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| LastUpdate 4/3/2025 |
| My first encounter with the mathematical function graphing software "GRAPES"
was when I partitipated in the 1993 Information Education Instructor Course
(sponsored by the Ministry of Education) , aimed at high school teachers
and held at the Kyoto Prefectural General Education Center. Three people
, including myself , participated from Gifu Prefecture. At this time , Katsuhisa Tomoda , the developer of "GRAPES" , participated as a representative from Osaka Prefecture. Participants brought and presented the learning software they had developed themselves. Many people , including myself , brought along presentation software on floppy disks. However , someone brought it with them on a large hard drive , and it made quite an impression on me. That person was Katsuhisa Tomoda , who released the MS-DOS version of "GRAPES". At that time , it wasn't like the portable disk that we have today , but the hard disk that Tomoda brought with him seemed to be one that came attached to a computer. That year was 1993 , Windows 95 was released in 1995 , and since then the internet has become widespread. This trainig in 1993 included training on computer communication over telephone lines , but there were still no lectures on the internet. The version I have installed on my computer now is "GRAPES 6.46". It has evolved significantly since the MS-DOS version of "GRAPES" that I first encountered. I would like to express my respect to Mr. Katsuhisa Tomoda. Furthermore , "GRAPES" is free software. |
| http://kn-makkun.com/MakkunWp/grapes.html |
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| To table of contents |
| 101. [Graph of quadratic function y=ax^2] |
| Investigate how the shape of the graph of y=ax^2 changes depending on
the value of a. ［File］ of data panel in the "GRAPES"→ Open → Folder"Sample data for considering how to use GRAPES" → Fplder "[100] Quadratic functions and Linear functions" → File name"y=ax^2.gps" Change the value of the parameter a to see how the shape of the graph changes. ＜About the script＞ Click on [Init (a=0.001)] and then click on [When a>0]. Next , click on [Init (a=-0.01)] and then click on [When a<0]. |
| 102. [Graph of quadratic function y=a(x-b)^2] |
| Investigate how the shape and position of the graph of y=a(x-b)^2 changes
depending on the values of a and b. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[100] Quadratic functions and Linear functions " → File name"y=a(x-b)^2.gps" Change the value of the parameter a to see how the shape of the graph changes. Change the value of the parameter b to see how the position of the graph changes. ＜About the script＞ Click on [Init (b=0)] and then click on (When b>0). Next , click on [Init (b=0)] and then click on (When b<0). |
| 103. [Graph of quadratic function y=ax2+c] |
| Investigate how the shape and position of the graph of y=ax^2+c changes
depending on the values of a and c. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[100] Quadratic functions and Linear functions " → File name"y=ax^2+c.gps" Change the value of the parameter a to see how the shape of the graph changes. Change the value of the parameter c to see how the position of the graph changes. ＜About the script＞ Click on [Init (c=0)] and then click on (When c>0). Next , click on [Init (c=0)] and then click on (When c<0). |
| 104. [Graph of quadratic function y=a(x-b)^2+c] |
| Investigate how the shape and position of the graph of y=a(x-b)^2+c changes
depending on the values of a , b , and c. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[100] Quadratic functions and Linear functions " → File name"y=a(x-b)^2+c.gps" Change the value of the parameter a to see how the shape of the graph changes. Change the value of the parameter b to see how the position of the graph changes. Change the value of the parameter c to see how the position of the graph changes. ＜About the script＞ Click on [Init (b=0,c=0)] and then click on [Parallel translation of a graph of a quadratic function]. |
| 105. [Graph of quadratic function y=ax^2+bx+c] |
| Investigate how the shape and position of the graph of y=ax^2+bx+c changes
depending on the values of a , b , and c. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[100] Quadratic functions and Linear functions " → File name"y=ax^2+bx+c.gps" Change the value of the parameter a to see how the shape of the graph changes. Change the value of the parameter b to see the relationship with the line y=bx+c Change the value of the parameter c to see how the position of the graph changes. ＜About the script＞ Click on [Init (a=2,b=5.5,c=-1)] and then click on [Regarding the coefficient b]. Let's see that the coefficient b is the slope of yhe tangent line at the point (0,-1) on the graph. |
| 106. Quadratic function y=x^2-2x minimum value |
| Investigate how the minimum value of y=x2-2x in the interval t≦x≦t+1 changes
depending on the value of t. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[100] Quadratic functions and Linear functions " → File name"intervalminimum.gps" Vary the parameter t to see how the minimum value changes. |
| 107. Quadratic function y=x^2-4x+5 maximum value |
| Investigate how the maximum value of y=x2-4x+5 in the interval t≦x≦t+1
changes depending on the value of t ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[100] Quadratic functions and Linear functions " → File name"intervalmaximum.gps" Vary the parameter t to see how the maximum value changes. |
| 108. A straight line passing through a point |
| Find that regardless of the value of a , the straight line y=a(x-b)+c
always passes through a single point (b,c). ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[100] Quadratic functions and Linear functions " → File name"y=a(x-b)+c.gps" Vary the value of the parameter a to see that the straight line y=a(x-b)+c always passes through a single point (b,c). ＜About the script＞ Click on [Init (a=1)] and then click on [When a>1]. Next , click on [Init (a=1)] and then click on [When a<1]. |
| To table of contents |
| 201. Graph of trigonometric function y=asinx |
| Investigate how the shape of the graph of y=asinx changes depending on
the value of a. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[200] Trigonometric functions , Exponential functions , and Logarithmic functions" → File name"y=asinx.gps" Vary the value of the parameter a to see how the shape of the graph changes. ＜About the script＞ Click on [Init (a=1)] and then click on [When a>1]. Next , click on [Init (a=1)] and then click on [When a<1]. |
| 202. Graph of trigonometric function y=acosx |
| Investigate how the shape of the graph of y=acosx changes depending on
the value of a. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[200] Trigonometric functions , Exponential functions , and Logarithmic functions" → File name"y=acosx.gps" Vary the value of the parameter a to see how the shape of the graph changes. ＜About the script＞ Click on [Init (a=1)] and then click on [When a>1]. Next , click on [Init (a=1)] and then click on [When a<1]. |
| 203. Graph of trigonometric function y=atanx |
| Investigate how the shape of the graph of y=atanx changes depending on
the value of a. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[200] Trigonometric functions , Exponential functions , and Logarithmic functions" → File name"y=atanx.gps" Vary the value of the parameter a to see how the shape of the graph changes. ＜About the script＞ Click on [Init (a=1)] and then click on [When a>1]. Next , click on [Init (a=1)] and then click on [When a<1] |
| 204. Graph of trigonometric function y=sinax |
