Download Calculus I - Fall 2011

Calculus I - Fall 2011
Midterm Exam, October 17, 2011
In all non-multiple choice problems you are required to show all your work and provide the necessary
explanations everywhere to get full credit. In all multiple choice problems you don’t have to show
your work.
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1. The graph of a function g is shown below. For each of the following, decide if the limit
exists. If it does, find the limit. If it does not, decide also if the limit is ∞, −∞, or neither.
No justification is necessary for full credit, but show your work for purposes of partial credit.
(a) lim g(x) =
x→0
(b) lim+ g(x) =
x→0
(c) lim− g(x) =
x→1
(d) lim+ g(x) =
x→1
(e) lim g(x) =
x→1
(f) lim g(x) =
x→2
(g) lim− g(x) =
x→3
(h) lim g(x) =
x→3
(i) lim− g(x) =
x→4
(j) lim+ g(x) =
x→4
(k) lim g(x) =
x→4
(l) lim− g(x) =
x→5
2
2. Find the following limits:
x2 − 4x
x→4 x2 − 3x − 4
(a) lim
√
√
x− 5
(b) lim
x→5
x−5
(c) lim t cot(3t)
t→0
3
3. Prove that the equation 3x5 − x3 = 1 has a solution.
4. Find a constant c that makes g(x) continuous on (−∞, ∞). Then use limits to prove that
g(x) is indeed continuous at x = 4.
{
x2 − c
if x < 4
g(x) =
cx + 20 if x ≥ 4
4
√
5. Find all horizontal and vertical asymptotes of the function f (x) =
5
2x2 + 1
.
3x − 5
6. Find lim f (x) if, for all x > 5,
x→∞
4x − 1
4x2 + 3x
< f (x) <
x
x2
7. Let f (x) =
√
4 − 3x. Use the definition of the derivative to find f ′ (x).
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8. Find ALL values of x at which the function f (depicted below) is not differentiable.
A
x = 1, 2
B
x = −1, 1, 2
C
x = −1, 0, 1, 2
D
x = 0, 1, 2
E
x = −1, 2
9. Let f (x) =
2x + 1
. Find f ′ (x).
x2
A
f ′ (x) = 2x−2 − 2x−3
B
f ′ (x) = −2x−2 + 2x−3
C
f ′ (x) = −2x−2 − 2x−3
D
f ′ (x) = 2x−2 + 2x−3
E
f ′ (x) = −2x−2 − x−3
10. Let f (x) = x cos2 (1 − x). Find f ′ (x).
A
f ′ (x) = cos2 (1 − x) − 2x cos(1 − x) sin(1 − x)
B
f ′ (x) = −2x sin(1 − x)
C
f ′ (x) = cos2 (1 − x) + 2x cos(1 − x)
D
f ′ (x) = cos2 (1 − x) + 2x cos(1 − x) sin(1 − x)
E
f ′ (x) = cos2 (1 − x)
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11. Let x3 + x2 y + 4y 2 = 6. Find dy/dx by implicit differentiation.
A
dy
3x2 + 2xy
= 2
dx
x + 8y
B
3x2 + 2xy
dy
=− 2
dx
x + 8y
C
dy
2x2 + 3xy
= 2
dx
x + 8y
D
dy
2x2 + 3xy
=− 2
dx
x + 8y
E
dy
3x + 2x2 y
=− 2
dx
x + 8y
12. Find the linearization L(x) of the function f (x) = cos x at a = π/2.
(
π)
A L(x) = 1 − sin x x −
2
)
(
π
B L(x) = − sin x x −
2
C
D
E
L(x) = 1
π
−x
2
π
L(x) = x −
2
L(x) =
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