Exercises

1.

Use Implicit under the Plot 2D submenu to plot the conic sections x² + y² = 1, x² - y² = 1, and x + y² = 0 all on the same coordinate axes. BITMAPSETAnswer0.2171in0.2006in0ina1

2.

Use Implicit under the Plot 2D submenu to plot the conic sections (x - 1)² + (y + 2)² = 1, (x - 1)² - (y + 2)² = 1, and (x - 1) + (y + 2)² = 0 on one pair of coordinate axes. With the hand symbol visible over the view, translate the view so that the curves match the curves in Exercise 1. In which direction did the axes move?BITMAPSETAnswer0.2171in0.2006in0ina2

3.

Plot x² + y² = 4 and x² - y² = 1 together. How many intersection points are there? Zoom in on the one in the first quadrant to estimate where the curves cross each other. Verify your estimate by typing the formulas into a matrix and choosing Numeric from the Solve submenu.BITMAPSETAnswer0.2171in0.2006in0ina3

4.

Plot the astroid x^2/3 + y^2/3 = 1.BITMAPSETAnswer0.2171in0.2006in0ina4

5.

Plot the folium of Descartes x³ + y³ = 6xy. BITMAPSETAnswer0.2171in0.2006in0ina5

6.

Plot the surface z = sin xy, with -4≤x≤4 and -4≤y≤4. Compare the location of the ridges with the implicit plot of the three curves xy = ${\frac{{\pi }}{{2}}}$, xy = ${\frac{{3\pi }}{{2}}}$, and xy = ${\frac{{% 5\pi }}{{2}}}$.BITMAPSETAnswer0.2171in0.2006in0ina6

7.

A standard calculus problem involves finding the intersection of two right circular cylinders of radius 1. View this problem by choosing Rectangular from the Plot 3D submenu to plot the two parametric surfaces $\left[\vphantom{ s,\cos t,\sin t}\right.$s, cos t, sin t$\left.\vphantom{ s,\cos t,\sin t}\right]$ and $\left[\vphantom{ \cos t,s,\sin t}\right.$cos t, s, sin t$\left.\vphantom{ \cos t,s,\sin t}\right]$.BITMAPSETAnswer0.2171in0.2006in0ina7

8.

Do the two space curves

$$\left[\vphantom{ (2+\sin t)10\cos t,\ (2+\cos t)10\sin t,3\sin 3t}\right.$$(2 + sin t)10 cos t, (2 + cos t)10 sin t, 3 sin 3t$$\left.\vphantom{ (2+\sin t)10\cos t,\ (2+\cos t)10\sin t,3\sin 3t}\right]$$

and 

$$\left[\vphantom{ 20\cos t,20\sin t,-3\sin 3t}\right.$$20 cos t, 20 sin t, -3 sin 3t$$\left.\vphantom{ 20\cos t,20\sin t,-3\sin 3t}\right]$$

intersect? Use Tube from the Plot 3D submenu and rotate the curves to find out.BITMAPSETAnswer0.2171in0.2006in0ina8

9.

View the intersection of the sphere x² + y² + z² = 1 and the plane x + y + z = ${\frac{{1}}{{2}}}$ by solving for z and choosing Rectangular from the Plot 3D submenu. Verify that the points of intersection lie on an ellipse (it is actually a circle) by solving x + y + z = ${\frac{{1}}{{2}}}$ for z, substituting this value into the equation x² + y² + z² = 1, and calculating the discriminantDM3-6.tex#Discriminant of the resulting equation.BITMAPSETAnswer0.2171in0.2006in0ina9

10.

Explore the meaning of contours by plotting the surface z = xy. Choose Patch & Contour as a Style in the Rectangular dialog box. Rotate the surface until only the top face of the cube is visible, and interpret the meaning of the curves that you see. Rotate the cube until the top face just disappears, and interpret the meaning of the contours that appear.BITMAPSETAnswer0.2171in0.2006in0ina10