Section 3.2 Rates of Change
Average Rate of Change The average rate of change of y = f (x) with respect to x from a to b
is the quotient
∆y
f (b) − f (a)
change in y
=
=
change in x
∆x
b−a
Note: The average rate of change is also the slope of the secant line that passes through the point
(a, f (a)) and (b, f (b)).
Example 1: Using f (x) defined below, determine the average rate of change of f (x) as x changes from
4 to 1.
20
f (x) = 7 −
x
Example 2: Calculate the average rate of change of the given function over the given interval.
Interval: [0, 3]
x
0
1
2
3
4
5
f (x)
5
16
13
14
21
22
Example 3: Calculate the average rate of change of the given function over the given interval. Specify
the units of measurement. Round your answer to three decimal places.
Interval: [2, 6]
t (months)
2
4
6
8
10
12
R(t) ($)
20,300
24,300
19,600
20,200
22,800
20,200
Example 4: Calculate the average rate of change of the given function over the given interval. Specify
the units of measurement. (Assume a = 10, b = 16, and c = 22. Round your answer to the nearest
cent.)
Interval: [1, 4]
Company A’s Stock Price ($)
Instantaneous Rate of Change The instantaneous rate of change of y = f (x) with respect to
x at x = a is given by
f (a + h) − f (a)
lim
h→0
h
Note: The instantaneous rate of change is also the slope of the tangent line at the point (a, f (a)).
Example 5: The cost (in dollars) of producing x units of a certain commodity is given below.
C(x) = 9000 + 12x + 0.1x2
(a) Find the average rate of change of C with respect to x when the production is changed from
x = 100 to the given value.
(i) x = 105
(ii) x = 101
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Maya Johnson
(b) Find the instantaneous rate of change of C with respect to x when x = 100. (This is called the
marginal cost.)
Example 6: Find the instantaneous rate of change of the given function at the given point.
f (x) = x2 − 1;x = 3
Slope of the Tangent Line The slope of the line tangent to the curve y = f (x) at the point x = a
is given by
f (a + h) − f (a)
lim
h→0
h
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Fall 2016,
©
Maya Johnson
Example 7: Consider the parabola y = 9x − x2 .
(a) Find the slope of the tangent line to the parabola at the point (1, 8).
(b) Find an equation of the tangent line in part (a).
Example 8: For the curve y =
1
x
(a) Find the slope of the tangent line to the curve at the point x = 1.
(b) Find an equation of the tangent line in part (a).
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Fall 2016,
©
Maya Johnson
Example 9: Three slopes are given. For each slope, determine at which of the labeled points on the
graph the tangent line has that slope. (Assume a = 7.)
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Fall 2016,
©
Maya Johnson