Math 111
Final Exam
1. Use the limit definition of derivative to find f 0 (x).
x2
(a) f (x) = 4x
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2. Evaluate each of the following limits by first writing it as the derivative of a function.
(1 + h)10
h!0
h
x
2
32
(b) lim
x!5 x
5
(a) lim
1
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3. Find the derivative of the following functions.
(a) f (x) = x ln x
x
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(b) g(x) = xcos x
Z 1p
1 + t2 dt
(c) h(x) =
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2x
4. Evaluate the following integrals
Z
(a)
tan(x) dx
(b)
(c)
Z
Z
ln(26)
ln(7)
p
3
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ex
dx
1 + ex
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(ln x)2
dx
x
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5. Evaluate the following limits if they exist.
p
3
x+5
(a) lim
x!4
x 4
3
x
7x
(b) lim
3
x!0
x
6x 6
(c) lim
x!1 |x
1|
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6. Use the "- definition of limit to show that f (x) = 1
2x is continuous at x = 3.
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7. Given
f (x) = e
x2
,
f 0 (x) =
2xe
x2
,
f 00 (x) = (4x2
2)e
x2
(a) What is the domain of f ? What are the x- and y-intercepts of f ?
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(b) Find all horizontal and vertical asymptotes of f .
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(c) When is f increasing and decreasing? Find any local maximums and minimums of f .
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(d) When is f concave up and concave down? Find all the inflection points of f .
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(e) Sketch a graph of f (x).
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8. Find the equation of the tangent line at the point (1, 1) to the curve given by equation
y3
y = x ln x.
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Math 111
Final Exam
1
9. Use a linear approximation to estimate p .
26
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10. A zoo keeper wants to build a panda enclosure in the shape of a right-angled triangle. The
hypotenuse of the triangle is to be along a river and does not require any fencing. The other
two sides require special panda-proof fencing and the zoo can provide 200 yards of this fencing.
What is the largest possible area of enclosure the keeper can build? (Make sure you explain
how you know your answer is the maximum.)
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11. Consider the region R bounded by the the graphs of y = 4 x2 , and y = x2 2x. Sketch this
region, then set up but do NOT evaluate an integral to determine the exact area of R.
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