MATH 137 Quiz 8
Wednesday, November 13. Duration: 35 minutes.
Notes:
1. Answer all questions in the space provided.
2. For multiple choice and true/false questions, answer by filling in the bubble on the last
page of the quiz.
3. Your grade will be influenced by how clearly you express your ideas and how well you
organize your solutions. Show all details to get full marks. Numerical answers should be
in exact values (no approximations).
4. There are a total of 24 possible points.
5. DO NOT write on the Crowdmark QR code at the top of the pages or your quiz will not
be scanned (and will receive a grade of zero).
6. Use a dark pen or pencil.
(MC) Answer the following multiple choice questions on the last page of the quiz. Bubble (a),
[6]
(b), (c), or (d). Note there is only one correct answer for each question.
1. If f 0 (x) = g 0 (x) for all x ∈ R, then f (x) − g(x) must be
(a)
(b)
(c)
(d)
zero.
a constant.
undefined.
None of the above.
2. For the function f (x) = 7,
(a)
(b)
(c)
(d)
f has no critical points.
every real number is a critical point of f .
f has no local extrema.
None of the above.
3. Consider the function f on [a, b]. Which of the following conditions is NOT an assumption needed for the Mean Value Theorem?
(a)
(b)
(c)
(d)
f is continuous on [a, b].
f is differentiable on (a, b).
Both f 0 (a) and f 0 (b) exist.
None of the above.
(TF) True/False, answer on the last page of the quiz. Bubble (a) for True, (b) for False.
[1]
4. TRUE or FALSE: The following are equivalent:
1. Rolle’s Theorem
2. the Mean Value Theorem
[1]
5. TRUE or FALSE: If f is a differentiable function on R and f 0 (0) = 0 then 0 is either
a local maximum or a local minimum of f .
(SA) Short answer questions, marks only awarded for a correct final answer, you do not need to
show any work.
[2]
(a) Use logarithmic differentiation to find the derivative of f (x) = xcos(x) for x > 0.
[2]
(b) List all of the critical points for f (x) = ln(|x|).
(LA) The remaining questions are long answer questions, please show all of your work.
[4]
1. Find
dy
for arctan(xy) + y = x
dx
[4]
2. Find the global maximum and global minimum for f (x) = 3x4 + 4x3 + 7 on the interval
[−2, 2].
[4]
3. Use the MVT to prove, for any a, b ∈ R with 0 < a < b, that
√
b−
√
a<
b−a
√
2 a
This page is for rough work.