| Investigate how the shape of the graph of y=sinax changes depending on
the value of a. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[200] Trigonometric functions , Exponential functions , and Logarithmic functions" → File name"y=sinax.gps" Vary the value of the parameter a to see how the shape of the graph changes. ＜About the script＞ Click on [Init (a=1)] and then click on [When a>1]. Next , click on [Init (a=1)] and then click on [When a<1]. |
| 205. Graph of trigonometric function y=cosax |
| Investigate how the shape of the graph of y=cosax changes depending on
the value of a. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[200] Trigonometric functions , Exponential functions , and Logarithmic functions" → File name"y=cosax.gps" Vary the value of the parameter a to see how the shape of the graph changes. ＜About the script＞ Click on [Init (a=1)] and then click on [When a>1]. Next , click on [Init (a=1)] and then click on [When a<1]. |
| 206. Graph of trigonometric function y=tanax |
| Investigate how the shape of the graph of y=tanax changes depending on
the value of a. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[200] Trigonometric functions , Exponential functions , and Logarithmic functions" → File name"y=tanax.gps" Vary the value of the parameter a to see how the shape of the graph changes. ＜About the script＞ Click on [Init (a=0)] and then click on [When a>1]. Next , click on [Init (a=1)] and then click on [When a<1] |
| 207. Graph of trigonometric function y=sin(x-a) and y=cos(x-b) |
| Investigate the shape and the positional relationship of the graph of
y=sin(x-a) and y=cos(x-b). ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[200] Trigonometric functions , Exponential functions , and Logarithmic functions" → File name"y=sn(x-a)_cos(x-b).gps" Vary the value of the parameter a to see the shape and the positional relationship of the graph. ＜About the script＞ Click on [Init (b=0)] and then click on [Translation]. Let's note that the graph of y=sinx is the graph of y=cosx translated by +90°along the x-axis. |
| 208. Graph of exponential function y=a^x |
| Investigate three characteristics of the graph of y=a^x by changing the
value of a. (①It passes through the point (0,1) regardless of the value
of a. ②When a>1 , it slopes upwards to the right. When 0<a<1 ,
it slopes downwards to the right. ③Regardless of the value of a , the asymptote
is the x-axis.) ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[200] Trigonometric functions , Exponential functions , and Logarithmic functions" → File name"y=a^x.gps" Vary the value of the parameter a to see the three characteristics of the graph. ＜About the script＞ Click on [Init (a=1.1)] and then click on [When a>1]. Next , click on [Init (a=1.1)] and then click on [When 1>a>0]. |
| 209. Graph of logarithmic function y=log(a,x) |
| Investigate three characteristics of the graph of y=log(a,x) by changing
the value of a. (①It passes through the point (1,0) regardless of the value
of a. ②When a>1 , it slopes upwards to the right. When 0<a<1 ,
it slopes downwards to the right. ③Regardless of the value of a , the asymptote
is the y-axis.) ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[200] Trigonometric functions , Exponential functions , and Logarithmic functions" → File name"y=log'(a,x).gps" Vary the value of the parameter a to see the three characteristics of the graph. ＜About the script＞ Click on [Init (a=1.1)] and then click on [When a>1]. Next , click on [Init (a=1.1)] and then click on [When 1>a>0] |
| To table of contents |
| 301. Descartes connects figures and equations (analytical geometry) | |
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| 302. Descartes connects figures and equations (analytical geometry) |
| Think about what shape the equation sinx+siny=sinax+sinay represents ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"y=equationdiagram2.gps" Vary the value of the parameter a to see how the shape changes. ＜About the script＞ Click on [Init (a=2)] and then click on [When a>2]. |
| 303. Descartes connects figures and equations (analytical geometry) |
| Think about what shape the equation sinax+sin(a+1)y=sin(a+2)x+sin(a+3)y
represents ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"y=equationdiagram3.gps" Vary the value of the parameter a to see how the shape changes. ＜About the script＞ Click on [Init (a=0)] and then click on [When a>0]. |
| 304. External division point on a line |
| Consider the case where m and n have opposite signs at point P((na+mb)/(m+n))
which divides two points A(a) and B(b) in the ratio m:n. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"externaldivisionpoint.gps" Vary the value of parameters m and n to see the position of point P relative to two points A and B when m and n have opposite signs. |
| 305. The three medians of a triangle intersect at exactly one point (centre of gravity) |
| Consider wheather the three medians of any triangles always intersect
at exactly one point. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"centreofgravity.gps" Move the vertices A , B , and C of a triangle and see that the three medians of any triangle always intersect at exactly one point. |
| 306. Perpendicular condition for two lines |
| Consider the cnditions under which two lines intersect perpendicularly. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"perpendicularconditionfortwolines.gps" Move the m and n of the slopes of the two lines to see at what values of m and n the two lines intersect perpendicularly. |
| 307. Tangent to a circle whose center is the origin. |
| Consider the equation of a tangent to a circle whose center is a point
other than the origin. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"tangenttoacircle1.gps" Move the tangent point P(p,q) and see that the tangent to the circle x^2+y^2=c^2 at tangent point P is px+qy=c^2. |
| 308. Tangent to a circle whose center is a point other than the origin. |
| Consider the equation of a tangent to a circle whose center is a point
other than the origin. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"tangenttoacircle2.gps" Move the center A(a,b) and the tangent point P(p,q) and see that the tangent to the circle (x-a)^2+(y-b)^2=c^2 at tangent point P is (p-a)(x-a)+(q-b)(y-b)=c^2. |
| 309. Equation of a circle through three points. |
| Consider the equation of a circle that passes through three points. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"circlethrough3points.gps" Find the equation of a circle that passses through three points O(0,0) , A(6,0) , B(-1,7) by solving a system of simultaneous equation. Use "GRAPES" to draw this circle and see that it passes through three points O(0,0) , A(6,0) , B(-1,7). |
| 310. Apollonius' Circle |
| Consider Apollonius' Circle. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"apolloniuscircle.gps" By moving the two points A and B , and changing the value of the parameter k , look at the locus of points whose distance ratio from the two points A and B is k : :1. See that when k=1 , it becomes the perpendicular bisector of two points A and B. When k≠1 , change the value of the parameter k and see that it becomes a circle. <About the script> Click on [Init(k=2)] and then click on [When k>2]. Next , click on [Init(k=2)] and then click on [When 0<k<2]. Furthermore , click on [When k=1] , too. |
| 311. Equation of a circle passing through the intersection of two circles |
| Consider equation of a circle passing through the intersection of two
