CH 3 Applications of Differentiation
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3-3 Increasing and Decreasing Functions (First
Derivative Test)
3-1 Extrema on an Interval
3-4 Concavity (Second Derivative Test)
3-6 A Summary of Curve Sketching
2-2/2-3 Velocity and Acceleration (Particle Motion)
3-2 Mean Value and Rolle’s Theorem
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3-5 Limits at Infinity (Already covered)
3-7 Optimization Problems (SKIP for now)
3-9 Differentials (SKIP for now)
Sect. 3-3
Increasing & Decreasing Fuctions
Goals:
• To determine the intervals on which a function is
increasing or decreasing.
• To apply the First Derivative Test to find relative
extrema of a function.
The inc\dec concept can be associated with
the slope of the tangent line. The slope of
the tangent line is positive when the function
is increasing and negative when decreasing
m0
m0
f ' ( x)  0
f ' ( x)  0
m0
f ' ( x)  0
3.3
Let x1 and x2 be numbers in the domain of a
function, f.
The function is
If f ' ( x)  0 for all x in (a, b),
increasing if for
every
x1 <
then f ( x) is increasing on [a, b].
x2 in an open
interval, f(x1) <
The function is
f(x2).
If f ' ( x)  0 for all x in (a, b),
decreasing
if for
every x1 < x2 in
then f ( x) is decreasing on [a, b].
an interval, f(x1) >
f(x2).
The function is
If f ' ( x)  0 for all x in (a, b),
constant if for every
then f ( x) is constant on [a, b].
x1 < x2 in the
interval, f(x1) =
f(x) is not defined at
x = 0, therefore, 0 is
NOT a critical #.
y = x2/3
Definition of critical numbers:
Let f be defined at c. If f(c)  0 or if f(x) is not
differentiable at c (f ’(c) DNE), then c is a critical #.
Back to
Warm Up
f '( x)  3x 4  4 x3
critical numbers x  - 1, 0
f (-1), f (0) defined &
f ' (1)  0, f ' (0)  0
-
+
0
-1
f(x)
0
+
0
Increasing [-1, ∞)
f’(x)
Back to Warm Up
2
f '( x)   3
x
f ' (0) DNE but
f (0) NOT defined
 x  0 is NOT a critical #.
+ DNE 0
Increasing (-∞, 0)
f(x)
f’(x)
Back to Warm Up
f '( x) 
2
33 x
critical number x  0
f (0) defined &
y = x2/3
-
f ' ( 0)  0
f(x)
DNE +
0
Increasing [0, ∞)
f’(x)
Ex 1: Find where the function f ( x)  x3  9 x 2  24 x
is increasing and decreasing.
f ( x)  x3  9 x 2  24 x
f ' ( x)  3x  18x  24
2
0  3( x  6 x  8)
0  ( x 2  6 x  8)
0  ( x  4)( x  2)
2
+ 0
-
2
f(x)
0
+
4
inc (,2]  [4, )
dec (2,4)
f’(x)
Example 2: Determine the intervals on which
f ( x)  x  3xis
. decreasing.
3
x2 1
f ( x) 
3
f +
f
3
x
3
 3x 
DNE
 3
critical points: x  1
2
+ 0
-1
x  0,  3
-
DNE
-
0
dec [1,0], [0,1]
0 +
1
DNE
3
+
Graph of f ( x)  x  3 x .
3
3
y
2
dec [1,0], [0,1]
1
x
-2
-1
1
-1
-2
-3
2
3
Example 3: Determine the intervals on which
x 1
is decreasing.
f ( x)  2
x
f ( x)  x 2  x 2
2
f '( x)  2 x  3
x
 x 4  1  2( x 2  1)( x 2  1)
1 

f '( x)  2  x  3   2  3  
3
x 
x

x


4
 f '( x)  0 at x  1
f '( x) DNE at x  0
x4  1
f ( x)  2
x
INT .
 , 1
f '( x) f '(2)  0
f (x)
Decreasing
2( x 2  1)( x 2  1)
f ' x 
x3
 1, 0
 0,1
f '(1/ 2)  0 f '(1/ 2)  0
Decreasing
Increasing
x  0, 1
1, 
f '(2)  0
Increasing
Example 4: Determine the intervals on (0, 2π)
1
where f ( x)  x  sin x is increasing.
2
1
 5
1
cos x   x  ,
f '( x)   cos x  0
2
3 3
2
Interval
 
