UC Merced: MATH 21 — Exam #3 — 30 November 2005
On the front of your bluebook print (1) your name, (2) your student ID number, (3) your instructor’s
name (Sprague) and (4) a grading table. Show all work in your bluebook and BOX IN YOUR FINAL
ANSWERS where appropriate. A correct answer with no supporting work may receive no credit
while an incorrect answer with some correct work may receive partial credit. Textbooks, class notes,
calculators and crib sheets are not permitted. There are a total of five problems and a total of 100
points. Please start each of the five problems on a new page.
1. (20 points) Answer the following Always True (T) or False (F). Only your final answer will be
graded on these problems.
(a) Rb
af(x)dxmay be described as the area between the function f(x) and the x-axis over the
interval a≤x≤b.
(b) Rcos(x/5)dx= 5 sin(x/5) + 5
(c) When marginal cost is equal to marginal revenue, profit is maximized.
(d) A set of parametric equations describing motion along the line y= 3x+ 4 is given by
y= 7 + 3tand x= 1 + t.
(e) All continuous functions are integrable.
2. (10 points) Using the definition of the derivative, show that d
dxsin(x) = cos(x), given that
lim
θ→0
sin(θ)
θ= 1 and lim
θ→0
cos(θ)−1
θ= 0
3. (30 points total) Find the requested information in the following unrelated problems:
(a) (8 points) A particle is located x= 1 at t= 0, and its velocity is given by v(t) = t2. Use
left-endpoint sum and ∆t= 1/2 to estimate the position of the particle at t= 3/2. Is your
approximation an over or under estimate for the true position?
(b) (6 points) Evaluate R(x+1)
x2dx
(c) (8 points) Evaluate the following limit:
lim
x→01
x−1
sin(x)
(d) (8 points) Given the plot of g(x) given below, calculate R4
0g(x) dx
0 1 2 3 4
-1
0
1
g(x)
x
4. (20 points) We wish to design a closed cylindrical container with a volume of 1 m3. It costs
twice as much for the material used for the lids than that used for the sidewalls. What is the
cylinder radius that will minimize the material cost.
5. (20 points) I was driving to work along Hwy 140 towards the intersection with Arboleda Drive,
along which a police car was approaching Hwy 140. Arboleda intersects Hwy 140 at a right
angle. The police car was approaching the intersection at a constant rate of 50 mph. When the
police officer was one mile from the intersection, and I was three miles from the intersection, his
laser indicated that the distance between us was decreasing at a rate of 70 mph. How fast was
I driving?