，・・・ |
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| ＜Procedure＞ (1) Create the following correspondence table on the worksheet.
＜Referrence＞ 《How to complete the worksheet above》 Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B46 , right-click , and select Paste. Enter =2^(B13-2) in half-width in cell C13. Right-click on the cell C13 and select Copy. Drag the cell range C14:C46 , right-click , and select Paste. Enter 3 in half-width in cell D12. Enter =D12+C13 in half-width in cell D13. Right-click on the cell D13 and select Copy. Drag the cell range D14:D46 , right-click , and select Paste. Enter =2^(B12-1)+2 in half-width in cell E12. Right-click on the cell E12 and select Copy. Drag the cell range E13:E46 , right-click , and select Paste. ＜Consideration＞ Make sure that the sum in column D of the worksheet matches 2^(n-1)+2 in column E. |
【９０８】Find the general term of the sequence {a(n)} defined by the following
recurrence formula.  |
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| ＜Procedure＞ When a(n+1) =2a(n)+3 is transformed , a(n+1)+3=2{a(n)+3} Sequence {a(n)+3} is a geometric progression where the first term is 4 and the common ratio is 2. Therefore , a(n)+3=4×2^(n-1)=2^(n+1) a(n)=2^(n+1)-3 (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B46 , right-click , and select Paste. Enter 1 in half-width in cell C12. Enter =2*C12+3 in half-width in cell C13. Right-click on the cell C13 and select Copy. Drag the cell range C14:C46 , right-click , annd select Paste. Enter =2^(B12+1)-3 in half-width in cell D12. Right-click on the cell D12 and select Copy. Drag the cell range D13:D46 , right-click , annd select Paste. ＜Reference＞ Make sure that the recurrence formula in column C of the worksheet matches 2^(n+1)-3 in column D. |
【９０９】 Find the general term of the sequence {a(n)} defined by the following
recurrence formula.   |
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| ＜Procedure＞ Solve this recurrence formula to find the general term.    (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the cell B13 and select Copy. Drag the cell range B14:B46 , right-click , and select Paste. Enter 0 in half-width in cell C12. Enter 1 in half-width in cell C13. Enter =(2*C13+C12)/2 in half-width in cell C14 Right-click on the cell C14 and select Copy. Drag the cell range C15:C46 , right-click , and select Paste. Enter =(((1+SQRT(3))/2)^(B12-1)-((1-SQRT(3))/2)^(B12-1))/SQRT(3) in half-width in cell D12. Right-click on the cell D12 and select Copy. Drag the cell range D13:D46 , right-click , and select Paste. ＜Consideration＞ Make sure that the recurrence formula in column C of the worksheet matches the general term in column D. |
【９１０】 Consider the Febonacci sequence determined by the following recurrence
formula.   |
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| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter 1 in half-width in cell B12. Enter =B12+1 in half-width in cell B13. Right-click on the B13 and select Copy. Drag the cell range B14:B61 , right-click , and select Paste. Enter 1 in half-width in cell C12. Enter 1 in half-width in cell C13. Enter =C12+C13 in half-width in cell C14. Right-click on the C14 and select Copy. Drag the cell range C15:C61 , right-click , and select Paste. Enter =C13/C12 in half-width in cell D13. Right-click on the D13 and select Copy. Drag the cell range D14:D61 , right-click , and select Paste. ＜Consideration＞ Make sure that the value of (the 50th term/ the 49th term) in column D of the worksheet is close to the golden ratio (1+√5)/2≒1.618033989. |
| To table of contens |
| 【1001】 Inverse trigonometric functions and Conversion from arc degree
method to degree method （ＡＳＩＮ function、ＡＣＯＳ function、ＡＴＡＮ function、ＤＥＧＲＥＥＳ function） |
