MTH 141 Applied Calculus
Exam 3 - Chapter 3 (Sections 3.5 to 3.7)
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1. (4.5 points) Find the number of units x that produces a maximum revenue R, where R = 400x x2
Solution:
R = 400x x2
R0 = 400 2x
R00 = 2
So
R0 = 0
400
Worksheet: Chapter 5 Section 2: Extrema and the First Derivative Test:
ANSWERS
2
1. Find any relative extrema (x, y) for the function f (x) = x2 3x.
f 0 (x) = x 3, 0 = x 3, x = 3. There is one critical value: x = 3. Test f 0 (x) at x = 0 and x = 4.
f 0 (0
Worksheet: Section 3.3: Concepts of Infinity and Division by Zero: ANSWERS
For the following (conceptual) forms, state whether they are determinate or indeterminate. In the case they are
determinate, reduce the form to 0 , , or .
1. 4 + =
2. 4 =
3. + =
Worksheet: Section 3.1: Limits from graphs: ANSWERS
1. The following represents a graph of a function y = g(x).
Find the following limits.
(a) lim g(x) = 3.5
x3
(b) lim+ g(x) = 6
x3
(c) lim g(x), does not exist
x3
(d) g(3) = 3.5
(e) lim+ g(x) = 1
x0
(f) l
Worksheet: Section 2.3: Integral EstimatesReimann Sums : ANSWERS
0
1. Determine the Riemann Sum for 4 x2 + 1 dx using n = 4 subintervals. Use evaluation at the right end of
the subintervals, then evaluation at the left end of the subinterval. Finally, giv
Worksheet: Section 3.4: Limits at Points of Possible Discontinuity: ANSWERS
1.
lim 2x3
x3/2 4x6
2(3/2)3
Evaluation at x = 3/2: 4(3/2)6
= 00 .
Divide out (x 3/2) or , using the denominator of the fraction 3/2, divide out 2(x 3/2) = 2x 3.
numerator: 2x3
2x3
Worksheet: Chapter 2 Section 2: Answers
1. Sketch the curves y = x3 3 and y = 2. Calculate the area bounded between the curves from x = 0 to x = 1.
Area:
5(0)
4
0
4
=
h
1
1
2 x3 3 dx = 0 5 x3 dx = 5x
0
x4
4
i1
= 5(1)
0
14
4
19
4
2. Sketch the curves y
MTH 141 Applied Calculus
Exam 2 - Chapter 2 (Sections 2.1 to 2.4)
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1. (5 points) Let f (x) = x2 + 5. Use the limit denition to nd the derivatives of f.
Solution:
)
limx0 f (x+xxf (x)
(x+x)2 +5(x2 +5)
limx0
x 2
2
2
x
limx
MTH 141 Applied Calculus
Exam 3 - Chapter 2 (Sections 2.5 to 2.8)
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1. (3 points) Let y = (2x2 5 x + 6)6 . Find f (x) and g (x) such that y = f (g (x).
Solution: f (x) = x6 and g (x) = 2x2 5 x + 6.
2. (3+3+3 points) Find
MTH 141 Applied Calculus
Exam 3 - Chapter 3 (Sections 3.1 to 3.4)
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_
1. (3 + 4 points) Find the critical numbers and the open intervals on which the function is increasing or
decreasing.
(a) g (x) = x2 + 3x 9
Solution: The
MTH 141 Applied Calculus
Exam 3 - Chapter 2 (Sections 2.5 to 2.7)
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1. (3 points) Let y = (x5 + 7x2 + 3 x3 + 6)8 . Find f (x) and g (x) such that y = f (g (x).
Solution: f (x) = x8 and g (x) = x5 + 7x2 + 3 x3 + 6.
2. (3+3
MTH 141 Applied Calculus
Exam 1 (Chapter 1) - Spring 2007
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1. (1.5 + 1.5 + 2 points) Find the domains of the following functions. Write your answer in the interval
form.
(a) f (x) = x2 3x + 4
Solution: f (x) is a polynom
MTH 141 Applied Calculus
Exam 2 - Chapter 2 (Sections 2.1 to 2.4)
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1. (5 points) Let f (x) = 2x 7. Use the limit denition to nd the derivatives of f.
Solution:
f 0 (x) =
=
=
=
=
f (x + x) f (x)
x
2(x + x) 7 (2x 7)
lim
x0
MTH 141 Applied Calculus
Exam 1 (Chapter 1) - Fall 2006
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_
1. (2 + 2 points) Find the x and y intercepts of the following graphs.
(a) y = x2 4x + 3
Solution: To nd the x-intercept, put y = 0. So
x2 4x + 3 = 0
(x 3)(x 1) =
MTH 141 Applied Calculus
Exam 3 - Chapter 3 (Sections 3.5 to 3.7)
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_
1. (3.5 points) Find the number of units x that produces a maximum revenue R, where R = 48x2 0.02x3
Solution:
R = 48x2 0.02x3
R0 = 96x 0.06x2
R00 = 96 0.
MTH 141 Applied Calculus
Exam 3 - Chapter 3 (Sections 3.1 to 3.4)
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_
1. (3 + 4 points) Find the critical numbers and the open intervals on which the function is increasing or
decreasing.
(a) g (x) = x2 2x + 3
Solution: The