Assume that a particle’s position on x-axis is given by x = 3 cos t + 4 sin t where x is measured in feet and t is measured in seconds.

Math 21A Name: Summer I 2018 Midterm 2 7/5/2018 Time Limit: 100 Minutes Instructor

1. Do not turn the page until told to do so. 2. It is a violation of the university honor code to, in any way assist another person in the completion of this exam. Please keep your own work covered up as much as possible during the exam so that others will not be tempted or distracted. Thank you for your cooperation. Violations can result in expulsion from the university. 3. No notes, no calculator, and no classmates may be used as resources for this exam. 4. Read directions to each problem carefully. Show all work and simplify your answers for full credit. In most cases, a correct answer with no supporting work will receive little to no credit. What you write down and how you write it are the most important means of you scoring well on this exam. Neatness and organization are also important. 5. You may NOT use L’Hopital’s Rule on this exam. 6. You may NOT use shortcuts for finding limits to infinity. Show enough work for full credit. 7. You will be graded on proper use of limit notation. 8. You have until 5:50 PM to finish the exam.

Grade Table (Instructor use only)

Question Points Score

1 10

2 10

3 10

4 10

5 10

6 10

7 10

8 10

9 10

10 10

Total: 100

Math 21A Midterm 2 7/5/2018

1. (10 points) Suppose we are pumping air into a spherical balloon at a rate of 50π ft3 per minute. How fast is the balloon’s diameter increasing at the instant the radius is 5 ft? How fast is the surface area increasing?

Math 21A Midterm 2 7/5/2018

2. (10 points) Write an equation that estimates the change that occurs in the lateral surface area of a right circular cone when the height changes from h0 to h0 + dh and the radius does not change. (Hint: V = 1

3 πr2h and surface area S = πr

√ r2 + h2)

Math 21A Midterm 2 7/5/2018

3. (10 points) Find the derivatives of the following functions:

y = xx+3

y = x 3 √

x

y = (log5 x) log3 x

Math 21A Midterm 2 7/5/2018

4. (10 points) Compute the derivatives of the following functions:

y = 4

√ 2x(x + 3)(x − 2) (3×2 + 1)(4x + 1)

y = log7

( sin θ cos θ

3θ5θ

)

Math 21A Midterm 2 7/5/2018

5. (10 points) Find equations for the tangent and normal to the equation y2(2 − x) = x3

at (1, 1) (Read up on normal on page 174 if you don’t know what that is and look at Figure 3.32).

Math 21A Midterm 2 7/5/2018

6. (10 points) Compute the derivatives of the following functions:

y = 4 sin

(√ 1 +

√ 3t

)

y =

3t +

√ 2 +

√ 1 − t

Math 21A Midterm 2 7/5/2018

7. (10 points) Assume that a particle’s position on x-axis is given by

x = 3 cos t + 4 sin t

where x is measured in feet and t is measured in seconds.

(a) Find the particle’s position when t = 0, t = π/2, and t = π

(b) Find the particles velocity when t = 0, t = π/2, t = π

(c) Do the same with speed and acceleration.

Math 21A Midterm 2 7/5/2018

8. (10 points) Find all points (x, y) on the graph of y = x x−2 with tangent lines orthogonal

(perpendicular) to the line y = 2x − 3.

Math 21A Midterm 2 7/5/2018

9. (10 points) Something here…..

Math 21A Midterm 2 7/5/2018

10. (10 points) Something here…..