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| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above.》 Drag the cell range B12:B20 , right-click and select Cell formatting. Select Display format → Classification string → OK Enter data in the cell range B12:B20 as shown in the table above Drag the cell range D12:D20 , right-click ,and select Cell formatting Select Display format → Classification string → OK Enter data in the cell range D12:D20 as shown in the table above Drag the cell range F12:F20 , right-click ,and select Cell formatting Select Display format → Classification string → OK Enter data in the cell range F12:F18 as shown in the table above Enter =DEGREES(ASIN(-1)) in cell C12. Similarly , enter =DEGREES(ASIN(value in the same row of column B)) in half-width from C13 to C20 Enter =DEGREES(ACOS(1)) in cell E12. Similarly , enter =DEGREES(ACOS(value in the same row of column D)) in half-width from E13 to E20 Enter =DEGREES(ATAN(-SQRT(3))) in cell G12. Similarly , enter =DEGREES(ACOS(value in the same row of column F)) in half-width from G13 to G18 ＜About functions＞ ＡＳＩＮ function･･･ ASIN(real number 1) Return the angle of SIN that has a real number 1 in radians. ＡＣＯＳ function･･･ ACOS(real number 1) Return the angle of COS that has a real number 1 in radians. ＡＴＡＮ function･･･ ATAN(real number 1) Return the angle of TAN that has a real number 1 in radians. ＤＥＧＲＥＥＳ function･･･ DEGREES(ラジアン) Converts an angle in radians to an angle in degrees. |
| 【1002】 Sum of squares of complex numbers （ＳＵＭＳＱ function） |
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| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Drag the cell range B11:B20 , right-click , and select Cell formatting. Select Display format → Classification string → OK. Enter data in the cell range B11:B20 as shown in the table above. Enter =SQRT(0) in half-width in cell C12. Enter =SQRT(1) in half-width in cell C13. Enter =SQRT(2) in half-width in cell C14. Enter =SQRT(3) in half-width in cell C15. Enter =SQRT(4) in half-width in cell C16. Enter =SQRT(5) in half-width in cell C17. Enter =SQRT(6) in half-width in cell C18. Enter =SQRT(7) in half-width in cell C19. Enter =SQRT(8) in half-width in cell C20. Enter =SUMSQ(C12,C13) in half-width in cell D13. Right-click on the cell D13 and select Copy. Drag the cell range D14:D20 , right-click , and select Paste. ＜About function＞ ＳＵＭＳＱ function･･･ SUMSQ(real number 1，real number 2) Returns the value of (real number 1)^2 + (real number 2)^2 |
| 【1003】 Four arithmetic calculations of complex numbers <sum , difference
, product , quotient> and display in complex number format （ＩＭＳＵＭ function 、ＩＭＳＵＢ function 、ＩＭＰＲＯＤＵＣＴ function 、ＩＭＤＩＶ function 、ＣＯＭＰＬＥＸ function） |
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| ＜Procedure＞ (1) Create the correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter A= in cell C11. Enter B= in cell C12. Enter =COMPLEX(5,-3) in half-width in cell D11. Enter =COMPLEX(-2,7) in half-width in cell D12. Enter C= in half-width in cell F11. Enter D= in half-width in cell F12. Enter =COMPLEX(2,1) in half-width in cell G11. Enter =COMPLEX(2,-1) in half-width in cell G12. Enter data in the cell range B14:B17 as shown in the table above. Enter data in the cell range C14:C17 as shown in the table above. Enter =IMSUM(D11,D12) in half-width in cell D14. Enter =IMSUB(D11,D12) in half-width in cell D15. Enter =IMPRODUCT(D11,D12) in half-width in cell D16. Enter =IMDIV(G11,G12) in half-width in cell D17. ＜About functions＞ ＩＭＳＵＭ function･･･ IMSUM(complex number 1，complex number 2) Returns the value of (complex number 1 + complex number 2) ＩＭＳＵＢ function･･･ IMSUB(complex number 1，complex number 2) Returns the value of (comlex number 1 - complex number 2) ＩＭＰＲＯＤＵＣＴ function･･･ IMPRODUCT(comlex number 1，complex number 2) Returns the value of (comlex number 1 × complex number 2) ＩＭＤＩＶ function･･･ IMDIV(comlex number 1，complex number 2) Returns the value of (comlex number 1 ÷ complex number 2) ＣＯＭＰＬＥＸ function･･･ COMPLEX(real number 1，real number 2) Returns the complex number form (real number 1) + (real number 2) i |
| 【1004】Conjugate complex number and Abloluete values （ＩＭＣＯＮＪＵＧＡＴＥ function、ＩＭＡＢＳ function） |
