# Section 5.1 - The First Derivative ```Math 142 Lecture Notes for Section 5.1
Section 5.1 -
1
The First Derivative
From previous lectures we can recall the following facts about the first derivative: Consequently, we
can use these facts to analyze the behavior of a function simply by understanding what the derivative
is telling us.
Definition 5.1.1:
A partition number of f 0 (x) is a value of x such that
Definition 5.1.2:
A critical value of f 0 (x) is a value of x such that
Strategy for using partition numbers and critical values of a function f (x):
• Step 1:
• Step 2:
• Step 3:
• Step 4:
Math 142 Lecture Notes for Section 5.1
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Example 5.1.3:
Find the critical values and partitions numbers for the following functions and then determine where
the function is increasing or decreasing.
(a) f (x) = 3x2 − 5x + 3
(b) f (x) =
x+4
x−2
(c) y = x ln x3 − 3x
(d) f (x) =
(x − 3)2
− ln(x − 3)
2
Math 142 Lecture Notes for Section 5.1
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Definition 5.1.4:
Relative (Local) Extrema:
• For a function f , a point f (c) is called a relative (local) maximum if there exists an interval
(m, n) containing c such that
• For a function f , a point f (c) is called a relative (local) minimum if there exists an interval
(m, n) containing c such that
• if f (c) is either a relative (local) maximum or minimum, then we can call f (c) a relative (local)
extremum of f .
First Derivative Test:
to
(1) If f 0 (x) changes from
and at x = c there is a
at x = c, then we have that f (x) is going from
.
to
to
(2) If f 0 (x) changes from
and at x = c there is a
at x = c, then we have that f (x) is going from
.
to
(3) If the sign of f 0 (x) is the same on both sides of x = c, then at x = c there is
.
All of this material adds a fifth step to the process discussed previously. This step is:
Step 5:
Math 142 Lecture Notes for Section 5.1
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Example 5.1.5:
Determine the intervals where the following functions are increasing and decreasing and find the local
extrema.
(a) f (x) =
3x4 + 8x3 − 30x2 − 72x
.
12
(b) y =
4
− 25
x2
Math 142 Lecture Notes for Section 5.1
(c) g(x) = (x − 3)ex
Suggested Homework: 1-41(odd), 45-53(odd), 61, 63, 73, 75, 87, 89.
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