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Practice Midterm Exam # 1
The first midterm will cover Chapter 0 (except section 0.2) and Chapter 1 (except sections 1.6 and 1.7). There will be 10 questions, and the test should be very doable in 50 minutes. Problems will be similar in difficulty to the homework. In particular, you should be familiar with the methods of solution for the following problems.
Problem 1: Suppose f (x) has the following graph:
0
20
40
60
80
100
โ3 โ2 โ1 1 2 3 x
Does f (x) have an inverse? Why or why not?
Problem 2: Suppose f (x) = x^2 + 2 for x โค 0. Find f โ^1 (x), and state its domain.
Problem 3: Carefully sketch the graph of the inverse of the function shown below (use the same axes):
2
4
6
โ2 2 4 6 x
Problem 4: Sketch a graph of y = 2 cos(3x), accurately showing the amplitude and period of the function.
Problem 5: Find all solutions of the equation
sin^2 x + 3 sin x โ 4 = 0.
Problem 6: Simplify the expression tan(secโ^1 x). (Hint: use a right
โ0.
โ0.
โ0.
โ0.
0
1
โ4 โ2 2 4 x
Problem 11: Complete the square and sketch the graph:
y = 2x^2 + 4x + 5.
Problem 12: Consider the graph y = f (x) below, and find the values listed. If a limit does not exist, say so.
(a) lim xโ 1 โ^
f (x)
(b) lim xโ 1 +^
f (x)
(c) lim xโ 1 f (x)
(d) lim xโ 2 โ^
f (x)
(e) lim xโ 2 +^
f (x)
(f) lim xโ 2 f (x)
(g) lim xโ 3 f (x)
Problem 13: Suppose
f (x) =
x^2 + 1 if x < โ 1 , 3 x + 1 if x โฅ โ 1.
Problem 17: Using the Intermediate Value Theorem, explain why any polynomial with odd degree has to have at least one real root.
Problem 18: Find all points of discontinuity for the function f (x) = x/ sin x.
Problem 19: Find all asymptotes (horizontal, vertical, or slant) of the function
f (x) =
x^3 โ 3 x^2 x^2 โ 4 x + 3
Problem 20: Determine each limit (write โ or โโ if it is appro- priate):
(a) lim xโ 0
x^2
(b) lim xโ 1
x โ 1
(c) lim xโ 0
sin x x (d) lim xโโ ln x
(e) lim xโ 0 +^
ln x
