College Mathematics: Learning Worksheets Chapter 11
303
Name ________________________________ Date ______________ Class ____________
Goal: To use the first derivative to analyze graphs
1. Given the function 3
() 3 81 13fx x x=−+, find
a) ‘( )
f
x
Section 11-1 First Derivatives and Graphs
Theorem 1: Increasing and Decreasing Functions
For the interval (a, b), if ‘( ) 0,fxthen ()
f
xis increasing and the graph of f rises.
If ‘( ) 0,fxthen ()
f
xis decreasing and the graph of f falls.
Definition: Critical Values
The values of x in the domain of f where ‘( ) 0fxor where ‘( )
f
xdoes not exist are
Procedure: First Derivative Test for Local Extrema
Let c be a critical value of f. Construct a sign chart for ‘( )
f
xclose to and on either
side of c. Then ()
f
cwill be
a) A local minimum if ‘( )
f
xchanges from negative to positive.
f
College Mathematics: Learning Worksheets Chapter 11
304
b) The critical values of the function
2
2
2
09 81
81 9
9
3
x
x
x
x
=−
=
=
±=
c) The partition numbers for the first derivative
2. Given the function 21
() 3
fx
x
=
+ find
a) ‘( )
f
x
2
(3)
‘( ) 21( 3) (1) x
fx x or
+
=− + −
b) The critical values of the function
c) The partition numbers for the first derivative
College Mathematics: Learning Worksheets Chapter 11
305
In Problems 3–5, for the functions given, find:
a) The intervals on which the function is increasing
b) The intervals on which the function is decreasing
c) The local extrema
3. 2
() 5 20 12fx x x=+−
First, find the derivative and the critical values as follows:
‘( ) 10 20
010 20
20 10
2
fx x
x
x
x
=+
=+
−=
−=
Choose a number on each side of the critical value to find the value of the first
derivative:
Test number: –3 Test number: 0
‘( 3) 10 20
‘( 3) 10( 3) 20
‘( 3) 30 20
‘( 3) 10
fx
f
f
f
−= +
−= −+
−=− +
−=−
‘(0) 10 20
‘(0) 10(0) 20
‘(0) 0 20
‘(0) 20
fx
f
f
f
=+
=+
=+
=
a) The test number 0 has a first derivative that is a positive value, therefore, the
College Mathematics: Learning Worksheets Chapter 11
306
4. 3
() 3 81 13fx x x=−+
First, find the derivative and the critical values as follows:
2
2
2
2
‘( ) 9 81
09 81
81 9
9
3
fx x
x
x
x
x
=−
=−
=
=
±=
The partition numbers (also critical values of f) are 3x and 3.x Choose a
number within each interval to find the value of the first derivative:
Test number: –4 Test number: 0 Test number: 4
2
‘( 4) 9( 4) 81
‘( 4) 9(16) 81
‘( 4) 144 81
‘( 4) 63
f
f
f
f
−=− −
−= −
−= −
−=
2
‘(0) 9(0) 81
‘(0) 9(0) 81
‘(0) 0 81
‘(0) 81
f
f
f
f
=−
=−
=−
=−
2
‘(4) 9(4) 81
‘(4) 9(16) 81
‘(4) 144 81
‘(4) 63
f
f
f
f
=−
=−
=−
=
a) The test numbers –4 and 4 have a first derivative that is a positive value,
College Mathematics: Learning Worksheets Chapter 11
5. 43
() 3 4 8fx x x=+−
First, find the derivative and the critical values as follows:
32
32
2
‘( ) 12 12
012 12
012 ( 1)
0, 1
f
xxx
x
x
xx
x
=+
=+
=+
−=
The partition numbers (also critical values of f ) are 0x and 1.x=− Choose a
number within each interval to find the value of the first derivative:
Test number: –2 Test number: –0.5 Test number: 2
32
‘( 2) 12( 2) 12( 2)
‘( 2) 12( 8) 12(4)
‘( 2) 96 48
‘( 2) 48
f
f
f
f
−= − + −
−= −+
−=− +
−=−
32
11 1
22 2
111
284
3
1
22
3
1
22
‘( ) 12( ) 12( )
‘( ) 12( ) 12( )
‘( ) 3
‘( )
f
f
f
f
−= − + −
−= −+
−=−+
−=
32
‘(2) 12(2) 12(2)
‘(2) 12(8) 12(4)
‘(2) 96 48
‘(2) 144
f
f
f
f
=+
=+
=+
=
College Mathematics: Learning Worksheets Chapter 11
In Problems 6–7, for the functions given, find
a) The intervals on which the function is increasing
b) The intervals on which the function is decreasing
c) Sketch the graph of the function
6. 32
() 2 3 12
f
xxx x
First, find the derivative and the critical values as follows:
2
2
2
‘( ) 6 6 12
06 6 12
06( 2)
06( 2)( 1)
2,1
fx x x
xx
xx
xx
x
The partition numbers (also critical values of f ) are 2x and 1.xChoose a
number within each interval to find the value of the first derivative:
Test number: –3 Test number: 0 Test number: 2
2
‘( 3) 6( 3) 6( 3) 12
‘( 3) 6(9) 6( 3) 12
‘( 3) 54 18 12
‘( 3) 24
f
f
f
f
2
‘(0) 6(0) 6(0) 12
‘(0) 6(0) 6(0) 12
‘(0) 0 0 12
‘(0) 12
f
f
f
f
2
‘(2) 6(2) 6(2) 12
‘(2) 6(4) 6(2) 12
‘(2) 24 12 12
‘(2) 24
f
f
f
f
a) The test numbers –3 and 2 have a first derivative that is a positive value,
309
7. 3
() 12 4
f
xx x
