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MAT137: Calculus with proofs

创建日期：2024-05-11 17:04:36

所属分类：数学mathematics

标签： MAT137

Calculus with proofs

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MAT137: Calculus with proofs

Test 1
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 + 6x2 − 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 = .

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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