circles. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"circlethrough2intersections.gps" The equation of the circle that passes through the intersection of two circles (x-a)^2+(y-b)^2=c^2 , (x-s)^2+(y-t)^2=u^2 is expressed as (x-a)^2+(y-b)^2-c^2+k{(x-s)^2+(y-t)^2-u^2}=0. . See that when k≠-1 , it is a circle , and when k=-1 , it is a straight line. Change the value of the parameter k and observe the movement of the circle. <About the script> Click on [Init(k=-10)] and then click on [When -10≦k≦-1] , [When -1≦k≦0] , [When 0≦k≦1] , and [When 1≦k≦10] in that order., |
| 312. Equation of an Ellipse |
| Consider the equation of an ellipse from the locus. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"ellipse.gps" See that the locus of point whose sum of distances from two points A(-c , 0) , B(c , 0) is constant (2a) is an ellipse. . Change the values of the parameters a and c to observe the movement of the ellipse. |
| 313. Tangent of an Ellipse |
| Consider the tangent of an ellipse. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"ellipsetangent.gps" Move the tangent point P(p,q) and see that the equation of the tangent line at tangent point P to the ellipse (x^2)/a^2+(y^2)/b^2=1 is (px)/a^2+(qy)/b^2=1. |
| 314. Equation of an Hyperbpla |
| Consider the equation of an hyperbola from the locus. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"hyperbola.gps" See that the locus of points whose difference in distance from two points A(-c , 0),B'(c , 0) is constant (2a) is a hyperbola. Change the values of the parameters a and c to observe the movement of the hyperbola. |
| 315. Tangent of a Hyperbola |
| Consider the tangent of a hyperbpla. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"hyperbolatangent.gps" Move the tangent point P(p,q) and see that the equation of the tangent line at tangent point P to the hyperbola (x^2)/a^2-(y^2)/b^2=1 is (px)/a^2-(qy)/b^2=1 |
| 316. Equation of an Parabola |
| Consider the equation of a parabola from the locus. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"parabola.gps" See that the locus of points equidistant from directrix x=-p and from the point A(p , 0) is a parabola. Change the value of the parameter p and observe the movement of the parabola. |
| 317. Conic Sections |
| Consider the curve that appears at the cut edge of a cone. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"conicsections.gps" <About the script> Click on [①an example of an ellipse] , [②an example of a parabola] , [③an example of a hyperbola] , and [④an example of a circle] respectively. Next , click on [rotate] to see that depending on how you cut a cone , it can become an ellipse , a parabola , a hyperbola , or a circle. |
| 318. Creating a foci of an ellipse |
| Consider the creating a foci of an ellipse. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"creatingfociellipse.gps" <About the script> Click on [Start] to see the construction that finds the focal position using half the length of the major axis. |
| 319. Properties of tangents to an ellipse |
| Consider the properties of tangents to an ellipse. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"propertiestangentellipse.gps" Notice that the two angles made by the tangent and the two line sequments connecting the two foci of ellipse and the tangent point of ellipse are always equal. Change the value of the parameter t and observe the two corners. <About the script> Click on [ Init (t=0°)] and then , click on [ When 0°< t <360°]. |
| 320. Create an ellipse from a circle |
| Consider how to create an ellipse from a circle. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"createellipsefromcircle.gps" See how to create an ellipse from a circle. Change the value of the parameter θ to observe how an ellipse is created. Take an arbitrary point A inside a circle whose center is point B , and an arbitrary point P on the circumference. Draw the perpendicular bisector KL of the line segment AP. Let Q be the intersection of KL and PB , fix point A, and move point P. In this case , the locus of point Q becomes an ellipse with foci B and A. This is because AQ=PQ , so AQ+QB=constant(radius of the circle). |
| 321. Curvature of a hyperbola |
| Consider the curvature of a hyperbola. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use GRAPES" → Folder "[300] Diagrams and Equations" → File name"curvaturehyperbola.gps" Place the cursor at the origin and right-click. → ZOOM → Repeat × 1/2 and use [Drag to move] to see that the part of the hyperbola away from the origin is almost a straight line. |
| 322. Eccentricity of a quadratic curve |
| Consider the eccentricity of a quadratic curve. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"eccentricityquadraticcurve.gps" By varying the value of the parameter k , we observe the locus of points where the ratio of the distance between the the directrix x=-c and the focal point A(c,0) is 1:k is as follows : ①When 0 < k < 1 , it becomes an ellips. ②When k = 1 , it becomes a parabola. ③When 1 < k , it becomes a hyperbola. <About the script> Click on [Init (c=1, k=1)] and then click on [ellipse (0<k<1)]. Next , Click on [parabola (k=1)]. Furthermore , click on [Init (c=1, k=1)] and then click on [hyperbola (1<k)]. |
| 323. Area represented by inequality 1 (straight line and clrcle) |
| Consider the area represented by inequality (x-y-2)(x^2+y^2-4)≧0. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name(x-y-2)(x^2+y^2-4)_0.gps" By varying the value of the parameter k , we observe the locus of points where the ratio of the distance between the the directrix x=-c and the focal point A(c,0) is 1:k is as follows : ①Line y=x-2 and circle x^2+y^2=4 form the boundaries of the area. ②Area where inquality y≦x-2 and inequality x^2+y^2≧4 are satisfied simultaneously. ③Area where inequality y≧x-2 and inequality x^2+y^2≦4 are satisfied simultaneously. See that the area represented are ① , ② , ③. |
| 324. Area represented by inequality 2 (straight line and parabola) |
| Consider the area represented by inequality (x+y-2)(y-x^2)<0. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"(x+y-2)(y-x^2)_0.gps" ①Line y=-x+2 and parabola y=x^2 form the boundary line. ②Iinquality y>x+2 and inequality y<x^2 are satisfied simultaneously. ③Inequality y<-x+2 and inequality y>x^2 are satisfied simultaneously. See that the area to be represented is an area that satisfies ① , ② , and ③. |
| 325. Area represented by inequality 3 (inequalities with absolute values) |
| Consider the area represented by inequality ｜ｘ｜＋｜ｙ｜≦１. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"abs(x)+abs(y)_1.gps" ①Straight lines y=x+1 , y=x-1 , y=-x+1and y=-x-1 are the boundaries of the area. ②This is an area where inequalities y≦x+1 , y≧x-1 , y≦-x+1, and y≧-x-1 are satisfied simultaneously. See ① and ②. Also , change the value of the parameter a , and observe the change in the area. <About the script> Click on [Init (a=1)] and then click on [When a>1)]. Next , Click on [Init (a=1)] and then click on [When 1>a>0]. |
| 326. Area represented by inequality 4 (inequalities with absolute values) |
| Consider the area represented by inequality x^2+y^2≦2｜x+y｜. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"x^2+y^2_2abs(x+y).gps" ①The circle (x-1)^2+(y-1)^2=2 and (x+1)^2+(y+1)^2=2 , and the line y=-x form the boundary of the area. ②Inequalities (x-1)^2+(y-1)^2≦2 and y≧-x are satisfies simultaneously.. ③Inequalities (x+1)^2+(y+1)^2≦2 and y<-x are satisfies simultaneously. See that it is the area ① , ② , and ③. Also , change the value of the parameter a , and observe the change in the area. <About the script> Click on [Init (a=2)] and then click on [When a>2)]. Next , Click on [Init (a=2)] and then click on [When 2>a>0] |
