MAT251 Multivariable Calculus III Test 1 Name ______________________

Prof. PORTER FALL 2006

 

This test is to be taken in the testing center with unlimited time. Students are allowed to use any calculator- including the TI-89, but must always document the steps used in the calculator to produce the answer. Credit is not given for unsubstantiated work.

 

 

1.       What is Multivariable Calculus?

 

 

 

Name five major areas of study that might require Multivariable calculus.

 

 

 

 

 

 

2.       Given that the plants Mars(M), Earth (E) and the Venus (V) at a specific time are located at the following coordinates:

 

Mars is at (1,0,2), Earth is at (-2,0,1), and Venus is at (-1,2,0):

 

 

1.       Find the angle that an observer on Earth must rotate to turn from Mars to Venus.

 

 

 

Answer____________

 

2.       Find the parametric equation of the line starting at Earth and heading toward Venus.

 

 

 

 

 

 

Answer____________

 

1.       Given the above locations for Mars, Earth, and Venus:


a) Find the equation of the plane containing the three planets

 

 

 

 

 

 

Answer____________

 

b) They say that almost all the planets are on the same plane. How would you show this is true if you knew the coordinates of the other planets? Would a planet at (1,2,3) be on the plane?

 

 

 

 

 

 

2.       Suppose a rival company finds the equation of the plane to be given below:

 

3x + 2y + 6z = 12

a) Find the intercepts and graph the plane. Graph the intercepts and the plane in the first octant.

 

 

 

 

 

b) What is the difference (or distance) between the above plane of your rival company and the parallel plane that your company found:

 

3x + 2y + 6z = 12.12

 

 

 

 

 

 

 

3.       You happen upon top secret plans for a new space station by your rivals. You must correctly identify the surface shape described by the following equations as either an ellipsoid, hyperboloid, elliptic cone, elliptic paraboloid, or hyperbolic paraboloid.

 

 

a) Answer____________

 

b) Answer____________

 

c) Answer____________

 

 

4.       Your allies have proposed a space ship that is so huge you can only view one level at a time. Draw the shape of the spaceship at levels z = 0, 1, 2 on the x-y plane if the shape of the ship is described by

 

 

 

 

5.       Change the planet coordinate located at (1,2,3) from rectangular coordinates to spherical coordinates.

 

 

 

Spherical Answer____________

 

 

Change the planet located at the point ( 2, p/4, 5) in cylindrical coordinates to rectangular coordinates.

 

 

 

Rectangular Answer____________

 

 

6.       Change the equation from rectangular to cylindrical coordinates.

 

 

 

 

Answer___________

Describe how the spherical function ρ = p/2 would be seen in 3D.

 

 

 

Answer___________

 

 

 

7.       Represent the curve formed by the intersection of the surfaces and y = x - 7 by a vector valued function using the parameter x = t.

 

 

Answer___________

 

 

 

What is the shape of the curve?

 

 

 

Answer___________

 

 

8.       Find r(t) under the conditions that r(t) = and r(0) =

 

 

 

 

 

 

 

Answer___________

 

 

9.       What is the length of the curve r(t) = <2t+1,0,t(t+1)> over the interval [0,3]?

 

 

 

 

 

 

Answer___________

 

10.    Given the position vector r(t) = <t, 1 - 2t, sin t>

 

a)       Find the acceleration at t = 0

 

 

 

 

Answer___________

 

b)       Find the speed at t = 0

 

 

 

 

Answer___________

 

 

1.       Does a spaceship leaving Earth at (-2,0,1) and heading towards Venus (-1,2,0) pass thru the trail of a comet that travels along the path x = 0, y = 2t, and z = t - 3 ? Explain.