Calculus I Exam 2: Derivatives
instructions:
• You will have 75 minutes to take this exam.
• There are 7 numbered problems.
• Show all work you needed to do to draw your conclusions, including reasoning, calculations,
and graphs.
• This is an in-class test! Except for the derivative interpretation questions, you do not need
to write in complete sentences, but you do need to show your work!
• There is scrap paper available at the front desk; do not use your own!
• Good luck!
1. (2 points each) Find the values requested for the graph of f (x) given below. You do not need
to show any work on this problem.
(a)
(b)
lim f (x)
x→−3−
lim f (x)
x→−3+
(c) lim f (x)
x→1
(d) f (1)
(e)
lim f (x)
x→−∞
(f) lim f (x)
x→∞
(g) List all values of a where lim f (x) does not exist
x→a
(h) List all values of x for which f (x) is not continuous
(i) List all values of x for which f (x) is not differentiable
2. (15 points) Find the derivative of f (x) =
quotient).
√
3x + 2 using the definition (limits and difference
3. This is a question about explaining the meaning of rates in context. Remember that in this
context you are expected for a ‘marginal’ type explanation:
W (v) will stand for how cold a person feels when the temperature outside is 0◦ F and there
is a wind blowing at v mph.
(a) (5 points) Explain what W 0 (5) = −1.2 means.
(b) (5 points) Explain what (W −1 )0 (−30) = −
1
means.
5
4. (5 points) Give a good numerical approximation of the derivative of y = (sin(t))t at t = 2,
with an error bound. Remember to show your work!
5. The function f (x) is graphed below.
(a) (3 points) Give the intervals on which f 0 > 0 and on which f 0 < 0.
(b) (3 points) Give the intervals on which f 00 > 0 and on which f 00 < 0.
(c) (12 points) Sketch the graph of f 0 (x). Be sure to show all your work!
6. Let f be the function:
f (x) =
 √
3


 2x
if x < 4



if x ≥ 4
mx + b
(a) (6 points) What equation must m and b satisfy in order for f to be continuous when
x = 4? Be sure to explain how you used the definition of continuity to come up with
your answer.
(b) (6 points) It is known that
d √
2
3
2x = √
.
3
dx
3 4x2
Use this along with your answer from part (a) to determine the values of m and b,
respectively, that will make f continuous and differentiable at x = 4. Be sure to explain
how you used the definition of differentiability to come up with your answer.
7. (5 points each) Evaluate each of the following limits algebraically:
1
1
−
2
9
(a) lim x
x→3 x − 3
(b)
(c)
lim
x→−1+
x2 − 4
x2 + 4x + 3
2x3 − x + 1
x→−∞ −5x3 + 4x2 + 3x + 1
lim