| 327. Area represented by inequality 5 (inequalities with absolute values) |
| Consider the area represented by inequality x^2+y^2≦2｜x｜+2｜y｜. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"x^2+y^2_2abs(x)+2abs(y).gps" ①The circle (x-1)^2+(y-1)^2=2 , (x-1)^2+(y+1)^2=2 , (x+1)^2+(y-1)^2=2 , (x+1)^2+(y+1)^2=2 , the lines x=0 and y=0 form the boundary of the area. ②Inequalities (x-1)^2+(y-1)^2≦2 , x≧0 , and y≧0 are satisfies simultaneously. ③Inequalities (x-1)^2+(y+1)^2≦2 , x≧0 , and y<0 are satisfies simultaneously. ④Inequalities (x+1)^2+(y-1)^2≦2 , x<0 , and y≧0 are satisfies simultaneously. ⑤Inequalities (x+1)^2+(y+1)^2≦2 , x<0 , and y<0 are satisfies simultaneously. See that it is in the area ① , ② , ③ , ④ and ⑤. Also , change the value of the parameter a , and observe the change in the area. <About the script> Click on [Init (a=2)] and then click on [When a>2)]. Next , Click on [Init (a=2)] and then click on [When 2>a>0] |
| 328. Maximum and minimum in an area |
| Consider the maximum and minimum values of x+y when 5x-3y≧0 , x-2y≦0 ,
and 3x+y-14≦0. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"xmaximumminimumarea.gps" In the area represented by inequalities 5x-3y≧0 , x-2y≦0 , and 3x+y-14≦0 , move point P(x,y) and observe the change in the value k of x+y. Also , change the value of the parameter k , and obseve x+y=k , where k is the y-intercept when considering the line y=-x+k. <About the script> Click on [Minimum value (k=0)]. Next , Click on [Maximum value (k=8)]. Furthermore , click on [When 0≦k≦8]. |
| 329. The area through which the graph of the quadratic function passes. |
| Consider the area through which the graph of the quadratic function y=x^2+2ax+2a^2
passes. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"y=x^2+2ax+2a^2.gps" The condition under which quadratic equation 2a^2+2xa+x^2-y=0. Condition : discriminant D=(2x)^2-4×2×(x^2-y)≧0 Let's rearrange this and look at the area represented by inequality y≧(1/2)x^2. Change the value of the parameter a to observe the area through which the graph of the quadratic function y=x^2+2ax+2a^2 passes. <About the script> Click on [Init (a=0)] and then click on [When a>0]. Next , Click on [When a<0]. |
| 330. Locus of the intersections of the two straight lines |
| Consider the locus of the intersection of the two straight lines tx+y=2
and x-ty=0. ［File］ of data panel in the "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[300] Diagrams and Equations" → File name"twolineintersectionlocus.gps" Change the value of the parameter t to see that the locus of intersection of two lines tx+y=2 and x-ty=0 is a circle with center (0,1) and radius 1. <About the script> Click on [Init (t=1)] and then click on [When 1<t<10]. Next , Click on [When 1>t>-10]. |
| To table of contents |
| 401. The slope of the tangent of the graph of a quadratic function y=x^2 |
| Find the slope of the rangent to y=x^2 at x=1. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"findslopetangenty=x^2.gps" Place the cursor at (1,1) and right-click → zoom → Enlarge by ×10 Change the value of the parameter m to find the slope m of the tangent. |
| 402. Graph of the derivative of a quadratic function |
| Investigate the relationship between the graph of a quadratic function
and the grapf of its derivative. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"quadraticderivativegraph.gps" Change the value of patameters a , b and c to look at the relationship between increse/decrease on the graph of a quadratic function and the positive/negative y value of the derivative. |
| 403. Graph of the derivative of a cubic function |
| Investigate the relationship between the graph of a cubic function and
the grapf of its derivative. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"cubicderivativegraph.gps" See the relationship between increse/decrease on the graph of a cubic function and the positive/negative y value of the derivative. |
| 404. Number of extreams of a cubic function |
| Investigate the number of extreams of a cubic function. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"cubicfunctionextreamnumber.gps" Change the value of the parameter a to see that the number of the extreams of a cubic function is either 2 or 0. <About the script> Click on [Init (a=-2) and then click on [When a<-2]. Next , click on [Init (a=-2)] and then click on [When -2<a≦0]. Furthermore , click on [When 0<a] |
| 405. Tangent to the graph of the quadratic function y=x^2-2x |
| Investigate how the tangent to y=x^2-2x at x=a changes depending on the
value of a. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"tangenttoy=x^2-2x.gps" Change the value of the parameter a to see the movement of the tangent. <About the script> Click on [Init (a=2) and then click on [When a>2]. Next , click on [Init (a=2)] and then click on [When a<2]. Furthermore , click on [When a=1] |
| 406. Tangent to the graph of the quadratic function y=-x^2+4x+2 |
| Investigate how the tangent to y=-x^2+4x+2 at x=a changes depending on
the value of a. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"tangenttoy=-x^2+4x+2.gps" Change the value of the parameter a to see the movement of the tangent. <About the script> Click on [Init (a=1) and then click on [When a>1]. Next , click on [Init (a=1)] and then click on [When a<1]. Furthermore , click on [When a=2] |
| 407. Tangent to the graph of the cubic function y=x^3-3x^2 |
| Investigate how the tangent to y=x^3-3x^2 at x=a changes depending on
the value of a. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"y=x^3-3x^2tangent.gps" Change the value of the parameter a to see the movement of the tangent. <About the script> Click on [Init (a=1) and then click on [When a>1]. Next , click on [Init (a=1)] and then click on [When a<1]. Furthermore , click on [When a=0] and then click on [When a=2]. |
| 408. Tangent to the graph of the cubic function y=-x^3+3x+1 |
| Investigate how the tangent to y=-x^3+3x+1 at x=a changes depending on
the value of a. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"y=-x^3+3x+1tangent.gps" Change the value of the parameter a to see the movement of the tangent. <About the script> Click on [Init (a=0) and then click on [When a>0]. Next , click on [Init (a=0)] and then click on [When a<0]. Furthermore , click on [When a=-1] and then click on [When a=1] |
| 409. A line passing through two points on the graph and a tangent |
| For a line m that passes through two points P'(1 , 1/4) , Q(a , (1/4)a^2)
on y=(1/4)x^2 , check that when a approaches 1 , the line m coincides with the tangent to y=(1/4)x^2 at x=1. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"linepassestwopointsandtangent.gps" Change the value of the parameter a to see the movement of the line m when a approaches 1. <About the script> Click on [Init (a=5) and then click on [When 5≧a≧1]. Next , click on [When a<1]. |
| 410. Implicit function graph 1 |
| Investigate the graph of the implicit function. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"x^2+y^2+x+y=x^2y^2.gps" Observe the graph of the implicit function x^2+y^2+x+y=x^2y^2. |
| 411 Implicit function graph 2 |
| Investigate the graph of the implicit function. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"sinxy=x+y.gps" Observe the graph of the implicit function sinxy=x+y. |
| 412 Implicit function graph 3 |
| Investigate the graph of the implicit function. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"x^3+y^3=txy.gps" Change the value of the parameter t to observe the graph of the implicit function x^3+y^3=txy. <About the script> Click on [Init (t=0) and then click on [When t>0]. Next , click on [Init (t=0)] and then click on [When t<0]. |
