Calculus 3

Exam File

Fall 2003

For #1 - 8, let u = 3i + 2j + k, v = 2i - j - 3k and w = 2i - 4k. Find the following, if possible. If it is not possible, state why it is not possible.

1.) 2u - 3v + 5w

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.) x² + z² - y² = 1

11.) How far is the point (1, 0, 2) 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________	equation________ name________
equations a.) x^2 + y^2 + z^2 = 1 b.) z^2 - x^2 - y^2 = 1 c.) y^2 - x^2 = z d.) x^2 - y^2 = z e.) y^2 = z f.) 2x + 3y + z = 6 g.) x^2 + y^2 = z h.) x^2 + y^2 - z^2 = 1 i.) 25y^2 + z^2 = 25 j.) x^2 + y^2 - z^2 = 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

Exam #2

1.) Consider f(x, y) = xy.

a.) For values of k = -1, 0, 1 draw level curves for f(x,y).

b.) Find f(1, 3).

2.) Consider f(x,y,z) = x² + y² - z².

a.) For values of k = -1, 0, 1 draw level surfaces for f(x, y, z).

b.) Find f(1, 2, 3)

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) = t² i + (3t³ + 4t) j. Find the time(s), when the velocity and acceleration vectors are orthogonal.

5.) Let r(t) = sin(t) i + cos²(t) j + 4 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 = 2p.

c.) Graph the path of the particle from t = 0 to t = 2p.

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 velocity at time t.

c.) Find the maximum altitude.

7.) Find the curvature and the equation of the circle of curvature for the curve given parametrically by x = t and y = 1/t at the point (1, 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.

Exam #3

1.) Determine if the following limits exist.

lim_(x,y)®(1,3) (x² - y²)

2.) Determine if the following limits exist.

3.) Find f_x, f_y, f_xx, f_yx and f_yy if f(x,y) = x²y³ - ln(xy).

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

a.) Is f continuous at (0, 0)?

b.) Is f differentiable at (0, 0)?

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

6.) Given that xz - xy = yz + 3, where z = f(x, y), find z_y.

7.) Let f(x, y) = x² - 7xy + y². Let P be the point (1, -1). Find the directional derivative at P in the direction of a = 3i - 4j.

8.) Let f(x, y) = 4x² - xy + y².

a.) Find any local extrema.

b.) Find the absolute extrema of f on the square with corners (1, 0), (0, 0), (0, 1) and (1, 1).

9.) Evaluate .

10.) Set up but DO NOT EVALUATE, the integral(s) necessary for finding the area bounded by the triangle with vertices (-1, 4), (3, 6) and (5, -2).

11.) Consider the following function.

f(x, y, z) = (x + y²) sin(xz)

a.) Find f(1, 3, -1).

b.) Find the gradient of f.

Exam #4

1.) Evaluate the line integral along the portion of the line y = 2x +3 from the point (0, 3) to the point (2, 7). Exact answer only.

2.) Set up (but do not evaluate) the integral necessary to find the area of the surface 3x + 2y + z = 6 that is in the first octant.

3.) Set up (but do not evaluate) the integral necessary to find the volume of the solid in the first octant bounded by the coordinate planes and 3x + 2y + z = 6.

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

5.) Use the Jacobian to derive the evaluation formula for triple integrals in spherical coordinates:

6.) Set up the integrals necessary to find the z-coordinate of the center of mass of the solid bounded by z = 1 - x² - y² and z = 0 if the density is given by r(x, y, z) = (x + y)².

7.) a.) Graph the vector field F(x, y) = yi – xj.

b.) Find an equation for the family of flow lines.

c.) Draw some flow lines on the graph from a.).

Final Exam

For #1 - 5, let u = 3i + 2j + k, v = 2i - j - 3k and w = 2i - 4k. Find the following, if possible. If it is not possible, state why it is not possible.

1.) 2u - 3v + 5w

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.) 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________	equation________ name________
equations a.) x^2 + y^2 + z^2 = 1 b.) z^2 - x^2 - y^2 = 1 c.) y^2 - x^2 = z d.) x^2 - y^2 = z e.) y^2 = z f.) 2x + 3y + z = 6 g.) x^2 + y^2 = z h.) x^2 + y^2 - z^2 = 1 i.) 25y^2 + z^2 = 25 j.) x^2 + y^2 - z^2 = 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.) thing-a-ma-jig J.) what-cha-ma-call-it

7.) Consider f(x, y) = x²y.

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

b.) Find f(1, 3).

8.) Let r(t) = t² i + (3t² - 4t) j. Find the time(s), when the velocity and acceleration vectors are orthogonal.

9.) 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 velocity at time t.

c.) Find the maximum altitude.

10.) Determine if the following limit exists.

lim_(x,y)®(1,3) (x² - y²)

11.) Determine if the following limits exist.

12.) Find f_x, f_y, and f_xy if f(x,y) = x²y³ - ln(xy).

13.) Given that sin(xz) - xy = yz + 3, where z = f(x, y), find z_y.

14.) Let f(x, y) = 4x² - xy + y².

a.) Find any local extrema.

b.) Find the absolute extrema of f on the square with corners (1, 0), (0, 0), (0, 1) and (1, 1).

15.) Evaluate .

16.) Set up but DO NOT EVALUATE, the integral(s) necessary for finding the area bounded by the triangle with vertices (-1, 4), (3, 6) and (5, -2).

17.) Evaluate the line integral along the portion of the line y = x² + 2 from the point (0, 2) to the point (2, 6).

18.) Set up (but do not evaluate) the integral necessary to find the volume of the solid in the first octant bounded by the coordinate planes and 3x + 2y + z = 6.

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

20.) Find the z-coordinate of the center of mass of the solid bounded by z = 1 - x² - y² and z = 0 if the density is given by d(x, y, z) = x² + y².