Calculus 3

Exam File

Fall 2004

Exam 1

For #1 - 8, let u = <3, 1, -2>, v = <2, -1, -3> and w = < -2, 0, 4>. Find the following, if possible. If it is not possible, state why it is not possible.

 

1.) 2u - 3v

 

2.) an equation for the plane through the point (1, 0, -1) with w as its normal vector

 

3.) two unit vectors parallel to v

 

4.) u (v w)

 

5.) a non-zero vector that is orthogonal to both v and w

 

6.) a set of parametric equations for the line through (1, 2, 3) with v as its direction vector

 

7.) symmetric form for the equations for the line on the point (3, 0, -1) with u as its direction vector

 

8.) the angle between v and w

 

In #9 - 10, given the equation, identify and graph the quadric surface.

 

9.) 2x2 + 3y2 + 4z2 = 1

 

10.) z2 - y2 = 1

 

11.) How far is the point (5, -1, 3) from the plane in problem #2?

 

12.) For each of the graphs below, select an equation from the list of equations, that could give the graph. Also choose the name of the surface from the list of names. Use the appropriate letter to indicate your choice.

 

 

equation________ name________

 

equation________ name________

 

equation________ name________

 

 

z

 

y

x

 

equation________ name________

equations

a.) x2 + y2 + z2 = 1

b.) z2 - x2 - y2 = 1

c.) y2 - x2 = z

d.) x2 - y2 = z

e.) y2 = z

f.) 2x + 3y + z = 6

g.) x2 + y2 = z

h.) x2 + y2 - z2 = 1

i.) 25y2 + z2 = 25

j.) x2 + y2 - z2 = 0

names

A.) paraboloid

B.) cylinder

C.) hyperboloid of two sheets

D.) ellipsoid

E.) hyperboloid of one sheet

F.) cone

G.) hyperbolic paraboloid

H.) plane

I.) thingamajig

J.) whatchamacallit

 

 

Exam #2

 

1.) Consider f(x, y) = x - y. For values of k = -1, 1 draw level curves for f(x,y).

 

2.) Consider f(x,y,z) = x2 - y2 - z2. For values of k = -1, 1, draw the level surfaces for f(x, y, z).

 

3.) Consider a function, f(x, y). Suppose, for real numbers k and m, the point (a. b) is on the level curves f(x, y) = k and f(x, y) = m. Explain why it must be true that k = m.

 

4.) Let r(t) = 3t2 i + (2t3 + 6t) j. Find the time(s), when the velocity and acceleration vectors are:

 

a.) orthogonal. b.) parallel.

 

5.) Let r(t) = sin2 (t) i + cos (t) j + 4t k.

a.) Using your calculator to do the work, find the arc length of the curve from t = 0 to t = 2p.

 

b.) Find the integral of r(t) from t = 0 to t = p.

 

c.) Graph the path of the particle from t = 0 to t = p.

 

6.) A projectile is fired from ground level with an initial speed of 1500 feet/sec at an angle of elevation of 60.

 

a.) How far had it gone when it hit the ground?

 

b.) Find the speed at the time it hits the ground.

 

7.) If y = sin px find an equation for the circle of curvature at the point (3/2, -1).

 

8.) The equation of motion of a particle is given by r(t) = sin 2t i + cos 2t j + t k. For t = p/2, find the following:

 

a.) principal unit tangent vector

b.) principal unit normal vector

c.) tangential component of acceleration

d.) normal component of acceleration.

 

9.) Determine if the following limit exists.

 

lim(x,y)(1,3) (3x2 + y)

 

10.) Determine if the following limit exists.

 

11.) Let f(x,y) be defined as follows:

 

Is f continuous at (0, 0)?

 

Exam 3

 

1.) Find fxx, fyx and fyy if f(x,y) = x2y3 - ln(xy).

 

2.) Find the volume of the solid in the first octant bounded by the cylinder x2 + y2 = 1 and 3x + 2y + z = 6.

 

3.) Suppose w = u2 cos v, u = x + 2y, and v = ln x. Use the chain rule to find w/x.

 

4.) For the function f(x, y, z) = (x + y2) sin (xyzp), find the direction of greatest increase at the point (2, 3, 4). Also find what that increase is.

 

5.) Let f(x, y) = x2 - 7xy + y2. Let P be the point (4, -2). Find the directional derivative at P in the direction of a = 3i + 4j.

