1. (8 points) The half-life of cesium-137 is 30 years. Suppose we have a 80 -mg
sample. Find the mass that remains after t years.
2. (8 points) Find the absolute maximum and absolute minimum values of
f (x) = x3 + 3x2 − 9x + 1 on [−1, 2].
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3. (16 points, 3/3/5/5) Determine the limit if it exists.
(a) lim
2 sin x + 5x
.
cos x
(b) lim
2x + sin 3x
.
3x + tan 2x
x→0
x→0
(c) lim+ sin x ln 4x
x→0
(d) lim+ (tan 5x)x
x→0
2
4. (10 points) Gravel is being dumped from a conveyor belt at a rate of 34 ft3 /min
, and its coarseness is such that if forms a pile in the shape of a cone whose
base diameter and height are always equal. How fast is the height of the pile
increasing when the pile is 12 ft high? (the volume of a right circular cone is
V = 31 πr2 h).
5. (10 points) Determine the exact value of x at which f (x) = (x2 +1)e−x increases
most rapidly.
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6. (8 points, 2/2/2/2) Consider the function f (x) = 13 x3 − x2 − 15x + 3.
(a) How does f behave as x → ±∞?
(b) Where are the critical points of f ? Where is f increasing, decreasing?
(c) Where is f 00 = 0? Where is f concave up, down?
(d) Which of the following graphs fit the data you have determined?
-5
5
-5
-5
-5
5
4
5
5
-5
5
-5
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7. (10 points) The top of a picture is B units above your head and the bottom is
A units above your head. Show
that that your viewing angle will be maximized
√
when you are a distance AB from the picture. Hint: In the diagram, the
viewing angle θ = θB − θA .
8. (8 points) Show that the equation 6x + cos x = 0 has exactly one real root.
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