Math 111 • Amherst College • Spring 2015
Final Exam, May 13, 2015
1. (5 points each) Evaluate the following limits.
√
√
x + 2 − 2x
(a) lim
x→2
x−2
1
4
(b) lim
−1
x→4 x − 4
x
x2 − 1 − |x − 1|
x→1
x2 − x
x
(d) lim ln 2
x→∞
x +1
(c) lim−
2. (5 points each) In parts (a) - (c), find the derivatives of the given functions. In part (d),
find dy/dx. You do not need to simplify your answers.
1
x3 + 3x + e
2x
(b) g(x) =
sin(3x) − x
Z x2 +6
2t
dt. Find h0 (x).
(c) Let h(x) =
(sin(3t) − t)
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(a) f (x) = √
3
(d) Find dy/dx, where y = ex+y .
3. (5 points each) Calculate the following integrals.
Z
(a)
sec2 (cos x) sin x dx
Z
2
(b)
1
Z
(c)
Z
2x + 3x2
dx
4x3
e−x
3 +3x
(x2 − 1) dx
0
(d)
|2x + 1| dx
−2
4. (10 points) Use the (limit) definition of the derivative to compute g 0 (x), where
g(x) =
1
3x
.
1 − 2x
5. (15 points) You’re watching the Boston Red Sox play a baseball game against the
Atlanta Braves. The baseball diamond the teams are playing on has the shape of a
square, with 80 ft. between bases, and the pitcher’s mound in the exact center. Boston
player David Ortiz runs from 2nd base to 3rd base at a rate of 20 ft./sec. When Ortiz
is 70 ft. away from 3rd base, at what rate is his distance to Atlanta pitcher Trevor
Cahill changing? Is this distance increasing or decreasing? (You may assume Cahill
does not move from the pitcher’s mound.)
6. (10 points) Use the -δ definition of the limit to prove that
lim (7 − 3x) = 1.
x→2
7. (a) (5 points) State the Mean Value Theorem.
(b) (10 points) Suppose that for all x, 0 < f 0 (x) ≤ 2, and f (1) = −1. Show that
−1 < f (4) ≤ 5.
R5
8. (10 points) The graph of y = f (x) is shown below. Estimate the value of −3 f (x) dx
by computing a Riemann sum with n = 4 rectangles and right endpoints as sample
points.
y
3
5 x
–3
–2
2
9. (15 points) A rectangular box with volume 25 ft3 must be built. The box will have a
square base and an open top. If the material for the base costs $2 per square foot and
the material for the sides costs $5 per square foot, what should the dimensions of the
box be to minimize the cost of the materials needed to build the box?
10. (15 points) Provide a detailed sketch of the graph of the function
f (x) =
x3 − 2
.
x
Determine the domain, asymptotes, intervals of increase/decrease, local maximum and
minimum values, concavity, and inflection points.
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