Math 111
Final Exam
1. Evaluate each of the following limits.
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x2 + 5x + 6
x! 3 x2 + 2x
3
4 2x
(b) lim
x! 1 5
x + 3x2
2
x
x 2
(c) lim 2
x!2 x
4x + 4
(a) lim
(d) lim
4
x
x! 4
1
x+3
x+4
2. Compute each of the following derivatives.
x
(a) f 0 (x), where f (x) = esin(x)
tan(x)
p
0
3
(b) g (x), where g(x) = ln(x + ex )
"
!#
p
2
1 + 2x ex
d
(c)
ln
dx
eln(x) (x3 + x)4
Z x
d
(d) dx
(cos t)sin t (sin t)cos t dt
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3. Evaluate each of the following integrals:
Z
e t
p
(a)
dt
1 2e t
Z 1
1 y2
(b)
dy
y
e
Z e2
1
(c)
dx
x(ln x)2
e
Z 3
(d)
|2 x| dx
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(e)
(e)
0
00
f (x),
Z
where f (x) = x4 ln(4x)
0
1
f 0 (t) dt, where f (t) =
0
4. Give an "- proof that lim 4
x!2
t(t3 + et )
2 t2
3x =
2.
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5. A small rectangular box is constructed out of cardboard. It has no top, a square base, and a
volume of 500 in3 . What dimensions give the box that requires the least amount of cardboard?
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6. A beach ball is losing air at the rate of 2 cm3 /s. How fast is the radius of the ball decreasing
at the point when that radius is 10 cm? (You may assume that the volume of a ball of radius
r cm is 43 ⇡r3 cm3 .)
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7. The position of a runner at time t (measured in hours after beginning a run) is 4 sin(⇡t) miles
from home.
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(a) Find a formula for the velocity of the runner at time t.
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Math 111
Final Exam
(b) What is the total distance traveled by the runner from t = 0 to t = 1?
8. Consider the region between the x-axis and the curve y = 1
(a) Draw a picture of the region.
x2 for 0  x  2.
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(b) Find the area of the region.
9. (a) State the limit definition of the derivative of a function f (x).
(b) Use the definition from part (a) to find the derivative of f (x) =
p
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x.
10. The function f and its derivatives are given by:
f (x) =
x2 + 1
,
x2 9
f 0 (x) =
20x
,
2
(x
9)2
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f 00 (x) =
60(x2 + 3)
.
(x2 9)3
Find any vertical or horizontal asymptotes for f , the intervals where f is increasing, decreasing,
concave up or concave down, and any inflection points for f . Then use this information to
present a detailed and labelled sketch of the curve on the axes provided on next page.
p
11. Find the equation of the tangent line to xy = x2 y 2 at the point (1,4).
12. Estimate the value of ln(.98) using the tangent line approximation, also called the linearization
or linear approximation.
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