Math 142 Lecture Notes for Section 5.2
Section 5.3 -
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The Second Derivative
Definition 5.3.1:
If f is a differentiable function, then its derivative f 0 is also a function, so f 0 may have a
derivative of its own, denoted by (f 0 )0 = f 00 . This new function f 00 is called the second
derivative of f because it is the derivative of the derivative of f .
f 00 (x) = y 00 =
d2 y
d dy
= 2.
dx dx
dx
Example 5.3.2:
Find the second derivative of the following functions:
(a) y = 4x4 − 2x3 + x2 + 5x
(b) 4x3 ln x
Math 142 Lecture Notes for Section 5.2
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Definition 5.3.3:
The graph of a function f is concave upward on the interval (a, b) if
on (a, b) and is concave downward on the interval (a, b) if
(a, b).
on
Example 5.3.4:
Given f (x) = 2x3 − 4x2 + 5x − 1, determine the interval where f 0 (x) is increasing and
decreasing.
Definition 5.3.5 (Test for Concavity):
(1) If f 00 (x) > 0, then
on (a, b).
for all x on (a, b), then f (x) is
(2) If f 00 (x) < 0, then
on (a, b).
for all x on (a, b), then f (x) is
Definition 5.3.6:
An inflection point is a point on the graph of a function where the concavity changes.
Math 142 Lecture Notes for Section 5.2
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Example 5.3.7 (Locating Inflection Points):
(1) Determine the values of x where the second derivative is zero, or the second
derivative is undefined.
(2) Using the tabular method (same as for the first derivative), Come up with a sign
chart for the second derivative.
(3) The point a is an inflection point if the second derivative changes sign AND a is in
the domain of f .
Example 5.3.8:
Determine the intervals where the following functions below are concave up and concave
down, locate any inflection points.
(a) f (x) = x5 − 4x3
Math 142 Lecture Notes for Section 5.2
(b) f (x) = ln(x2 + 7x + 5)
Definition 5.3.9 (Second Derivative Test for Local Extrema):
f 0 (c) f 00 (c) Concavity?
f (c) is
0
+
0
0
0
4
Math 142 Lecture Notes for Section 5.2
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Example 5.3.10:
Determine the type and location of the local extrema of the function
f (x) = −x3 + 24x + 17.
Section 5.2 Suggested Homework: 1-35(odd), 41-45(odd), 57, 63, 71, 73