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| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to compleete the worksheet above》 Enter 3+4i in half-width in cell B12. Enter 3 in half-width in cell B13. Enter 3i in half-width in cell B14. Enter =IMCONJUGATE(B12) in half-width in cell C12. Enter =IMCONJUGATE(B13) in half-width in cell C13. Enter =IMCONJUGATE(B14) in half-width in cell C14. Enter =IMABS(B12) in half-width in cell D12. Enter =IMABS(B13) in half-width in cell D13. Enter =IMABS(B14) in half-width in cell D14. ＜About functions＞ ＩＭＣＯＮＪＵＧＡＴＥ function･･･ IMCONJUGATE(complex number 1) Returns complex number that is the conjugate of complex number 1. ＩＭＡＢＳ function･･･ IMABS(complex number 1) Returns the absolute value of a complex number 1. |
| 【1005】argument , real part , imaginary part of complex number （ＩＭＡＲＧＵＭＥＮＴ function、ＩＭＲＥＡＬ function、ＩＭＡＧＩＮＡＲＹ function） |
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| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Drag the cell range B11:B20 , right-click and select Cell formatting. Select Display format → Classification string → OK Enter data in the ccll range B11:B20 as shown in the table above. Enter =COMPLEX(1,0) in half-width in cell C12. Enter =COMPLEX(SQRT(3),1) in half-width in cell C13. Enter =COMPLEX(1,1) in half-width in cell C14. Enter =COMPLEX(1,SQRT(3)) in half-width in cell C15. Enter =COMPLEX(0,1) in half-width in cell C16. Enter =COMPLEX(-1,SQRT(3)) in half-width in cell C17. Enter =COMPLEX(SQRT(-1,1) in half-width in cell C18. Enter =COMPLEX(-SQRT(3),1) in half-width in cell C19. Enter =COMPLEX(-1,0) in half-width in cell C20. Enter =DEGREES(IMARGUMENT(C12)) in half-width in cell D12. Right-click on the cell D12 and select Copy. Drag the cell range D13:D20 , right-click and select Paste. Enter =IMREAL(C12) in half-width in cell E12. Right-click on the cell E12 and select Copy. Drag the cell range E13:E20 , right-click and select Paste. Enter =IMAGINARY(C12) in half-width in cell F12. Right-click on the cell F12 and select Copy. Drag the cell range F13:F20 , right-click and select Paste. ＜About functions＞ ＩＭＡＲＧＵＭＥＮＴ function･･･ IMARGUMENT(complex number 1) Returns the argument of a complex number 1 in radians. unit ＩＭＲＥＡＬ function･･･ IMREAL(complex number 1) Returns the real part of a complex number 1. ＩＭＡＧＩＮＡＲＹ function･･･ IMAGINARY(complex number 1) Returns the imaginary part of a complex number 1. |
【1006】Think of  de Moivle's theorem  （ＩＭＰＯＷＥＲ function） |
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| ＜Procedure＞ (1) Create the following correspondence table on a worksheet. ＜Reference＞ Refer to 【401】 how to complete the worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter data in the cell range B9:B23 as shown in the table above. Enter =COMPLEX(1,SQRT(3)) in half-width in cell C9. Enter data in the cell C11 as shown in the table above. Enter =IMPOWER($C$9,B12) in half-width in cell C12. Right-click on the cell C12 and select Copy. Drag the cell range C13:C23 , right-click and select Paste. Enter =DEGREES(IMAGUMENT(C12)) in half-width in cell D12. Right-click on the cell D12 and select Copy. Drag the cell range D13:D23 , right-click and select Paste. Enter =IMABS(C12) in half-width in cell E12. Right-click on the cell E12 and select Copy. Drag the cell range E13:E23 , right-click and select Paste. ＜About functions＞ ＩＭＰＯＷＥＲ function･･･ IMPOWER(complex number 1，ｎ) Returns (complex number 1)^n. |
| 【1007】 When the points represented by α,β, and γare A , B ,and C respectively , find the magnitude of ∠ＡＢＣ. | ||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ For points Ａ(α)、Ｂ(β)、and Ｃ(γ)  (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter data in the cell range B11:B18 as shown in the table above. Enter =COMPLEX(1+SQRT(3),-2+SQRT(3)) in half-width in cell C11. Enter =COMPLEX(0,-1) in half-width in cell C12. Enter =COMPLEX(1,-2) in half-width in cell C13. Enter =IMSUB(C11,C13) in half-width in cell C15. Enter =IMSUB(C12,C13) in half-width in cell C16. Enter data in the cell C18 as shown in the table above. Enter =DEGREES(IMARGUMENT(IMDIV(C16,C15))) in half-width in cell D18. ＜Consideration＞  β－γ＝－ｉ－(１－２ｉ)＝－１＋ｉ Calculate and find arg{(β－γ)/(α－γ)}. |
| To table of contents |
| 【1101】 When the integer part of √5 is a and the decimal part of √5 is b , find the value of (a^2-b^2). | |||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet above.
