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Midterm 1

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University of Connecticut Department of Mathematics

Math 1131

Sample Exam 1

Spring 2014

Name:

Question: Points: Score: 1 12 2 6 3 9 4 12 5 9 6 6 7 24 8 10 9 6 10 6 Total 100

Math 1131

Sample Exam 1

1. If the statement is always true, circle the printed capital T. If the statement is sometimes false, circle the printed capital F. In each case, write a careful and clear justiﬁcation or a counterexample. (a) The line x = 1 is a vertical asymptote of the graph of y = Justiﬁcation: x2 − 1 . x2 − 2x + 1 T F [3]

(b) The function y(t) = 3 + 6e−kt , with k a positive constant, has horizontal asymptote y = 6. Justiﬁcation:

T

F

[3]

(c) A function that is not continuous at a point can not be deﬁned at that point. Justiﬁcation:

T

F

[3]

(d) If f (x) is diﬀerentiable then f (x) and f (x) − 2 have the same derivative. Justiﬁcation:

T

F

[3]

Page 1 of 10

Math 1131 2. (a) If f (x) =

Sample Exam 1 [3]

x for x = −3, then ﬁnd its inverse function f −1 (x) and determine the x+3 domain of the inverse function.

(b) If log2 a = .83 and log2 b = .56, compute log2 (a3 /b2 ) and give your numerical answer exactly.

[3]

Page 2 of 10

Math 1131 3. Let f (x) = ln (4x + 1) + 1, g(x) = (a) Find the domain of f (x). √ 3x + 6, and h(x) = x3 .

Sample Exam 1

[3]

(b) Find the domain of (g ◦ h)(x)

[3]

(c) Find f −1 (x) and its domain.

[3]

Page 3 of 10

Math 1131

Sample Exam 1

4. Determine the following limits, using algebraic methods to simplify the expression before ﬁnding the limit. Evaluate each of the following limits. If the limit does not exist but goes to ∞ or −∞, indicate so. If the limit does not exist for any other reason, write DNE with a justiﬁcation. (a) lim 3−x x→2 1/3 − 1/x [4]

(b) lim

x2 − 1 x→0 x2 − x

[4]

x−4 (c) lim √ x→4 x−2

[4]

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Math 1131 5. Let f (x) = √ 2x3 + x2 − 7 . 49x6 + 9x2 + x (a) Compute lim f (x). x→∞ Sample Exam 1

[3]

(b) Compute lim f (x). x→−∞ [3]

(c) What are all the horizontal asymptotes of the graph of y = f (x)?

[3]

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Math 1131 3 − kx, k − x, if x > 1, if x ≤ 1.

Sample Exam 1

6. Let f (x) =

(a) lim f (x) = x→1− [2]

(b) lim f (x) = x→1+ [2]

(c) Find the constant k so that f (x) is continuous at every point.

k=

[2]

Page 6 of 10

Math 1131

Sample Exam 1

7. Find the derivatives of the following functions. Simplify your answers (e.g., combine like powers of x in a polynomial). √ (a) f (x) = 9x − 7 x.

[6]

(b) f (x) = x4 ex .

[6]

(c) f (x) =

x2 + x . x3 − 3

[6]

(d) f (x) =

ex . x2 + 1

[6]

Page 7 of 10

Math 1131

Sample Exam 1 [6]

8. (a) Find the slope of the tangent line to the graph of y = (x2 − 1)2 at (2, 27).

(b) Find the equation of the tangent line in part a, writing the answer as y = mx + b.

[4]

Page 8 of 10

Math 1131 9. The picture below is the graph of f (x) = x3 − x, −x2 + ax + b, if x ≤ 1, if x > 1,

Sample Exam 1 [6]

where a and b are chosen to make the function diﬀerentiable at x = 1. Determine a and b. (Hint: the function has to be continuous at x = 1 also, since it is diﬀerentiable.) y

x 1

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Math 1131

Sample Exam 1 [6]

10. In the graph of y = x2 /3, which is illustrated below, ﬁnd the number c such that the tangent lines to the graph at the points where x = 1 and x = c are perpendicular. y (c, c2 /3) (1, 1/3) x c 1

Page 10 of 10

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