MAT137: Calculus with proofs
Test 1
June 20, 2022
Full Name:
Student Number:
Instructions
Write clearly and concisely in a linear fashion.
Explain all your steps. Unless stated otherwise, show all of your work. You must give all the details of
how to solve the problem by hand.
How to submit your answers. You may use either of the following three methods.
1. Write digitally: using a tablet or digital pen to write directly onto the pdf
2. Print + Scan: printing the pages, writing your responses into the appropriate places using a pen or
pencil, and scanning them.
3. Write on blank paper + Scan: writing your answers on blank/lined paper, then scanning them. Please
indicate clearly which page corresponds to which question. When uploading your answers to Gradescope,
match your answers to the appropriate questions.
You may use a scanner or a smartphone scanner app (such as Scannable by Evernote).
The test starts at 6pm ET. You have until 8pm ET to submit your responses on Gradescope.
There are 7 questions, for a total of 50 points.
Page 1 of 11
1. Consider the following statement:
“For every differentiable function f on R, if f is increasing on [a, b], then x ∈ (a, b), f ′(x) > 0.”
(a) (2 points) In words and symbols, write the negation of this statement without using “no”,“not”,
“none”, “ ”, etc.
.
(b) (4 points) If the original statement is true, then prove it. If the statement is false, then prove
its negation.
Page 2 of 11
2. (8 points) Prove directly from the definitions of the derivative and of the limit that if f(x) =
3
x 2,
then f ′(1) = 3. Note that this is an ε-δ proof, so you are not allowed to use limit or differentiation
rules.
Page 3 of 11
3. Compute the following limits using any method that you want. Show all of your work.
(a) (4
points)
lim
x→1
x2 + 4x 5√
x 1
Page 4 of 11
(b) (4 points)
lim
x→0
sin(7 tan(3x))
tan(4 sin(5x))
Page 5 of 11
(c) (4 points)
lim
x→0+
(x+ ex)
1
x
Page 6 of 11
4. Let f(x) = x3 + 6×2 36x+ 5.
(a) (3 points) What is the largest interval containing 0 on which f is one-to-one Write the interval
below. No justification is needed.
The interval is .
(b) (3 points) Let g be the inverse of f on the interval of the previous part. What is g′(5) No
justification is needed
g′(5) = .
Page 7 of 11
5. (5 points) Find the equation of the line tangent to the curve x2+y3 = x3+y at (1, 1). No justification
is needed.
The equation of the tangent line is .
Page 8 of 11
6. Let f be a differentiable function on R. The table below lists the values of f at 6 different points.
x -2 -1 0 1 2 3
f(x) 3 3 -1 0 2 -2
(a) (2 points) What is the smallest number of zeros that f must have No justification is needed.
The function f must have at least zero(s).
(b) (2 points) What is the smallest number of zeroes that f ′ must have No justification is needed.
The function f ′ must have at least zero(s).
Page 9 of 11
7. Let f be a continuous function on [0, 14]. The graph of its derivative is sketched below.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
4
3
2
1
0
1
2
3
4
f ′(x)
(a) (3 points) Find all the local maxima of f . Don’t forget to include endpoints as potential local
maxima. No justification is needed.
The local maxima of f are located at x = .
(b) (3 points) Find all the local minima of f . Don’t forget to include endpoints as potential local
minima. No justification is needed
The local minima of f are located at x = .
(c) (3 points) Find all the inflection points of f . No justification is needed.
The inflection point(s) of f are located at x = .
Page 10 of 11
Extra page.
Page 11 of 11