First, find the derivative and the critical values as follows:
2
2
2
2
‘( ) 3 12
03 12
12 3
4
2
fx x
x
x
x
x
The partition numbers (also critical values of f ) are 2x and 2.xChoose a
number within each interval to find the value of the first derivative:
Test number: –3 Test number: 0 Test number: 3
2
‘( 3) 3( 3) 12
‘( 3) 27 12
‘( 3) 15
f
f
f
2
‘(0) 3(0) 12
‘(0) 0 12
‘(0) 12
f
f
f
2
‘(3) 3(3) 12
‘(3) 27 12
‘(3) 15
f
f
f
a) The test numbers –3 and 3 have a first derivative that is a positive value,
College Mathematics: Learning Worksheets Chapter 11
310
College Mathematics: Learning Worksheets Chapter 11
311
Name ________________________________ Date ______________ Class ____________
Goal: To use the second derivative to analyze graphs
Section 11-2 Second Derivatives and Graphs
Notation: Second Derivative
For (),yfxthe second derivative of f, provided that it exists, is ”( ) ‘( ).
d
f
xfx
dx
Definition: Concavity
The graph of a function f is concave upward on the interval (a, b) if ‘( )
f
xis
increasing on (a b) and is concave downward on the interval (a, b) if ‘( )
f
xis
decreasing on (a, b).
Summary: Concavity
For the interval (a,b), if ”( ) 0,fxthen ‘( )
f
xis increasing and the graph of f is
f
Theorem: Inflection Point
If ( )
y
fxis continuous on (a, b) and has an inflection point at ,
x
cthen either
”( ) 0fcor ”( )
f
cdoes not exist.
Procedure: Graphing Strategy (first version)
f
Step 2. Analyze ‘( ).
f
xFind the partition numbers for, and the critical values of,
‘( ),
f
x and determine local extrema.
f
f
Step 4. Sketch the graph of the function f.
College Mathematics: Learning Worksheets Chapter 11
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1. Given the function 43
() 7 11 15 3,fx x x x=− −+find ”( ).
f
x
2. Given the function 185
5
() 4 12 19,fx x x=+ −find ”( ).
f
x
3. Given the function 32
() 2 18 7 11,fx x x x=− +−find the x and y coordinates of all
inflection points.
Inflection points are found by setting the second derivative equal to zero.
Now check the value of the second derivative on both sides of 3.
College Mathematics: Learning Worksheets Chapter 11
In Problems 4–6, for the functions given, find
a) The intervals on which the function is concave upward
b) The intervals on which the function is concave downward
c) The inflection points
4. 42
() 2 48
f
xx x=−
First, find the second derivative and the partition numbers as follows:
3
2
2
2
2
‘( ) 8 96
”( ) 24 96
024 96
96 24
4
2
f
xx x
fx x
x
x
x
x
=−
=−
=−
=
=
±=
The partition numbers are 2x=− and 2.x= Choose a number on each side of the
partition numbers. Find the value of the second derivative at these test numbers:
Test number: –4 Test number: 0 Test number: 4
2
‘‘( 4) 24( 4) 96
‘‘( 4) 24(16) 96
‘‘( 4) 384 96
‘‘( 4) 288
f
f
f
f
−= − −
−= −
−= −
−=
2
‘‘(0) 24(0) 96
‘‘(0) 24(0) 96
”(0) 0 96
‘‘(0) 96
f
f
f
f
=−
=−
=−
=−
2
‘‘(4) 24(4) 96
‘‘(4) 24(16) 96
‘‘(4) 384 96
‘‘(4) 288
f
f
f
f
=−
=−
=−
=
a) The test numbers –4 and 4 have a second derivative that is a positive value,
College Mathematics: Learning Worksheets Chapter 11
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5. 32
() 3 18 4 10fx x x x=− +−
First, find the second derivative and the partition numbers as follows:
2
‘( ) 9 36 4
”( ) 18 36
018 36
36 18
2
fx x x
fx x
x
x
x
=−+
=−
=−
=
=
The partition number is 2.x= Choose a number on each side. Find the value of the
second derivative at these test numbers:
Test number: 0 Test number: 4
‘‘(0) 18(0) 36
”(0) 0 36
‘‘(0) 36
f
f
f
=−
=−
=−
‘‘(4) 18(4) 36
‘‘(4) 72 36
‘‘(4) 36
f
f
f
=−
=−
=
315
6. 2
() 16 2
x
x
f
xee=−
First, find the second derivative and the partition numbers as follows:
2
2
2
2
‘( ) 16 4
”( ) 16 8
016 8
816
2
ln 2
x
x
x
x
x
x
xx
x
f
xee
f
xee
ee
ee
e
x
=−
=−
=−
=
=
=
The partition number is ln 2.x= Choose a number on each side. Find the value of the
second derivative at these test numbers:
Test number: 0 Test number: 2
(0) 2(0)
00
‘‘(0) 16 8
‘‘(0) 16 8
‘‘(0) 16 8
”(0) 8
fee
fee
f
f
=−
=−
=−
=
(2) 2(2)
24
‘‘(2) 16 8
‘‘(2) 16 8
‘‘(2) 118.225 436.785
‘‘(2) 318.56
fee
fee
f
f
=−
=−
=−
=−
College Mathematics: Learning Worksheets Chapter 11
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In Problems 7–9, apply the four-step graphing procedure and sketch the graph of the
following functions.
