Mathematics 54 Exercise Set

Part I

I. Evaluate the following integrals.

1.∫

sin6 x

cos4 xdx

2.∫ tan−1 1√

2x

x3dx

3.∫

ln (sin θ) cot θ csc2 θ dθ

4.∫ 1

0

dx

2√x√x+ 2

√x

II. Use Comparison Theorems to determine the convergence of the following.

1.∫ ∞

0

1 + sec2 x√x

dx 2.∫ ln 2

0

ex

2√x+ x5

dx

III. The rate of growth of a flu epidemic is jointly proportional to the fraction y of the population who have theflu and the fraction 1− y who do not have it.

1. Set up and solve the differential equation satisfied by y.

2. Five days ago,134

of the people in a certain town had the flu and today,112

have it. How many days

from now will the fraction of infected people be314

?

Part II

I. Consider the region bounded by the circle r = 1 and the cardioid r = 1 + cos θ. Find the following.

1. the perimeter P of the region R.

2. the area A of the region R.

II. Consider the limacon with a loop with equation r = 1 + 5 cos θ.

1. Sketch the polar curve.

2. Give a set of parametric equations for the given polar curve.

III. A particle moves along a curve C described by the parametric equations{x = (1 + cos t) cos ty = (1 + cos t) sin t

1. Find the slope of the line tangent to C at any time t.

2. Determine the points at which the curve C has a vertical tangent line.

3. Determine the points at which the curve C has a horizontal tangent line.

IV. The foci of a hyperbola are the endpoints of the latus rectum of the parabola x2 = 4(y − 1) whose vertex isone of the endpoints of the conjugate axis of the hyperbola.

1. Find an equation of the hyperbola.

2. Give the coordinates of the vertices, foci, and conjugate axis endpoints of the hyperbola.

3. Give the equations of the asymptotes.

4. Sketch the hyperbola and label all significant points.

V. An ellipse has a focus at the pole and vertices at the points (8, π) and (−2, π).

1. Find the eccentricity e of the ellipse.

2. Give the polar equation of the ellipse.

3. Sketch the ellipse with the directrices.

Part III

I. Write TRUE if the statement is correct. Otherwise, write FALSE.

1. If−→A and

−→B are unit vectors in V3, then

−→A ×

−→B is also a unit vector.

2. If |−→A ·−→B | = ‖

−→A‖ · ‖

−→B‖, then

−→A ×

−→B =

−→0 .

3. If−→A and

−→B are orthogonal and

−→C =

−→A ×

−→B , then

−→A ×

−→B is parallel to

−→B .

4. The axis of revolution of the solid of revolution with equation x2 − (y − 3)4 + z2 = 0 is the y-axis.

5. In R3, two nonparallel lines must intersect.

II. Consider the surface S with equation 9x2 − 4y2 + 36z = 0.

1. Determine the traces of S with the coordinate planes.

2. Identify and sketch the surface S. Label the points of intersection with the coordinate axes.

III. Consider the sphere with equation x2 − 6x+ y2 − 8y + z2 − 10z + 116 = 0.

1. Find the midpoint Q of the center of the sphere and the point P(5,2,7).

2. What is the minimum distance of the point Q from the surface of the sphere?

IV. Assume that−→A ,−→B ,−→C ∈ V3 where

−→B = 2i + 3j,

−→C = i + 4j + 3k, and ‖

−→A‖ =

√142. If

−→A makes an angle

of π6 with the plane determined by the vectors

−→B and

−→C , find the volume of the parallelipiped formed by the

three vectors.

V. Give the total surface area of the parallelipiped determined by the vectors −→a = 2i− 4j + 5k,−→b = 3i+ j + k,

and−→C = −3i− j + 1k.

VI. Find an equation of the plane through the point ( 2 , -1 , 3 ) and containing the line given by the parametricequations

x = 1 + 2t

y = 7− t

z = 2− 5t

VII. Find the angle between a diagonal of a cube and one of its edges.

2

Part IV

I. Reparametrize the curve represented by−→R (t) =< 3 cos t, 4t, 3 sin t > with respect to arclength s measured

from the point where t = 0 in the direction of increasing t.

II. Suppose−→R (t) =< cos 3t, 4, sin 3t > where 0 ≤ t ≤ π

2 .

1. Find the moving trihedral at the point (1,4,0).

2. Give the radius of curvature at (1,4,0).

3. Give an equation of the osculating plane at (1,4,0).

III. Find the tangential and normal components of the acceleration vector of a particle traveling along the curvetraced out by

−→R (t) =< t, t2, 3t >.

IV. A particle is moving with acceleration vector−→A (t) =< t2, t, 1 >. At t = 0, its velocity vector is < 1,−1,−1 >

and its position vector is j.

1. Give the position vector ⇀ R(t) that represents the path of the particle.

2. Give a definite integral that represents the distance traveled by the particle from t = 0 to t = 1. SETUP ONLY.

V. A ball is thrown at an angle of 60◦ from the ground. If the ball lands 90 feet away, what is the initial speed ofthe ball?

VI. A particle moves along a curve C given by−→R (t). At t = 1, the normal component of the acceleration vector of

the particle is 7. If−→R ′(1) ×

−→R ′′(1) =< 3,−4,

√24 >, what is the curvature of C at t = 1.

PS: For part V, review your 5th exam.o