 

6.) Find all critical points on the graph of f(x, y) = 2x2 - 2xy - y3, and use the second partials test to classify each point as a relative extremum or a saddle point.

 

7.) Let f(x, y) = 5 + 4x - 2x2 + 3y - y2. Find the absolute extrema of f on the region bounded by the lines y = 2, y = x and y = -x.

 

8.) For the following integral, switch the order of integration and then evaluate the resulting integral.

 

Exam #4

1.) Use a triple integral to find the volume of the solid in the first octant bounded by the coordinate planes and 3x + 2y + z = 6.

 

2.) Graph the vector field F(x, y) = yi xj. Also, calculate the equation for the family of flow lines.

 

3.) Set up all of the integrals (BUT DO NOT EVALUATE THEM) necessary for finding the centroid of the region in the plane bounded by y = x2 and y = 2x + 3, if the density is constant.

 

4.) Find the volume of the solid that is inside the sphere x2 + y2 + z2 = 9 and outside the cylinder x2 + y2 = 1.

 

5.) Evaluate the line integral along the line segment from the point (0, 0) to the point (2, 4). Exact answer only.

 

6.) Use a transformation of axes and one double integral to find the area of the region bounded by x + y = 1, x + y = 7, x - 2y = 0 and x - 2y = 6

 

7.) Find the volume of the solid above the cone z = and below the sphere x2 + y2 + z2 = 8. Exact answer only. (Hint : Spherical coordinates might be a VERY good idea to consider.)

 

8.) Set up all of the integrals (BUT DO NOT EVALUATE THEM) necessary for finding the z-coordinate of the centroid of the solid bounded by z = 1 - x2 - y2 and and z = 0 if the density is given by r(x, y, z) = (x + y)2.

 

Final Exam

For #1 - 6, let u = <3, 1, -2>, v = <2, -1, -3> and w = < -2, 0, 4>. Find the following, if possible. If it is not possible, state why it is not possible.

 

1.) 2u - 3v

 

2.) an equation for the plane through the point (1, 0, -1) with w as its normal vector

 

3.) two unit vectors parallel to v

 

4.) a non-zero vector that is orthogonal to both v and w

 

5.) a set of parametric equations for the line through (1, 2, 3) with v as its direction vector

 

6.) the angle between v and w

 

7.) Use a triple integral to find the volume of the solid in the first octant bounded by the coordinate planes and 3x + 2y + z = 6.

 

8.) Set up all of the integrals (BUT DO NOT EVALUATE THEM) necessary for finding the centroid of the region in the plane bounded by y = x2 and y = 2x + 3, if the density is constant.

 

9.) Find the volume of the solid that is inside the sphere x2 + y2 + z2 = 9 and outside the cylinder x2 + y2 = 1.

 

10.) For the following integral, switch the order of integration and then evaluate the resulting integral.

 

11.) Find the volume of the solid above the cone z = and below the sphere x2 + y2 + z2 = 8. Exact answer only.

 

12.) Find the z-coordinate of the centroid of the solid bounded by z = 1 - x2 - y2 and and z = 0 if the density is given by r(x, y, z) = x2 + y2.

 

13.) Given the equation, identify and graph the quadric surface. z2 - y2 = 1

 

14.) Let r(t) = t2 i + (t3 + 4t) j with t > 0. Find the time(s), when the velocity and acceleration vectors are: a.) orthogonal. b.) parallel.

 

15.) A projectile is fired from ground level with an initial speed of 500 feet/sec at an angle of elevation of 30.

 

a.) How far had it gone when it hit the ground?

b.) Find the speed at the time it hits the ground.

 

16.) Determine if the following limit exists.

 

17.) Find fxx, fyx and fyy if f(x,y) = x2y3 - ln(xy).

 

18.) For the function f(x, y, z) = (x + y2) sin (xyzp), find the direction of greatest increase at the point (2, 3, 4). Also find what that increase is.

 

19.) Find all critical points on the graph of f(x, y) = 2x2 - 2xy - y3, and use the second partials test to classify each point as a relative extremum or a saddle point.

 

20.) Let f(x, y) = 5 + 4x - 2x2 + 3y - y2. Find the absolute extrema of f on the region bounded by the lines y = 2, y = x and y = -x.