Math 1325
Professor Merrill
Lab 1
Show all work
Name ____________________________________
Due on Exam 1 day or class after Exam 1
Show all work on this paper. Box/Circle your answers.
The graphs of the functions f and g are given below.
graph of f
graph of g
1. Find each of the following values.
f (2)  __________
g(2)  __________
f (1)  __________
g(1)  __________
f (1)  __________
g(1)  __________
lim f (x)  __________
x  2
lim g(x)  __________
x  2
lim f (x)  __________
x  2
lim g(x)  __________
x  2
lim f (x)  __________
x  2
lim g(x)  __________
x  2
lim f (x)  __________
x  1
lim g(x)  __________
x  1
lim f (x)  __________
x  1
lim g(x)  __________
x  1
lim f (x)  __________
x  1
lim g(x)  __________
x  1
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lim f (x)  __________
x  +1
lim g(x)  __________
x  +1
lim f (x)  __________
x  +1
lim g(x)  __________
x  +1
lim f (x)  __________
x  +1
lim g(x)  __________
x  +1
2. If a function is continuous at a point, does this mean that the limit must exist at this point? Explain.
3. If the limit exists at a point, does this mean that it must be continuous at that point? Explain.
4. If a function is continuous at a point, must the function be defined at that point? Explain.
5. If a function is defined at a point, must the function be continuous at that point? Explain.
6. The graph of a function f is given below. Find all points at which the following function is discontinuous
at x. Explain your answer.
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7.
Find the following limits.
a.
c.
e.
x 3
lim
x 3 x 2  4x  3
b.
2
x
lim
d.
x  4  2
x2
3
1
lim
x  2  1
x
3
2x  7
lim
x  x 3  x 2  x  7
f.
x 2  5x  6
x 2
x 2
h. lim
4
9x  x
lim
x  2x 4  5x 2  x  6
4x 2  5x  2
x  7x 2  1000
g. lim
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lim  x  31973
x 4
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8. Find the first derivative of the function f(x) = 3x2 – x + 27 using the limit definition.
9. An object is dropped from the top of a 100m tower. Its height above the ground after t seconds is given
by f(x) = 100-4.9t2m.
a. Find the average rate of change from 1-3 seconds.
b. Find the instantaneous rate of change at t = 2 seconds.
10. The graph of a function f is given below. Find all points in [4, 4] at which f is not differentiable at x
(no derivative exists). Explain your answer.
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11. Approximate the slope of each tangent line in the graphs below.
m tan  __________
m tan  __________
m tan  __________
m tan  __________
12. Draw a line that is tangent to the graph of y  f (x) at ( (3, 2) and approximate the slope of this line.
Then find an equation of the line that is tangent to the graph of y  f (x) at (3, 2)
m tan  __________
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13. Match the graph of each function in (a) – (d) with the graph of its derivative in I – IV.
(a) _____
(b) _____
(c) _____
(d) ____
14. Find the derivative of each function. Your final answer should not have fractional or negative
exponents.
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2
a. f ( x)  x3  4 x 2  3 x  4
b. f (x )  4
x
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15. Suppose that f (x)  1  2x 1  2x  . Find f ( x) .
16. Suppose that f (x)  2x  1 . Find f ( x) .
2x  1
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17. For f (x)  2x  x 2 , answer the following questions:
a. Find the average rate of change of f (x)  2x  x 2 from x  2 to x  3 .
b. Find the instantaneous rate of change of f (x)  2x  x 2 at x  2 .
c.
Find f ( x) .
d. Find f  ( 2)
18. For f (x)  4  3x  x 3 :
a. Find the slope of the tangent line at x  1
b. Find an equation of the tangent line to the graph of f (x)  4  3x  x 3 at the point (1, 0) .
19. Describe what is happening and determine if the limit exists.
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1000
 1000 where q represents the
q2
demand for the produce. Find the marginal revenue when the demand is 10.
20. If the price in dollars of a stereo system is given by p(q) 
21. An analyst has found that a company’s costs and revenues in dollars for one product are given by
x2
, respectively, where x is the number of items produced.
C(x)  2x
and
R(x)  6x 
1000
a.
Find the marginal cost
b. Find the marginal revenue
c.
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Using the fact that profit is the difference between revenue and costs, find the marginal profit
function.
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