 0, 
 3
Sign of f '( x)
negative
Conclusion
Decreasing
positive
 5

, 2 

 3

negative
Increasing
Decreasing
  5 
 , 
3 3 
  5 
inc  , 
3 3 
1
y  x  sin x
2
1

3
5
3
Example 5: Determine the intervals on which
x
is increasing.
h( x ) 
x 1

x  11  x 1
1
h( x) 

2
2
x  1
x  1
0
Which of the following statements is correct? Explain.
a) x  1 makes f ' ( x) discontinuous,
therefore it is a critical point.
b) x  1 makes f ' ( x) discontinuous,
but it is NOT a critical point.
Although there are no critical points, you still set up the
intervals using the undefined point x = -1 and create test
points.
+ DNE +
-2
-1
f(x)
f’(x)
0
inc  ,1   1,  
Guidelines for Finding Intervals on Which a
Function Is Increasing or Decreasing
You Try…
Find the intervals of inc/dec for each function.
1. f ( x )  x  1
2
3. f ( x)  2 x  3 x
2
3
x2
2. f ( x) 
x 1
4. f ( x)  2sin x  cos 2x, [0, 2]
Solution 2
x2
f ( x) 
x 1
Domain:
x ≠ 1, xR
2 x( x  1)  x 2 x 2  2 x x( x  2)


f ' ( x) 
2
2
( x  1)
( x  1)
( x  1)2
Critical number:
x = 1, x = 0, and x = 2
f ' (0)  f ' (2)  0, f ' (1) does not exist
inc [-, 0]  [2, ]
dec [0, 1)  (1, 2]
f ’(0) does not exist
Solution 3
f ( x)  2 x  3x
2
3
 x 3 1
2
f ' ( x)  2  1  2 1 
 3 
3
x
 x 
1
Critical number:
x
f ‘(x)
x = 0 and x = 1
(-∞, 0)
f ‘(–1) > 0
inc [-, 0]  [1, ]
(0, 1)
(1, ∞)
f ‘(0.5) < 0
dec [0, 1]
f’ (2) > 0
Solution 4.
4. f ( x)  2sin x  cos 2x, [0, 2]
Where is f '( x) undefined or where does f '( x)  0?
f '( x)  2 cos x  ( sin 2 x)(2)
 2cos x  2sin 2x  0
cos x  sin 2x  0
cos x  2sin x cos x  0
cos x(1  2sin x)  0
 cos x  0 or 1 2sin x  0

3
x
or
2
2
1
7
11
sin x 
x
or
2
6
6
    7 3  11

inc 0,    ,   
,2 
 2  6 2   6

  7   3 11 
dec  ,    ,

2
6
2
6

 