| 413 Implicit function graph 4 |
| Investigate the graph of the implicit function. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"sinx+siny=sinnx+sinny.gps" Change the value of the parameter n to observe the graph of the implicit function sinx+siny=sinnx+sinny. <About the script> Click on [Init (n=2) and then click on [When n≧3]. Next , click on [Init (n=2)] and then click on [When n≦-2]. Furthermore , Click on [When n=1] , [When n=0] , and [When n=-1]. |
| 414 Implicit function graph 5 |
| Investigate the graph of the implicit function. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"x2+y^2=x^4+y^4+a.gps" Change the value of the parameter a to observe the graph of the implicit function x^2+y^2=x^4+y^4+a. <About the script> Click on [Init (a=0.3) and then click on [When 0.3≦a≦0.5]. Next , click on [Init (a=0.3) and then click on [When 0.3≧a≧0]. Furthermore , Click on [When a≦0] |
| 415 Tangent to the graph of the implicit function |
| Investigate the slope of the tangent to the graph of the implicit function. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"implicitfunctiongraphtangent.gps" Move the tangency point P(a,b) and see that the slope of the graph of the implicit function x^2+xy+y^2=28 is -(2a+b)/(a+2b) and the equation of the tangent is y-b=-(2a+b)/(a+2b)(x-a). |
 |
| Investigate the slope of the tangent to the graph of y=sinx at x=0. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"slopetangenty=sinxx=0.gps" Place the cursor at the origin and right-click → zoom → ×10 . See that the slope of the tangent to the graph of y=sinx at x=0 is 1. |
| 417 Definition of e |
| In the graph of y=(a^x-1)/x , find the value of a where the value of y
is closest to 1 when x=0. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"definitione1.gps" Change the value of the parameter a to see that when x=0 on the graph of y=(a^x-1)/x , the value of y approaches 1 , and the value of a becomes 2.718. |
| 418 Definition of e |
| Find the value of y when x=0 on a graph of y=(1+x)^(1/x). ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"definitione2.gps" Read the y-intercept of a graph of y=(1+x)^(1/x). Place the cursor on the y-intercept and right-click → zoom → Repeat ×10. See that the y-intercept is 2.718. |
| 419 The positional relationship between the graph of an exponential function and the graph of a logarithmic function |
| Investigate the positional relationship between the graph of y=a^x and
the graph of y=log(a,x). ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"exponentiallogarithmpositionalrelationship.gps" Change the value of the parameter a to see that graphs y=a^x and y=log(a,x) are symmetrical about the line y=x. <About the script> Click on [Init (a=2) and then click on [When a>2]. Next , click on [Init (a=2) and then click on [When 2>a>0]. |
| 420 Parametric graph 1 |
| Investigate the graph represented by the parameters x=2t and y=t+2. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"parametricgraph1gps" Change the value of the parameter t to observe what the graph looks like. <About the script> Click on [Init (t=1) and then click on [When t>1]. Next , click on [When t<1]. |
| 421 Parametric graph 2 |
| Investigate the graph represented by the parameters x=cost and y=sint. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"parametricgraph2gps" Change the value of the parameter t to observe what the graph looks like. <About the script> Click on [Init (t=0) and then click on [When 0≦t≦2π]. |
| 422 Parametric graph 3 |
| Investigate the graph represented by the parameters x=t^2 and y=t^2-2t. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"parametricgraph3gps" Change the value of the parameter t to observe what the graph looks like. <About the script> Click on [Init (t=2) and then click on [When t>2] Next , click on [When 2>t>-2]. |
| 423 Cycloid |
| Investigate the graph represented by the cycloid : parameters x=t-sint
and y=1-cost. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"cycloidgps" Change the value of the parameter t to observe what the graph looks like. <About the script> Click on [Init (t=0) and then click on [When 0≦t≦6π] |
| 424 Tangent of the cycloid |
| Find out that the equation of the tangent at t=a of the cycloid x=t-sint
, y=1-cost is expressed as y={sina/(1-cosa)}{x-(a-sina)}+1-cosa. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"cycloidtangentgps" Change the value of the parameter t to see that it is a tangent. <About the script> Click on [Init (t=0) and then click on [When 0<t<2π] |
| 425 Normal of the cycloid |
| Find out that the equation of the normal at t=a of the cycloid x=t-sint
, y=1-cost is expressed as y=-{(1-cosa)/sina}{x-(a-sina)}+1-cosa. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"cycloidnormalgps" Change the value of the parameter t to see that it is a normal. Also , observe the shapes created by the afterimage of normals. <About the script> Click on [Init (t=0) and then click on [When 0<t<10π]. |
| 426 Quadrature of pieces |
| The area enclosed by the parabola y=x^2 , the x-axis , and the line x=1
is calculated using the quadrature method. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"quadraturegps" <About the script> Click on [Perform Left End Partial Quadrature] and [Perform Right End Partial Quadrature] to check the approximate values. See how the difference between the approximate values obtained by [Performing the left end section quadrature] and [Performing the right end section quadrature] becomes smaller as the value of n increases. |
| 427 Area of a part enclosed by a curve and straight lines 1 |
| Find the approximate value of the area of a part enclosed by the curve y=1/(1+x^2) , x-axis , the straight lines x=-1 , and x=1 , using "GRAPES". ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"area1gps" → [Background / Tools] → [Display the definite integral value] → Enter 1 for the upper limit and -1 for the lower limit Check that the approximate value of the area of a part enclosed by the curve y=1/(1+x^2) , x-axis , the straight lines x=-1 , and x=1 is 1.57079633. (The integral value , 1.57079633 , is displayed.) |
| 428 Area of a part enclosed by a curve and straight lines 2 |
| Find the area of a part enclosed by the curve x=t^2 , y=-t^2+2t , and
x-axis . ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"area2gps" Change the value of the parameter t to check that t=0 when x=0 , and t=2 when x=4. <About the script> Click on [Init (t=0)] and then click on [When 0≦t≦2] ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"area2-1gps" → [Background/Tools] → [Display the definite integral value] → Enter 2 for the upper limit and 0 for the lower limit Check that the approximate value of the area of a part enclosed by the curve y=2x(-x^2+2x) and x-axis is 2.66666667. (The integral value , 2.66666667 , is displayed.) |
| 429 Area of a part enclosed by a curve and straight line 3 |
| Find the area of a part enclosed by the cycloid x=t-sint , y=1-cost ,
and x-axis . ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"area3gps" Change the value of the parameter t to check that t=0 when x=0 , and t=6.28 when x=2π. <About the script> Click on [Init (t=0)]. Next , click on [When t=2π]. Furthermore , click on [When t=0] and then click on [When 0<t<2π] ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"area3-1gps" → [Background/Tools] → [Display the definite integral value] → Enter 2π for the upper limit and 0 for the lower limit Check that the approximate value of the area of a part enclosed by the curve y=(1-cosx)^2 and x-axis is 9.42477796. (The integral value , 9.42477796 , is displayed.) |
