Download Math 165 Exam 1 Study Guide Name: Date: 1. Find the limit (if it

Math 165
Exam 1 Study Guide
Name:
Date:
1. Find the limit (if it exists).
(a)
x 3
x!0 x + 3
(b)
lim
x2 + 2x 3
(c) lim 2
x! 3 x + 4x + 3
(d)
x+4
x!4 x
4
lim
lim
x!0
p
x+9
x
3
2. State the interval(s) on which the function is continuous. Explain why the function is
continuous on the interval(s). If there are any discontinuities, determine whether they are
removable.
x2 25
3
(a) f (x) =
(b) g(x) =
x 5
x+1
(c) f (x) =
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(
1
x
x,
x2 ,
x<2
x 2
(d) f (x) =
1/5
|x
x
3|
3
Math 165 Exam 1 Study Guide.pdf (#23)
3. Use the limit definition to find the derivative of the function.
p
(a) f (x) = 2x2 3
(b) f (x) = x + 4
4. Find the derivative of each function.
(a) f (t) = t4 + 3t
(b) h(x) = (x
(c) g(x) = x3/4
(d) f (x) = (5x2 + 7)3
(e) f (x) =
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2
(2 9x)3
(f) f (x) =
2/5
2)(x2
4x + 1)
(7x + 3)3
2x 1
Math 165 Exam 1 Study Guide.pdf (2/5)
5. Find an equation of the tangent line to the graph of
f (x) = 2x
2
x+1
at the point (1, 1).
6. The monthly demand function p and cost function C for x newspapers are given by
p=5
0.001x and C = 35 + 1.5x.
(a) Find the monthly revenue R as a function of x.
(b) Find the monthly profit P as a function of x.
(c) Find the marginal profit when x = 1200.
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3/5
Math 165 Exam 1 Study Guide.pdf (3/5)
7. The weekly total cost in dollars incurred by a company in manufacturing x compact discs
is
C(x) = 4000 + 3x
0.0001x2
0  x  10, 000
(a) Find the additional cost when the manufacturing increases from 2000 to 2001 discs.
(b) Find the marginal cost when x = 2000.
8. Find the third derivative of each function.
p
(a) f (x) = 5 x
(b) f (x) =
3x + 1
4x 1
9. Use implicit di↵erentiation to find dy/dx.
(a) y 2
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x2 + 8x
9y
(b) x2 + 9xy + y 2 = 3
1=5
4/5
Math 165 Exam 1 Study Guide.pdf (4/5)
10. A (square) baseball diamond has sides that are 90 feet long. A player 26 feet from third
base is running at a speed of 30 feet per second. At what rate is the player’s distance from
home plate changing?
x
•
s
90 ft
11. A company is increasing the production of a product at the rate of 25 units per week.
The demand and cost functions for the product are given by p = 50 0.01x and C =
4000 + 40x 0.02x2 . Find the rate of change of the profit with respect to time when the
weekly sales are x = 800 units.
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5/5
Math 165 Exam 1 Study Guide.pdf (5/5)