Northwestern University
Math 220
Midterm 2
February 29, 2016
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Instructions:
• This exam has 7 problems and 10 pages, where the last page is blank.
• Make sure that all pages are included.
• Read each problem carefully.
• Write legibly.
• DO NOT write outside of the box on each page.
• Cross off anything you do not wish to be graded.
• Write in complete sentences.
• You may not use books, notes or calculators.
• You have 60 minutes to complete this exam.
Question 1 (12 points, 3 points each) True or False? Circle the correct answer. No justification required.
A arctan(tan(−
True
11
11
π )) = − π.
12
12
False
B If f 00 (c) = 0 then (c, f (c)) is an inflection point for f .
True
False
C It is always possible to divide an equation by e x .
True
False
D x = 0 is a vertical asymptote of the graph of the function f ( x ) =
True
False
ln( x + 1)
.
x
Question 2 Evaluate the following limits or explain why they do not exist. Show all your work
and justify each step of your computations.
1
1
−
x →0 sin x
x
(a) (6 points) lim
1
(b) (8 points) lim (3x − 1) ln x
x →0+
Question 3 For each of the following functions, find its derivative. You do not need to simplify
your answers.
(a) (6 points) f ( x ) = 2arcsin( x
2)
√
(b) (8 points) f ( x ) = (ln x + x )
x
Question 4 (10 points) Use linearization to approximate the value arctan(0.99). You may leave
your answer in terms of π. You must state which function you are using, and for which value a
you use linearization.
Question 5 Consider the function f ( x ) = ln(e x − 3).
(a) (5 points) Find the domain and range of f ( x ).
(b) (2 points) State the domain and range of f −1 ( x ).
(c) (5 points) Find a formula for f −1 ( x ).
Question 6 (10 points) Show that the equation x3 + 2x − 5 = 0 does not have 2 roots/solutions.
(Hint: What would happen if the equation had two roots a and b?) Make sure you cite any
theorem you use, and verify that the hypotheses for said theorem hold.
Question 7 Consider the function f ( x ) = x ln( x2 ) whose derivatives are f 0 ( x ) = ln( x2 ) + 2
2
and f 00 ( x ) = .
x
(a) (5 points) State the domain of f , and find all x − and y− intercepts of the graph of f ( x ).
(b) (8 points) Find all vertical and horizontal asymptotes of the graph of f ( x ). Justify your
answers using limits.
Recall that f ( x ) = x ln( x2 ), f 0 ( x ) = ln( x2 ) + 2, f 00 ( x ) =
2
.
x
(c) (10 points) On what interval(s) is f increasing? Decreasing? At what value(s) of x (if any)
does f has a local maximum? local minimum?
(d) (5 points) On what interval(s) is the graph of f ( x ) concave upward? Downward? What are
the inflection points (if any) of the graph of f ( x )?
This page is left intentionally blank. Anything written here will not be graded.