＜Reference＞ 《How to complete the worksheet above》 Enter data in the cell range B11:B16 as shown in the table above. Enter =TRUNC(SQRT(5)) in half-width in cell C11. Enter =SQRT(5)-C11 in half-width in cell C12. Enter =POWER(C11,2)-POWER(C12,2) in half-width in cell C14. Enter =4*SQRT(5)-5 in half-width in cell C16. ＜About functions＞ ＰＯＷＥＲ function･･･ POWER(real number 1，ｎ) Returns (real number 1)^n ＴＲＵＮＣ function･･･ TRUNC(real number 1) Returns a real number 1 with the decimal part truncated. ＜Consideration＞ Since a=2 , b=√5 - 2  Make sure that the value of a^2-b^2 on the worksheet matches the value of 4√5 - 5. |
| 【1102】 Find the value of a^2+8a when the decimal part of √23 is a. | |||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter data in the cell range B11:B14 as shown in the table above. Enter =TRUNC(SQRT(23)) in half-width in cell C11. Enter =SQRT(23)-C11 in half-width in cell C12. Enter =POWER(C12,2)+8*C12 in half-width in cell C14. ＜About functions＞ ＰＯＷＥＲ function･･･ POWER(real number 1，n) Returns (real nimber 1)^n ＴＲＵＮＣ function ･･･ TRUNC(real number 1) Returns a real number 1 with the decimal part truncated. ＜Consideration＞ Since a=√２３ － ４  Make sure that this matches the value of a^2+8a on the worksheet. |
| To table of contens |
| 【1201】 Regarding the tangent at point (t , t^3+t^2-2t) on the cubic function y=x^3+x^2-2x , observe the state of the tangent when t moves within the range of -2.8 to 2.1. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter data in the cell range A2:A62 in increments of 0.1 as shown in the table above. Enter ＝A2^3+A2^2-2*A2 in half-width in cell B2. Copy cell B2 to the cell range B3:B62. Enter ＝(3*$F$2^2+2*$F$2-2)*A2-2*$F$2^3-$F$2^2 in half-width in cell C2. Copy cell C2 to the cell range C3:C62. Enter ＝F2^3+F2^2-2*F2 in half-width in cell G2. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range A1:C62 in the correspondence table of (1) above. ② Left-click ［Insert］→［Scatter plot］→［Scatter line］ ③ Delete the legend (3) Arrange the xy coordinate plane on which the graph is drawn. ① Set the x-axis scale to -3 or more and 3 or less , and set the scale interval to 1. ② Set the y-axis scale to -10 or more and 10 or less , and set the scale interval to 1 ③ Set the scale line style to dotted lines for both the x-axis and y-axis.  ＜About Macro＞ Sub Sessen() Dim t As Single 'Declare it as single-precision floating point type DoEvents 'Allow you to force exit from ForNext repetition. For t = -2.8 To 2.1 Step 0.01 Cells(2, "F") = t 'Assign the the value of t to row 2 and column F. Calculate 'Recalculate Next End Sub ＜Consideration＞ Let's run the macro and observe the movement of the tangent line. You can observe the state of the tangent line as the x-coordinate of the contact point changes from -2.8 to 2.1. |
| 【1202】 Observe that the graph of the quadratic function y=a(x-p)^2+q is obtained by translaing the graph of y=ax^2 by +p in the x-axis direction and +q in the y-axis direction. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter data in the cell range B3:B63 in increments of 0.1 as shown in the table above. Enter ＝$G$2*(B3-$G$3)^2+$G$4 in half-width in cell C3 Copy the cell C3 to the cell range C4:C63. Enter ＝$G$2*(B3-$I$8)^2+$J$8 in half-width in cell D3. Copy the cell D3 to the cell range D4:D63. Enter =I8 in half-width in cell I7. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range B2:D63 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Scatter line］ ③ Delete the legend. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Set the x-axis scale to -3 or more and 3 or less , and set the scale interval to 1. ② Set the y-axis scale to -10 or more and 10 or less , and set the scale interval to 1. ③ Set the scale line style to dotted lines for the x-axis and y-axis.  ＜About macro＞ Option Explicit 'Force bariable declaration Sub apq() Dim a, p, q As Single 'Declare as simple-precision floating point type Dim sp, sq As Single 'Declare as simple-precision floating point type sp = Cells(8, "I") 'Assign the value of row 8 and column I to sp sq = Cells(8, "J") 'Assign the value of row 8 and column J to sq Cells(3, "G") = 0 'Write 0 in row 3 and couimn G Cells(4, "G") = 0 'Write 0 in row 4 and column G DoEvents 'Allow forcible exit from ForNext statement If sp > 0 And sq > 0 Then 'If the vertex is in the first quadrant Cells(3, "G") = 0 'Write 0 in row 3 and column G Cells(4, "G") = 0 'Write 0 in row 4 and column G For p = 0 To sp Step 0.05 Cells(3, "G") = p 'Write the value of p in row 3 and column G Calculate 'Recaluculate Next p For q = 0 To sq Step 0.05 Cells(4, "G") = q 'Write the value of q in row 4 and column G Calculate 'Recalculate Next q End If If sp > 0 And sq < 0 Then 'If the vertex is in the 4th quadrant Cells(3, "G") = 0 'Write 0 in row 3 and column G Cells(4, "G") = 0 'Write 0 in row 4 and column G For p = 0 To sp Step 0.05 Cells(3, "G") = p 'Write the value of p in row 3 and column G Calculate 'Recalculate Next p For q = 0 To sq Step -0.05 Cells(4, "G") = q 'Write the value of q in row 4 and column G Calculate 'Recalculate Next q End If If sp < 0 And sq > 0 Then 'If the vertex is in the 2nd quadrant Cells(3, "G") = 0 'Write 0 in row 3 and column G Cells(4, "G") = 0 'Write 0 in row 4 and column G For p = 0 To sp Step -0.05 Cells(3, "G") = p 'Write the value of p in row 3 and column G Calculate 'Recalculate Next p For q = 0 To sq Step 0.05 Cells(4, "G") = q 'Write the value of q in row 4 and column G Calculate 'Recalculate Next q End If If sp < 0 And sq < 0 Then ''If the vertex is in the 3rd quadrant Cells(3, "G") = 0 'Write 0 in row 3 and column G Cells(4, "G") = 0 'Write 0 in row 4 and column G For p = 0 To sp Step -0.05 Cells(3, "G") = p 'Write the value of p in row 3 and column G Calculate 'Recalculate Next p For q = 0 To sq Step -0.05 Cells(4, "G") = q 'Write the value of q in row 4 and column G Calculate 'Recalculate Next q End If End Sub ＜Consideration＞ Let's run the macro and observe that the quadratic function y=a(x-p)^2+q is obtained by translating y=ax^2 by +p in the x-axis direction and +q in the y-axis direction. |
| To table of contents |
| 【1301】 Find prime numbers less than or equal to 20 using a macro. |
＜About prime number＞ A prime number is a natural number greater than or equal to 2 that has no positive divisors other than 1 and itself. The following macro determines whether a number is prime by dividing it one by one until 2 or more and 1 smaller than itself and checking whether it is divisible. ＜About macro＞ Sub Sosu2() For n = 2 To 20 d = 0 'Initialize d For k = 2 To n - 1 If n Mod k = 0 Then 'If k is a divisor of n d = d + 1 'Add 1 to d. End If Next k If d = 0 Then 'If d=0 MsgBox (n & "is prime！") 'Display the message that d is a prime number. End If Next n MsgBox ((n - 1) & " We have now completed our search for the following prime numbers.") 'Display the "End" message. End Sub ＜Consideration＞ Let's run the macro and find prime numbers less than or equal to 20. Next , let's rewrite the macro and find prime numbers less than or equal to 30. |
| 【1302】 Find perfect numbers using a method other than brute force. | ||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
※ Follow the steps below so that column D does not display "･････E-12". Select column D → Right+click → Format cells → Display format → Numerical value → ＯＫ ＜About perfect number＞ A perfect number is a natural number whose sum of positive divisors , excluding itself , is equal to itself. For example , the divisors of 6 , excluding itself , are 1,2,3 , so 1+2+3=6 , thefore , 6 is a perfect number. When n is a natural number greater than or equal to 2 and  is a prime number ,  is a perfect number. (This can be easily proven using the formula for
