Calculus 3

Exam File

Fall 2002

 

Exam #1

 

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

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

3.)        a unit vector in the same direction as 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.)        x2 + y2 + z2 = 1

10.)     x2 + y2 - z2 = 1

For #11 - 12, given the points A(1, 0, 2), B(4, 5, 0), C(0, -3, 5), D(-2, 0, 7) and E(0, -1, 7) do the following:

11.)     How far is the point A from the plane in problem #2.

12.)     Find the distance between the line joining B and D and the line joining C and E.

13.)     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.)        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

 

 

Exam #2

1.)        For values of k = -1, 0, 1 draw :

            a.)        level curves for f(x, y) = x + y.

            b.)        level surfaces for f(x, y, z) = x2 - y2 - z2.

2.)        Using the definition of limit, prove that lim(x,y)®(0,0) (4x2 + 4y2) = 0.

3.)        Determine if the following limits exist.

            a.)        lim(x,y)®(1,3) (x2 - y2)                

            b.)       

4.)        Find fx, fy, fxx, fyx and fyy if f(x,y) = xy - sin(xy).

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

 

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

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

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

7.)        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.

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

9.)        Let r(t) = 2 sin(3t) i + 2 cos(3t) j + 4t k.

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

            b.)        Find the integral of r(t) from t = 0 to t = 2 p.

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

10.)     Consider a function, f(x, y, z).  Let k and m be real numbers.  Consider the level surfaces f(x, y, z) = k and f(x, y, z) = m.  Discuss what happens if there is a point that is on both of the two level surfaces.

 

Exam #3

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

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

3.)        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.

4.)        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).

5.)        Find the area bounded by r = 3 cos 2Θ.

6.)        For the following integral, switch the order of integration.  DO NOT EVALUATE THE RESULTING INTEGRAL!!

7.)        Evaluate .

8.)        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).

 

Exam #4                                           

1.)        Find the area of the surface 3x + 2y + z = 6 that is in the first octant.  Exact answer only.

2.)        Use a triple integral to find the volume of the solid bounded by x = 0, y = 0, z = 0, and 3x + 2y + z = 6.  Exact answer only.

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

4.)        Find the centroid of the region in the plane bounded by y = x2 and y = 2x + 3, if the density is constant.  Exact answers only.

5.)        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 (x, y, z) = (x + y)2.  Exact answer only.

6.)        Find the volume of the solid in the first octant that is bounded by the cylinder      x2 + y2 = 2y, the cone z = and the xy-plane.  Exact answer only.

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

8.)        Evaluate the line integral  along the portion of the parabola y = x2 from the point (0, 0) to the point (2, 4).  Exact answer only.

 

Final Exam

 

1.)        Graph F(t) = (cos t)i + (sin2 t)j, 0 < t < 2p.

[image]

 

 

2.)        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.

3.)        Consider f(x, y, z) = xy cos(yz).  Find fx, fxy, fxx.

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

5.)        Set up (but do not evaluate ) the integral necessary to find the area bounded by  y = 1 - x2 and y = x. 

6.)        Set up (but do not evaluate ) the integral necessary to find the volume of the solid in the first octant that is bounded by the xy-plane, the cone z = and the cylinder x2 + y2 = 2y.

7.)        Consider the following vectors.

                        u = 2i - 3j + k, v = i + 4j - k, w = i - j + 3k

Find the following:

a.)        2u - 3v + 5w

b.)        two unit vectors parallel to w

c.)        u ∙ w

d.)        u x v

8.)        Consider the following vector valued function.

f(x, y, z) = (x + y2) sin (xyzp)

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

b.)        Find the gradient of f.

9.)        Find all critical points on the graph of f(x, y) = 8x3 - 24xy + y3, and use the second partials test to classify each point as a relative extremum or a saddle point.

10.)     Find the centroid of the plane region bounded by y = x, y = -x and y = 2x + 3, if the density is constant.

11.)     Graph x2 + y2 - z2 = 1.

12.)     Graph x2 - y2 - z2 = 0.