| 430 Area of a part enclosed by a curve and straight lines 4 |
| Find the area of a part enclosed by a curve x=(cost)^2 , y=(sint)^2 ,
x-axis , and y-axis. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"area4gps" Change the value of the parameter t to check that t=1.57(π/2) when x=0 , and t=0 when x=1. <About the script> Click on [When t=π/2]. Next , click on [When t=0]. Furthermore , click on [When t=π/2] and then click on [When π/2>t>0] ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"area4-1gps" → [Background/Tools] → [Display the definite integral value] → Enter π/2 for the upper limit and 0 for the lower limit Check that the approximate value of the area of a part enclosed by a curve y=3{(sinx)^4}{(cosx)^2} and x-axis is 0.29452431. (The integral value , 0.29452431 , is displayed.) |
| 431 Volume of a soild of revolution (cone) |
| Consider how to find the volume of a solid (cone) created by rotating
a line around the x-axis. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"volumesolidrevolution(cone)gps" Change the value of the parameter t to notice that the cone is made up of many disks stacked on top of each-other. You can also click "Draw Rotating Body" to see that the cone is made up of many disks stacked on top of each-other. Use the definite integrals to find the volume of the solid (cone) created by rotating the line y=x/2 (0≦x≦4) around the x-axis. <About the script> Click on [Init] and then click on [Draw Rotating Body]. |
| 432 Volume of a soild of revolution (sphere) |
| Consider how to find the volume of a solid (sphere) created by rotating
a circle around the x-axis.. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"volumesolidrevolution(sphere)gps" Change the value of the parameter t to notice that the sphere is made up of many disks stacked on top of each-other. You can also click "Draw Rotating Body" to see that the sphere is made up of many disks stacked on top of each-other. Use the definite integrals to find the volume of the solid (sphere) created by rotating the circle x^2+y^2=a^2 (-a≦x≦a) around the x-axis. <About the script> Click on [Init] and then click on [Draw Rotating Body]. |
| 433 Volume of a soild of revolution (y=√x) |
| Find the volume of a solid created by rotating the curve y=√x (0≦x≦2)
around the x-axis. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"volumesolidrevolution(y=√x).gps" Change the value of the parameter t to notice that this solid is made up of many disks stacked on top of each other. You can also click "Draw Rotating Body" to see that this solid is made up of many disks stacked on top of each other. Use definite integrals to find the volume of the solid created by roptating the curve y=√x (0≦x≦2) around the x-axis <About the script> Click on [Init] and then click on [Draw Rotating Body]. |
| 434 Volume of a soild of revolution (y=√(x^2+1)) |
| Find the volume of a solid created by rotating the curve y=√(x^2+1) (-2≦x≦2)
around the x-axis. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"volumesolidrevolution(y=√(x2+1)).gps" Change the value of the parameter t to notice that this solid is made up of many disks stacked on top of each other. You can also click "Draw Rotating Body" to see that this solid is made up of many disks stacked on top of each other. Use definite integrals to find the volume of the solid created by roptating the curve y=√(x^2+1) (-2≦x≦2) around the x-axis <About the script> Click on [Init] and then click on [Draw Rotating Body] |
| 435 Length of the curve (cycloid) |
| Find the length of the cycloid x=t-sint , y=1-cost (0≦t≦2π). ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"curvelength1.gps" Change the value of the parameter b to see that the change in length of the cycloid as 0≦t≦b. Use the definite integrals to find the length of the cycloid x=t-sint , y=1-cost (0≦t≦2π). <About the script> Click on [When b=0]. Next ,click on [When b=2π]. Furthermore , click on [When b=0] and then click on [When 0≦b≦2π]. |
| 436 Length of the curve { x=(cost)^3 , y=(sint)^3 } |
| Find the length of the curve x=(cost)^3 , y=(sint)^3 (0≦t≦π/2). ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"curvelength2.gps" Change the value of the parameter b to see that the change in the length of the curve as 0≦t≦b. Use the definite integrals to find the length of the curve x=(cost)^3 , y=(sint)^3 (0≦t≦π/2). <About the script> Click on [When b=0]. Next , click on [When b=π/2]. Furthermore , click on [When b=0] and then click on [When 0≦b≦π/2]. |
| 437 Length of the curve x=t , y={e^t+e^(-t)}/2 |
| Find the length of the curve x=t , y={e^t+e^(-t)}/2 (0≦t≦1). ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[400] Differenntiation and Integration" → File name"curvelength3.gps" Change the value of the parameter b to see that the change in the length of the curve as 0≦t≦b. Use the definite integrals to find the length of the curve x=t , y={e^t+e^(-t)}/2 (0≦t≦1). <About the script> Click on [When b=0]. Next , click on [When b=1]. Furthermore , click on [When b=0] and then click on [When 0≦b≦1] |
| To table of contents |
| 501 Parametric curves |
| Consider the curve x=2cost+sint , y=cost+2sint expressed by parameters. ［File］ of data panel in "GRAPES" → Open → Folder "Sample data for considering how to use the GRAPES" → Folder "[500] Various curves" → File name"parametriccurve.gps" See that the equation 5x^2+5y^2-8xy-9=0 , which is obtained by eliminating t from x=2cost+sint , y=cost+2sint is equal to the curve x=2cost+sint , y=cost+2sint expressed in terms of the parameter t. Click the show/hide button to observe. <About the script> Click on [Init (t=0)] and then click on [When 0<t<2π]. |
| To table of contents |
| 601 [Problem 01] |
| There is a circle C: x^2+y^2+2x-6y=0 and a line L: 3x-y+k=0 (k is a constant)
on a cordinate plane with O as the origin. The circle C and the line L
intersect at two distinct points P and Q. (1) Find the range of possible values of k. (2) Find the value of k when △OPQ is a right-angled triangle. [Answer] (1) -4 < k <16 (2) k = 6 , 12 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "circlelineintersect.gps" ↓ Click Display (When not in presentation mode) ↓ In (1) Increase the parameter k to confirm that the circle and the line intersect at two different points within the value of -4<k<16. ↓ In (2) When k=6 , click on the T and U parts of T(0,6) and U(-2,0) to display points P and Q , and confirm that △OPQ is a right-angleed triangle. Next , when k=12 , click on the Q and S parts of Q(-2,6) and S(a , 3a+12) to display points P and Q , and confirm that △OPQ is a right-angled triangle. (Using a script , it is possible to automatically vary k and observe △OPQ.) <Remarks> By clicking the letters P , Q , R , , , to the left of each frame of a basic shape , you can show /hide the basic shape. |
| 602 [Problem 02] |
| On a coordinate plane, there is a circle C1whose diameter is defined by two points A(3,4) and B(5,8). Also , there is a circle C2: x^2+y^2-4ax-2ay+5a^2-5=0. Here , a is a constant. (1) Find the equation for C1. (2) Find the range of the values of a such that C1 and C2 intersect at two distinct points. [Answer] (1) (x-4)^2+(y-6)^2=5 (2) 8/5 < a <4 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "twocirclesintersect.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ In (2) Increase the parameter a to check that the two circles intersect at two different points in the value of 1.6<a<4. Verify that when the parameter a moves , the center of circle C2 moves on the straight line y=(1/2)x. (By using a script , it is possible to automatically change a and observe.) |