the sum of geometric progressions.)So , are all perfect numbers expressed as when is a prime number ?It has been proven that all even perfect numbers are denoted by when is prime.However , the odd perfect number has not yet been discovered. It has not been proven that the odd perfect numbers don't exist. The following macro uses the fact that "When n is a natural number greater than or equal to 2 and is a prime number , is a perfect number" to find a perfect number.＜About macro＞ Sub Kanzensusp() Dim n As Long 'Declare as long integer type Dim k As Long 'Declare as long integer type Dim d As Long 'Declare as long integer type 'Cells.Clear DoEvents 'Make it possible to stop even in the middle of a ForLoop For n = 2 To 31 Cells(n, "A") = n 'Assign the value of n to row n and column A Cells(n, "B") = 2 ^ n - 1 'Assign the value of 2^n-1 to row n and column B. d = 0 'initialize d. For k = 2 To 2 ^ n - 1 - 1 If (2 ^ n - 1) Mod k = 0 Then 'If k is a divisor of 2^n-1 , then d = d + 1 'Add 1 to d End If Next k If d = 0 Then 'If d=0 then , then Cells(n, "C") = "Prime" 'Assign "Prime number " to row n and column C. End If Cells(n, "D") = (2 ^ (n - 1)) * (2 ^ n - 1) 'Assign the value of 2^(n-1)*(2^n-1) to row n and column D If Cells(n, "C") = "Prime" Then 'If row n and column C is "Prime number" , then Cells(n, "E") = "Perfect" 'Assign "Perfect number" to row n and column E End If Next n MsgBox ("The search of perfect numbers has ended...") 'Display a completion message. End Sub ＜Consideration＞ Let's run the macro and find the perfect numbers. It will take some time to find 8 perfect numbers. |
| To table of contents |
| 【1401】 Draw a figure (positive leaf curve) repesented by the polar equetion ｒ＝ｓｉｎａθ. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a warksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter data in the cell range A2:A362 in 1 complements as shown in the the table above. Enter ＝SIN($F$2*RADIANS(A2)) in half-width in cell B2. Copy cell B2 to the cell range B3:B362. Enter ＝B2*COS(RADIANS(A2)) in half-width in cell C21. Copy cell C2 to cell range C3:D362. Enter ＝B2*SIN(RADIANS(A2)) in half-width in cell D2. Copy cell D2 to the cell range D3:D362. Enter "a=" in cell E2. Enter "2" in cell F2. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range C2:D362 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Scatter plot (smooth line)］ ③ Delete the legend. (3) Arrange the xy coordinate plane on which the graph is drawn. ① Set the scale interval of the x-axis to 0.1 between -1 and 1. ② Set the scale interval of the y-axis to 0.1 between -1 and 1. ③ Set the scale line style to dotted lines for both the x-axis and y-axis.  ＲＡＤＩＡＮＳ function ･･･ ＲＡＤＩＡＮＳ(Angle in degrees) Convert an angle in degrees to arc degrees. ＜About macro＞ Sub Seiyou() Dim a As Single 'Declare a as a single-precision floating point type. For a = 0 To 10.05 Step 0.05 Cells(2, "F") = a 'Assign the value of a to row 2 and column F. Calculate 'Recalculate Next a End Sub ＜Consideration＞ Let's sequentially substitute 2 , 3 , 4 , ･･･,10 into cell F2 (value of a) and observe the drawn figure (positive leaf curve). Also , let's run the macro and observe the figures drawn as the value of a changes from 0 to 10 in 0.05 increments. |
| 【1402】 Draw the figure (Archimedes spiral line) represented by the polar equation r=aθ. | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| ＜Procedure＞ (1) Create the following correspondence table on a worksheet.