Relative Extrema
A function f has a relative (local) maximum at x  c if
there exists an open interval (r, s) containing c such that
f (x) < f (c) for all x in (r, s).
Relative
Maxima
Relative
Minima
A function f has a relative (local) minimum at x  c if
there exists an open interval (r, s) containing c such that
f (x) > f (c) for all x in (r, s).
NOTE: The relative extrema occur only at the critical numbers.
The First Derivative Test
Back to
Warm Up
f '( x)  3x 4  4 x3
-
0
-1
relative
min
(-1, -1)
+
f(x)
0
0
+
f’(x)
NO
relative
max
Back to
Warm Up
2
f '( x)   3
x
+ DNE 0
relative
max ??
f(x)
f’(x)
NO
relative
max or
min
Back to
Warm Up
y = x2/3
f '( x) 
2
33 x
-
f(x)
0
0
relative
min
(0, 0)
+
f’(x)
Back to Ex 1: Find where the function f ( x)  x3  9 x 2  24 x
has a relative max or min.
f ' ( x)  3x  18x  24
2
+ 0
2
relative
max (2,20)
-
f(x)
0
+
4
relative
min (4, 16)
f’(x)
Back to Ex 2: Find all the relative extrema of
f ( x)  3 x 3  3 x .
x2 1
f ( x) 
3
x
3
 3x 
2
Relative min.
Relative max.
f (1)   3 2
f (1)  3 2
f
f
+
ND + 0
 3
-1
+
ND
0
1
-
0
+ ND
3
Graph of f ( x)  x  3 x .
3
3
y
2
Local max.
(1, 3 2)
1
x
-2
-1
1
2
3
-1
-2
-3
Local min.
(1,3 2 )
Back to Ex 3: Determine the relative minima of
x 1
f ( x)  2 .
x
1 

f '( x)  2  x  3 
x 

4
INT .
 , 1
f '( x) f '(2)  0
f (x)
Decreasing
 1, 0
 0,1
1, 
f '(1/ 2)  0 f '(1/ 2)  0
Decreasing
Increasing
relative
min (-1,2)
f '(2)  0
Increasing
relative
min (1,2)
Back to Ex 4: Determine the relative extrema of
1
f ( x)  x  sin x on (0, 2π).
2
1
f '( x)   cos x  0
2
Interval
positive
 5

, 2 

 3

negative
Increasing
Decreasing
  5 
 , 
3 3 
 
 0, 
 3
Sign of f '( x)
negative
Conclusion
Decreasing
relative
min
relative
min
3  3 3
  
f   

6
3 6 2
 5
f
 3
1
Relative Max
 5 
,?

 3 
3 5  3 3
 5



2
6
 6

3
 
 ,? 
3 
Relative Min
1
y  x  sin x
2
5
3
Back to Ex 5: Determine the relative extrema of
x
h( x ) 
.
x 1
1
h( x) 
2
x  1
+ DNE +
-2
-1
0
no relative
extrema
f(x)
f’(x)
You Try…
Find the relative max/min for each function.
1. f ( x )  x  1
2
3. f ( x)  2 x  3 x
2
3
x2
2. f ( x) 
x 1
4. f ( x)  2sin x  cos 2x, [0, 2]
You Try…
Determine the intervals on which the following
functions are increasing. Calculate the relative
max/min for each function.
1.
f (x)  (x  2)2 3  1
2.
1
f ( x)  x 
x
3.
1 4
f ( x)  x  3x 2  4 x  8
2
Solution 1
f (x)  (x  2)
2 3
1
1 3
2
f (x)  x  2 
3
2
f (x)  3
3 x2
0
f(x)
DNE +
2
4
f’(x)
f (x) DNE when x = 2
f (x) ≠ 0 thus, x = 2 is
the only critical value.
inc [2, ]
rel min (2,1)
1
1 2
x
2
x 1
Solution 2
1
f ( x)  x 
x
1
f ( x)  1  2  0
x
-
+

x  1, x  1
-1
relative
max (-1,-2)
0
+
1
relative
min (1,2)
inc  ,1  (1, )
f (x)

1
f ( x)  x 
x
-1
(1,2) is a relative min
1
(-1,-2) is a relative max
Solution 3
1 4
2
f ( x)  x  3x  4 x  8
2
f ( x)  2 x 3  6 x  4  0
-
x 3  3x  2  0
+
-2
1 1 0 -3 2
+1 +1 -2
1 1 -2 0
+
f (x)
1
relative
min
x2  x  2  0
x  2x  1  0
x  2, x  1
critical points
x = 1 and -2
f (2) 
1
 24  3 22  4(2)  8  20
2
4y
3
2
1
4
3
2
1
Rel Min (-2, 20)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
1
Neither
2
3
4
x
5
Closure
Explain the steps for finding
the relative extrema of a function.