| 603 [Problem 03] |
| Find the range of values of r for which the two circles x^2+y^2=r^2 (r>0)
･･･① and x^2+y^2-8x-4y+15=0 ･･･② have a common point. [Answer]  <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "twocirclesintersection.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ In (2) Change the parameter k to check that the two circles have common points in the range of 2.2≦k≦6.7. (By using a script , it is possible to automatically change k and observe). |
| 604 [Problem 04] |
| Find the equation of the line that passes through the intersection of
the two circles x^2+y^2=5 and (x-1)^2+(y-2)^2=4. [Answer] x+2y-3=0 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "twocirclesintersectionline.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter k and verify that (x-1)^2+(y-2)^2-4+k(x^2+y^2-5)=0 is as follows ; When k≠-1 , it is a circle that passes through the intersections of the two circles. When k=-1 , it is a line that passes through the intersections of the two clrcles. (By using a script , it is possible to automatically change k and observe.) |
| 605 [Problem 05] |
| Find the equation of the line that passes through the intersection of
the two lines 2x+3y=7・・・ ① and 4x+11y=19・・ ・② and through the point (5,4). [Answer] x-y-1=0 <How to use GRAPES> Click on the file. ↓a Click Open. ↓ Double-click the file-name "twolinesintersectionline.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter k and verify that k(2x+3y-7)+4x+11y-19=0 is a straight line passing through the intersection of the two straight innes , regardless of the value of k.follows ; In paticular , confirm that when k=-3 , it passes through the point (5,4). (By using a script , it is possible to automatically change k and observe.) |
| 606 [Problem 06] |
| Find the value of a when the circle x^2+y2=1 and the parabola y=ax2-2
shsre only two distinct points. [Answer]  <How to use GRAPES> Click on the file. ↓a Click Open. ↓ Double-click the file-name "circleparabolaintersection.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to check the change in the number of common points. (By using a script , it is possible to automatically change a and observe.) |
| 607 [Problem 07] |
| About circle C1: (x-5)^2+(y-2)^2=16 and circle C2: (x-1)^2+(y+1)^2=a+2 (1)Find the range of values of a when each is outside the other. (2)Find the range of values of a when one is inside the other. [Answer] (1) -2<a<-1 (2) a>79 <How to use GRAPES> Click on the file. ↓a Click Open. ↓ Double-click the file-name "twocirclespositionalrelationship.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to see what the values of a when they are outside each other and when one is inside the other. (By using a script , it is possible to automatically change a and observe.) |
| 608 [Problem 08] |
| About parabola y=x^2+2(a-2)x-4a+5 When a is changed , the vertex of the parabola forms a single curve. Find the equation of this curve. [Answer] y=-x^2+4x-3 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "parabolaapextrajectory.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to confirm that the vertex moves on the parabola y=-x^2+4x-3. (By using a script , it is possible to automatically change a and observe.) |
| 609 [Problem 09] |
| Find the number of points that the line y=mx+1 and the circle x2+y2-2x+2y+1=0
share. [Answer] When m < -3/4 , 2 When m = -3/4 , 1 When m > -3/4 , 0 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "linecircleintersectionnumber.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter m to confirm that when m<-0.75 , there are two common points , when m=-0.75 , there is one common point , and when m>-0.75 , there are no common points. (By using a script , it is possible to automatically change m and observe.) |
| 610 [Problem 10] |
| When t varies with real values , what kind of figure does the intersection
P(x,y) of the two lines L: tx-y=t and M(: x-ty=2t+1 form. ? [Answer] circle (x-1)^2+(y-1)^2=1 However , the point (1 , 2) is excluded. <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "twolinesintersectionlocus.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter t to confirm that the intersection of the two lines moves along the circle (x-1)^2+(y-1)^2=1 , except for the point (1 , 2). (By using a script , it is possible to automatically change t and observe.) |
| 611 [Problem 11] |
| When t varies with real values , about two lines L: x+t(y-3)=0 and M(:
tx-(y+3)=0. (1) Show that the line L passes through the fixed point regardless of the value of t. (2) When t moves through the entire real numbers , what kind of figure does the intersection of lines L and M form ? [Answer] circle x^2+y^2=9 However , the point (0 , 3) is excluded. <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "twolinesintersectionlocus2.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter t to confirm that the intersection of the two lines moves along the circle x^2+y^2=9 , except for the point (0 , 3). (By using a script , it is possible to automatically change t and observe.) |
| 612 [Problem 12] |
| Suppose that the line y=mx intersects with the parabola y=x^2+1 at two distinct points P and Q. When m changes while satisfying this condition , find the range of possible values for m. Also , at this time , find the locus of the miidpoint M of the line segment PO. [Answer] m < -2 , m > 2 parabola y=2x^2 (x < -1 , x > 1) <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "lineparabolamidpointlocus.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Change the parameter m to confirm that the midpoint M moves along a parabola y=2x^2 (x<-1 , x>1). (By using a script , it is possible to automatically change m and observe.) |
| 613 [Problem 13] |
| Find the value of the constant k when the two lines: 2x+5y-3=0 ･･･① and
5x+ky-2=0 ･･･② are parallel and when they are parpendicular. [Answer] When parallel , k=25/2 (12.5) When parpendicular , k=-2 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "parallelparpendicularcondition.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter k to confirm that the two lines are parallel when k=12.5 , and they are parpendicular when k=-2. (By using a script , it is possible to automatically change k and observe.) |
| 614 [Problem 14] |
| When the three points A(a , -2) , B(3 , 2) , C(-1 , 4) are on the same
line , find the value of the constant a. [Answer] a=11 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "3pointsline.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to confirm that the three points are on the same line. (By using a script , it is possible to automatically change a and observe.) |
| 615 [Problem 15] |
| Find the value of the fixed point a when the three lines 2x+y+3=0 , x-y+6=0
, and ax+y+24=0 intersect at one point. [Answer] a=9 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "2linesintersectionline.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to confirm that when a=9 , three linrs intersect at one point (By using a script , it is possible to automatically change a and observe.) |
| 616 [Problem 16] |
| Find the equation of a circle that has its center on a line y=-4x+5 and
is tangent to bboth the x-axis and the y-axis.. [Answer] Since a=1 , 5/3(1.66・・・・) , (x-1)^2 + (y-1)^2 = 1 (x-5/3)^2 + (y+5/3)^2 = 25/9 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "bothxaxisyaxistangentcircle.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to confirm that when a=1 and a=1.66 , the circle is tangent to the x-axis and the y-axis. (By using a script , it is possible to automatically change a and observe.) |
