Download 1. (8 points) Use the definition of the derivative to find f

1. (8 points) Use the definition of the derivative to find f ′ (x), where f (x) =
√
1 − 2x.
2. (3 points each) Find y ′ . You may use any rules of differentiation. DO NOT SIMPLIFY
your answers.
(a) y =
√
2
3
+ 4x − 6
x
√
(b) y = (2x + 1) sin x
1
(c) y = xπ + π x
(d) y =
9
(3 sec t − 1)3
(e) y = ln(x3 − 2x)
(f) y = ecosh(5x)
(g) y = tan−1 (x2 ), equivalently y = arctan(x2 ).
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3. (4 points) Use logarithmic differentiation to find y ′ . Your answer must be in terms of x
but need not be simplified.
√
y = (3x − 1) x
4. (4 points) Find
d
tan(sin−1 x) and simplify your result.
dx
3
5. Refer to the graph of y = f (x) to answer (a) and (b).
y
2
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-1
1
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3
4
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x
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(a) (1 point each) Find the indicated quantity, if possible. If it is not possible, write
“DOES NOT EXIST”.
i. lim− f (x)
x→0
ii. f (1)
iii. lim f (x)
x→3
iv. lim f (x)
x→1
v. f ′ (2)
(b) (2 points) Is f continuous at x = 1? Justify your answer using the definition of
continuity.
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6. (3 points each) Find the indicated limits, and be sure to justify your answers. If the limit
does not exist, distinguish between +∞, −∞, and “DOES NOT EXIST”.
(a) lim
x→0
(b) lim
sin x2
x
x3 − 8
+ x − 10
x→2 2x2
(c) lim
1 − 4x3
x + 3x3
(d) lim−
t2 + 9
t2 − 9
x→∞
t→3
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7. Answer (a) and (b) with regard to the curve described by:
xy = sin y + 2x.
(a) (5 points) Use implicit differentiation to find an expression for
dy
.
dx
(b) (3 points) Find the equation of the line tangent to the curve at the point (0, 0).
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8. (2 points each) Let f be a function that is continuous on the interval [0, 4] and differentiable
on (0, 4). Answer “TRUE” or “FALSE” to the statements below. If you believe it is true,
state in one sentence why it is true. If you believe it is false, provide one counter-example.
(a) It is always possible to find a number c in [0, 4] such that f (x) ≤ f (c) for all x in [0, 4].
(b) It is always possible to find a number c in [0, 4] such that the instantaneous rate of
change of f at c equals the average rate of change of f over the interval [0, 4].
(c) It is always possible to find a number c in [0, 4] such that the line tangent to the graph
of f at (c, f (c)) is horizontal.
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9. (8 points) A camera mounted on top of a pole, 6 m. from the ground, is videotaping a car
that is travelling 100 km/h ( 250
9 m/s). How fast is the camera angle changing when the
car is directly below the camera?
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Camera
θ
Car •
6m
x
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10. (10 points) Let f be a function with the following properties.
• f is continuous on the interval [−1, 5] and differentiable on (−1, 0) ∪ (0, 5).
• f is not differentiable at 0.
• f is increasing on [1, 3] and decreasing on [3, 5].
• f (0) = 2
• f is concave down on (2, 4) and concave up on (0, 2) ∪ (4, 5).
Answer (a)–(e) with regard to f , and make sure to justify your answers.
(a) Sketch a possible graph of f between x = −1 and x = 5.
(b) Label all relative maxima and relative minima on your graph.
(c) Find the critical numbers of f on [−1, 5].
(d) On what interval(s) is f ′ (x) > 0?
(e) On what interval(s) is f ′′ (x) < 0?
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11. (12 points) You are designing a rectangular poster to contain 50 square inches of printing
with a 4-inch margin at the top and bottom, and a 2-inch margin on each side. What
overall dimensions will minimize the amount of paper used?
Hint: Use w and l for the width and length, respectively, of the printed portion of the
paper.
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