APPM 1350 EXAM 1 FALL 2006
INSTRUCTIONS: Books, notes, flying monkeys and electronic devices are not permitted. Write
your (1) name, (2) instructor’s name, and (3) recitation number on the front of your bluebook.
Work all 6 problems. Start each problem on a new page. Show your work clearly and box your
final answer. A correct answer with incorrect or no supporting work may receive no credit, while
an incorrect answer with relevant work may receive partial credit.
1. (20 points) Using appropriate rules of differentiation, find the following.
(a) f0(x) where f(x) = 2x+ 1
x2−1
(b) d
dx ³1−2x3−3x−6−π2´
(c) If u(0) = 5, u0(0) = −3, v(0) = −1 and
v0(0) = 2,find d
dx(uv) at x= 0.
2. (15 points) Answer the following questions as either TRUE meaning always true or FALSE
meaning not always true. For this problem only, you do not need to justify your answer.
(a) A function can be both odd and even.
(b) If fis continuous at x0then fis differentiable at x0.
(c) If fis differentiable on (−∞,∞), f0(a)≥1 and f0(b)≤ −1 then there must be a point
cwith f0(c) = 0.
(d) If f(x) is even then g(x) = f(x) + 1 is even.
(e) sin(πt
2) is an odd function.
3. (20 points) Find the following limits, if they exist. If the limit does not exist or is infinite,
clearly state this. Show all your work and be sure to justify your answers.
(a) lim
x→0x2cos µ9
x¶
(b) lim
x→−2
(x+ 3)|x+ 2|
x+ 2
(c) lim
h→0+
√h2+ 4h+ 5 −√5
h
(d) lim
x→0sin(π
2+ tan x)
4. (20 points)
(a) State the definition of the derivative, dy
dx.
(b) Using this definition, calculate the derivative with respect to xof y=1
x−1.
(c) Find equations of all lines having slope −1 that are tangent to the curve y=1
x−1.
5. (10 points) Let f(x) = (cos x)2−x. Show that fmust have a root (or zero). You do not need
to find the value of the root.
6. (15 points) A ball thrown vertically upward from a platform 56 feet above the ground at a
velocity of 96 feet/second reaches a height of s=−16t2+ 96t+ 56 feet in tseconds. Please
state the units in your answer.
(a) Find the ball’s velocity and acceleration at time t.
(b) How long does it take the ball to reach its maximum height?
(c) How high does the ball go?