| 617 [Problem 17] |
| Find the equation of the line that passes through point(3,1) and is tangent
to the circle x^2+y^2=2 , and the coordinates of the tangency point. [Answer] Since m=1 , -1/7 (-0.1428・・・・) , (-1 , -3) (1/5 , 49/35) <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "fixedpointcircletangentline.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter m to confirm that it is tangent to the circle when m=1 and m=-1/7 (-0.14). (By using a script , it is possible to automatically change m and observe.) |
| 618 [Problem 18] |
| Find the minimum value of y=x^2-2x (a≦x≦a+1). [Answer] When a < 0 , minimum value a^2-1 (x=a+1) When 0≦ａ≦1 , minimum value -1 (x=1) When 1 < a . minimum value a^2-2a (x=a) <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "cubicinequalityproof.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to confirm that the minimum value is reached at x=a+1 when a<0 , it is reached at x=1 when 0≦ａ≦1, and it is reached at x=a when 1<a. (By using a script , it is possible to automatically change a and observe.) |
| 619 [Problem 19] |
| Find the range of values of the constant a such that the cubic inequality
x^3-3a^2x+2>0 holds for all x≧0. [Answer] -1<a <1 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "quadraticfunctionminimum.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to confirm that when -1<a<1 for all x≧0 , the cubic inequality x^3-3a^2x+2>0 holds. (By using a script , it is possible to automatically change a and observe.) |
| 620 [Problem 20] |
| Let a be a constant. Find the number of solutions to the equation x^2｜x-3｜=a. [Answer] When a>4 , 2 When a=4 , 3 When 0<a<4 , 4 When a=0 , 2 When a<0 , 0 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "absvaluegraph.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to confirm that the number of intersections between the graph of y=x^2｜x-3｜and the graph of y=a is 2 when a>4 , 3 when a=4 , 4 when 0<a<4 , 2 when a=0 , and 0 when a<0. (By using a script , it is possible to automatically change a and observe.) |
| 621 [Problem 21] Tohoku University |
| Let D be the region on the coordinate plane represented by the inequality
2y>x+1+3｜x-1｜. For real number a , define parabora C as y=x^2-2ax+a^2+a+2. In this case , find the range of a such that all points on C are points on D. [Answer] 1/8≦a≦2 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "touhoku.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter a to confirm that all points on C are points in D when 1/8≦a≦2. (By using a script , it is possible to automatically change a and observe.) |
| 622 [Problem 22] Kansei Gakuin University |
| Let r >0. Consider the number of common points between the parabola y=x^2-1 and the circle x^2+y^2=r^2 on the xy plane. Find the range ofvalues of r that maximizes the number of common points. Also , find the value of r when the number of common points is odd. ［Answer］  ｒ＝１ <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "kansei.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter r to confirm that when √3/2<r<1 , the number of common points becomes four. And comfirm that when r=1 , the number of common points becomes three. (By using a script , it is possible to automatically change r and observe.) |
| 623 [Problem 23] Gifu Shotoku Gakuen University |
| About the parabola y=(x-p)^2-3 Find the range of values of real number p such that this parabola intersects with the triangle with three vertices (0 , 0) , (0 , -2) , (2 , 0). ［Answer］   <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "syoutoku.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter p to confirm that when -√3≦p≦2-√3 , 1≦p≦2+√3 , the parabola intersects with the triangle. (By using a script , it is possible to automatically change p and observe.) |
| 624 [Problem 24] Soka University |
| If a circle passing through the two intersection points of two circles
x^2+y^2=2 , (x-1)^2+(y+1)^2=1 is tangent to the line y=x , find the center
and radius of that circle. ［Answer］ center (2/3 , -2/3) radius 2√2 / 3 <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "souka.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter k to confirm that when k=0.5 , the circle passing through the two intersection points of the two circles is tangent to the line y=x. (By using a script , it is possible to automatically change k and observe.) |
| 625 [Problem 25] Fukuoka University |
When k is a constant satisfying k≠－√2 ,  ① represents a circle that passes through the fixed point A regardless of the value of k. In this case , find the coordinates of fixed point A. Also , if circle ① and circle (x-1)^2+(y-1)^2==9 have only one common point and k>0 , find the value of k. ［Answer］   <How to use GRAPES> Click on the file. ↓ Click Open. ↓ Double-click the file-name "fukuoka.gps" ↓ Click View (When not in Presentation Mode) ↓ Click on Presentation Mode (When not in Presentation Mode) ↓ Vary the parameter k to confirm that circle ① passes through the fixed point (√2/2 , -√2/2) , regardless of the value of k. Also , confirm that when k=√2/2 , verify that the two circles have only one common point and are tangent. (By using a script , it is possible to automatically change k and observe.) |
| ※ The above sample data file "～.gps"can be downloaded below. |
| To table of contents |
| When you download the sample data below that shows how to use "GRAPES" , a comoressed file "sample_grapes.lzh"will be downloaded. When you unzip the downloaded compressed file "sample_grapes.lzh" , the folder "sample data for considering how to use GRAPES" will be created. |
| When you download the sample data 2 below for using "GRAPES" to solve exam questions , a compressed file "sample_grapes2.lzh" will be downloaded. When you unzip the download compressed file "samplegrapes2.lzh" , the folder "sample data 2 for using "GRAPES" to solve exam questions" will be created. |
| When you download the sample data 3 below for using "GRAPES" to solve exam questions , a compressed file "sample_grapes3.lzh" will be downloaded. When you unzip the download compressed file "samplegrapes3.lzh" , the folder "sample data 3 for using "GRAPES" to solve exam questions" will be created. |
| GRAPES Sample Data 1 | Sample data for considering how to use "GRAPES" |
| GRAPES Sample Data 2 | Sample data 2 for using "GRAPES" to solve exam questions |
| GRAPES Sample Data 3 | Sample data 3 for using "GRAPES" to solve exam questions |
| Function graph software "GRAPES" | Mathematical function graph software "GRAPES" |
| Click on the Mathematical function graph software "GRAPES" above
to open the GRAPES official website. When you download the function graph software "GRAPES" from the the official GRAPES website , a compressed file "grps662.exe" is downloaded. When you double-click the downloaded compressed file "grps662.exe" , it will be self-decompressed and the folder "GRAPES" will be created. Double-click "grapes.exe" in the folder to launch "GRAPES". However , the 662 in the downloaded compressed file "grps662.exe" represents the version. (As of August 3.2008) |
| To table of contents |
| Inside the folder "Sample data for considering how to use "GRAPES" , are five folders "[100] Quadratic functions and Liner functions","[200] Trigonometric functions , Exponential functions , and Logarithmic functions", "[300] Diagrams and Equations" , "[400] Differentiation and Integration", "[500] Various curves" Open (load) the file "～.gps" in each folder in "GRAPES". |
| The folder "Sample data for using "GRAPES" to solve exam questions contains 20 files with the name "～.gps. Open (load) these in GRAPES. |