＜Reference＞ 《How to complete the worksheet above》 Enter data in the cell range A2:A722 in increments of 1 as shown in the table above. Enter ＝$F$2*RADIANS(A2) in half-width in cell B2. Copy cell B2 to the cell range B3:B722. Enter ＝B2*COS(RADIANS(A2)) in half-width in cell C2. Copy cell C2 to the cell range C3:C722. Enter ＝B2*SIN(RADIANS(A2)) in half-width in cell D2. Copy cell D2 to the cell range D3:D722. Enter "a=" in cell E2. Enter "2" in cell F2. (2) Draw a graph from the correspondence table in (1) above. ① Drag and select the cell range C2:D722 in the correspondence table of (1) above. ② Left-click ［Incert］→［Scatter plot］→［Scatter plot（Smooth line）］ ③ Delete legend. (3) Arrange the xy-coordinate plane on which the graph is drawn. ① Set the x-axis scale to -150 or more and 150 or less , and set the scale interval to 30. ② Set the y-axis scale to -150 or more and 150 or less , and set the scale interval to 30. ③ Set the scale line style to the dotted lines for both the x-axis and y-axis.  ＜About functions＞ ＲＡＤＩＡＮＳ function･･･ ＲＡＤＩＡＮＳ(Angle in degrees) Convert angles in degrees to arc degrees. ＜About macro＞ Sub Aruki() Dim a As Single 'Declare a as a singe-precision floating point type For a = 0 To 10.05 Step 0.05 Cells(2, "F") = a 'Assign the value of a to row 2 and column F. Calculate 'Recalculate Next a End Sub ＜Consideration＞ Let's sequentially assign 2 , 3 , 4 , ･･･, 10 to cell F2 (value of a) and obseve the drawn figure (Archmedean spiral line). Also , let's run the macro and observe the figures drawn as the value of a changes from 0 to 10 in 0.05 increments. |
| ※ You can download the above sample data files "～.xls" below. |
| To table contents |
| When you downloaded the data below for how to use Excel , compressed file
"sample_excel.lzh" will be downloaded. When the downloaded compressed file "sample_excel.lzh" is decompressed , a folder "Sample data for thinking about how to use "Excel" is created. When you download the sample data for "simulation" , "Prime and Perfect Numbers" , and "Polar Equation"below , copmressed file "ExcelSample2.lzh" will be downloaded. When the downloaded compressed file "ExcelSample2.lzh" is downloaded , the sample data "Simulation" , "Prime and Perfect Number" , and "Polor Equation" are created. |
| Ｅｘｃｅｌ sample data | Sample data for thinking about how to use "Excel" |
| Ｅｘｃｅｌ sample data 2 | "Simulation" , "Prime and Perfect Number" , "Polar Equation "sample data |
| To table of contents |
| There are 11 folders "[0100]Quadratic function" , "[0200]Trigonometric function" , "[0300]Exponential function" , "[0400]Logarithmic function" , "[0500]Various curves" , "[0600]Differentiation(Part 1)" , "[0700]Differentiation (Part 2)" , "[0800]Integral" , "[0900]Number sequence" , "[1000]Complex number" , "[1100]Application of square roots" in the folder "Sample data for thinking about how to use Excel". Open the files "～.xlsm" in each folder in Excel. |
| There are six files in the folder "Simulation " , "Prime and Perfect Numbers" , "Polor Equations" sample data. Open each file "～.xlsm" in Excel. |