7. 2
() ( 2)( 4)fx x x x
Step 1: Analyze ().
f
x The domain of the function is all real numbers. To find the
intercepts, first find (0).f Next, find the solutions of () 0.fx
Step 2: Analyze the first derivative.
Test number: –2 Test number: 0 Test number: 2
2
‘( 2) 3(4) 4 6
‘( 2) 2
f
f
‘(0) 3(0) 0 6
‘(0) 6
f
f
‘(2) 3(4) 4 6
‘(2) 10
f
f
3
3
and has a local maximum at 119
3.x
College Mathematics: Learning Worksheets Chapter 11
317
Step 3: Analyze the second derivative
1
3
”( ) 6 2
06 2
26
fx x
x
x
x
The partition number is 1
3.x Choose a number on each side of the partition
number. Find the value of the second derivative at these numbers:
Test number: –1 Test number: 0
Step 4: Sketch the function
318
8. 43
() 5 10
f
xx x
Step 1: Analyze ().
f
x The domain of the function is all real numbers. To find the
intercepts, first find (0).f Next, find the solution of () 0.fx
43
(0) 5(0) 10(0)
f
43
05 10
x
x
Step 2: Analyze the first derivative.
Test number: –1 Test number: 1 Test number: 2
32
‘( 1) 20( 1) 30( 1)
f
32
‘(1) 20(1) 30(1)
f
32
‘(2) 20(2) 30(2)
f
Therefore, the function is increasing on the interval 3
2
(, )and decreasing on the intervals
College Mathematics: Learning Worksheets Chapter 11
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Step 3: Analyze the second derivative
32
f
‘( ) 20 30
f
xxx
Test number: –1 Test number: 0.5 Test number: 2
2
”( 1) 60( 1) 60( 1)
f
2
‘‘(0.5) 60(0.5) 60(0.5)
f
2
‘‘(2) 60(2) 60(2)
f
Step 4: Sketch the function
320
9. 2
() 8
x
x
f
xee
Step 1: Analyze ().
f
x The domain of the function is all real numbers. To find the
intercepts, first find (0).f Next, find the solutions of () 0.fx
02(0)
(0) 8
x
fee
2
08
x
x
ee
Step 2: Analyze the first derivative.
2
2
‘( ) 8 2
08 2
x
x
x
x
x
f
xee
ee
Test number: 0 Test number: 2
02(0)
‘(0) 8 2
‘(0) 4
fee
f
22(2)
‘(2) 8 2
‘(2) 50.08
fee
f
College Mathematics: Learning Worksheets Chapter 11
Step 3: Analyze the second derivative
2
2
x
‘( ) 8 2
”( ) 8 4
x
x
x
x
x
f
xee
f
xee
322
10. A company estimates that it will sell ( )Nxunits of a product after spending x
thousand dollars on advertising, as given by
43 2
( ) 0.25 23 360 45,000Nx x x x 25 54.x
When is the rate of change of sales increasing and when is it decreasing? What is the
point of diminishing returns and the maximum rate of change of sales?
To find when the rate of change is increasing and decreasing, find the second
derivative and use the second derivative test.
2
2
‘‘( ) 3 138 720
0 3( 46 240)
0 3( 40)( 6)
6, 40
Nx x x
xx
xx
x
The partition numbers are 6x and 40.x Choose a number on each side of the
partition numbers within the given domain. Find the value of the second derivative at these
test numbers:
Test number: 39 Test number: 41
2
‘‘(39) 3(39) 138(39) 720
‘‘(39) 4563 5382 720
‘‘(39) 99
f
f
f
2
‘‘(41) 3(41) 138(41) 720
‘‘(41) 5043 5658 720
‘‘(41) 105
f
f
f