Math 1205
Common Final Exam
Spring 2014
Test Version: A
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will be the number of correct answers. You have one hour to complete this portion of the exam.
Turn in the opscan sheet with your answers and the question sheets at the end of this part of the
final exam.
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1. Let y = f (x) be the function shown below. Suppose = 1. Find the largest value of δ > 0
such that 0 < |x − 2| < δ =⇒ |f (x) − 3| < .
y
4
3
2
1
x
−1
1
2
3
−1
−2
(A) 0.5
(C) 1.5
(B) 1
(D) No such δ exists
2. Evaluate lim
x→0−
1
1
−
.
x |x|
(A) 0
(B) 1
(D) −∞
(C) +∞
3. Consider the graph of y = f (x) given below. Assume that the graph extends out of view only
as indicated by the arrows.
y
6
4
2
x
−4
−2
2
4
−2
−4
−6
Which of the following statements is FALSE?
(A) lim f (x) does not exist
(C) lim f (x) does not exist
(B) lim f (x) = −∞
(D) lim f (x) = f (0)
x→−4
x→0−
x→2
x→0+
4. If f 00 (x) = ex (x2 − 1), then f (x) is concave down on the interval
(A) (−∞, 0)
(B) (−1, 1)
(C) (1, ∞)
(D) (0, 2)
5. Let P2 (x) be the second degree Taylor polynomial of f (x) = x ln(x) centered at the point
a = 1. Then, P2 (1.1) is
(A) 0.125
(B) 0.15
(C) 0.015
6. Use the graphs of f (x) and g(x) below to find
(A) 0
(B) 2
d
dx
(D) 0.105
[g (f (x))] at x = 4.
(C) 4
(D) Does not exist
7. A rock thrown vertically upward from the surface of the moon reaches a height of
s = 24t − 2t2
meters in t seconds. How high does the rock get before falling back toward the moon?
(A) 72 m
(B) 6 m
(C) 132 m
(D) 24 m
8. For f (x) defined below, determine which of the following statements are true.
f (x) =
x2 + 2x + 1
x2 + 2x − 8
A) f (x) has a vertical asymptote at 2.
B) f (x) is continuous at x = 4
C) f (x) has a horizontal asymptote at y = −1
D) lim f (x) = 1
x→1
(A) A & B
(C) C & D
(B) A & C
(D) All of the above.
9. What is the rate of change of the area of a square, with respect to the length of the base,
when the base has length 4?
(A) 16
(B) 32
(C) 4
10. Suppose that y is a differentiable function of x. Find
tan(y) =
(A) −1
(B)
11. Find the derivative
dy
dx
dy
(B)
dx
dy
(C)
dx
dy
(D)
dx
(A)
1
2
(D) 8
dy
evaluated at the point (1, π/4) when
dx
4x
.
+3
x2
(C)
1
4
dy
when y = (sec(x))tan(x) .
dx
= sec(x)tan(x) · sec2 (x) ln (sec(x)) + tan2 (x)
= tan(x) · (sec(x))tan(x)
1
tan(x)
2
= sec(x)
· sec (x) ln (sec(x)) +
sec(x) tan(x)
1
tan(x)
2
= sec(x)
· sec (x) ln (sec(x)) +
sec(x)
(D) 3
12. A cylindrical oil storage tank has a known height of 5 m. Suppose the radius is measured to
be 8 m with a possible error of ±0.25 m. Use differentials to estimate the maximum error in
calculating the volume using this radius value. (V = πr2 h)
(A) dV = 20π m3
(C) dV = 50π m
(B) dV = 80π m3
(D) dV = 20π m
√
13. Suppose that Newton’s method is to be used to compute 3 4. That is, the solution to x3 −4 = 0
is sought. If we start the Newton’s method iterations at x0 = 1, then
5
3
(B) x2 = 1.587
(C) x2 =
(A) x2 =
√
3
(D) x2 =
3
4
9
14. Consider the following function with domain [−2, 4).
(
1/x,
if −2 ≤ x < 0
f (x) =
.
3 − x, if 0 ≤ x < 4
Which of the following statements is true?
(A) f (x) has a global maximum and a global minumum.
(B) f (x) has a global maximum but no global minimum.
(C) f (x) has a global minimum but no global maximum.
(D) f (x) has neither a global minimum nor a global maximum.
15. Which of the following functions satisfies the hypothesis of Rolle’s Theorem on the interval
[−1, 1]?
(A) f (x) = |x|
(B) f (x) =
√
x
(C) f (x) =
1
x2
(D) f (x) = cos(x)