MATH 1411 notes part 2 PDF

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Notes from second part of semester. Dr Pownuk...

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Math 1411, Calculus I Andrew Pownuk

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Table of contents Table of contents .......................................................................................................................................... 2 1

2

Fermat's theorem (stationary points, critical points) ........................................................................... 9 1.1

(*) Proof ...................................................................................................................................... 27

1.2

Find the critical numbers of f x 

1.3

2  x  Find the critical numbers of f x  2  2x   , if any. ...................................................... 29 2 

1.4

Find the critical numbers of f x  x 2 , if any. ........................................................................ 29

1.5

Find the critical numbers of f x 

1.6

Find the critical numbers of f x 

1.7

Find the critical numbers of f x 

1.8

Find the critical numbers of f x 

1.9

x  16x , if any. ........................................................... 35 Find the critical numbers of f x  3

1.10

Find the critical numbers of f x  2  2 x 

1.11

Find the critical numbers of f x  2  12x 

1.12

Find the critical numbers of

f (x)  12x  3x 2  2x 3

1.13

Find the critical numbers of

f (x)  3x3  9x2  27 x , if any. ................................................. 38

1.14

Find the critical numbers of

f (x)  3sin2x , if any. ................................................................ 39



1 2 x  x , if any. .............................................................. 29 2



 



x2  2x , if any. .............................................................. 30 2



x3  2x , if any. .............................................................. 31 3



x3  4x , if any. .............................................................. 32 3



x3  9x , if any. .............................................................. 34 3



3



3x 2 x 3  , if any............................................. 35 2 3

 

x2 x3  , if any. ......................................... 36 2 3 , if any. .............................................. 37

(*) Extreme Value Theorem ................................................................................................................ 43

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3

 

2.1

Example f x  x 2, a,b   1,2  ....................................................................................... 44    

2.2

Example f x   1 , a,b    1,2  ....................................................................................... 45    x 

Rolle's Theorem the Mean Value Theorem ........................................................................................ 46 3.1

 

Explain why the Roll’s theorem doesn’t apply to the following functions.

f x  3.2

1 , x   1, 1 ......................................................................................................................... 51 x

Explain why the Roll’s theorem doesn’t apply to the following function.

 

f x  1  x  1 , x  0, 2  ............................................................................................................... 53

3.3

Find two x-intercepts of the function f and show that f’(x)=0 at some point between two

 

intercepts f x  x 2  x  2 ................................................................................................................ 55 4

5

The Mean Value Theorem................................................................................................................... 56 4.1

The Mean Value Theorem........................................................................................................... 56

4.2

Example ....................................................................................................................................... 59

4.3

Example ....................................................................................................................................... 61

4.4

Example ....................................................................................................................................... 62

4.5

(**) The Cauchy Mean Value Theorem....................................................................................... 64

First derivative test ............................................................................................................................. 66 5.1

First derivative test ..................................................................................................................... 68

5.2

(*) First derivative test - proof .................................................................................................... 70

5.3

Example ....................................................................................................................................... 78

5.4

Consider the function f x  5  36x  3x2  2x3 (a) Find the critical numbers of (𝑥), if

 

any. (b) Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. ........................................................................... 79 5.5

 

Consider the function f x  1  15x  6x 2  x 3 (a) Find the critical numbers of (𝑥), if any.

(b) Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. ........................................................................... 81 5.6

 

Consider the function f x  3x3  9x2 - 27x (a) Find the critical numbers of (𝑥), if any. (b)

Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. ........................................................................... 83

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5.7

 

Consider the function f x  5  120x  27x2  2x3 (a) Find the critical numbers of (𝑥), if

any. (b) Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. ........................................................................... 86 5.8

 

Consider the function f x  11520 x  13440x2  7600x3  2250x4  336x5  20x6

(a) Find the critical numbers of (𝑥), if any. (b) Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. .. 98 5.9

 

Consider the function f x  12x  3x 2  2x 3 (a) Find the critical numbers of (𝑥), if any.

(b) Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. ......................................................................... 101 5.10

 

Consider the function f x  12x  3x2  2x 3 (a) Find the critical numbers of (𝑥), if any.

(b) Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. ......................................................................... 103 5.11

 

Consider the function f x  2  4x 2  4x 3  x 4 (a) Find the critical numbers of (𝑥), if any.

(b) Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. ......................................................................... 105 5.12

 

Consider the function f x  2  8x 2  x 4 (a) Find the critical numbers of (𝑥), if any. (b)

Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. ......................................................................... 108 5.13

 

Consider the function f x  180x  210x 2  115x 3  30x 4  3x 5 (a) Find the critical

numbers of (𝑥), if any. (b) Find the open intervals on which the function is increasing or decreasing. (c) Apply the First Derivative Test to identify all relative extrema, if any. ........................................... 111

6

 

5.14

Example f x  x  2 sin x .................................................................................................. 114

5.15

(*) Minimum of the function y  x ...................................................................................... 116

Related rates (page 148, section 2.6) ............................................................................................... 117 6.1 All the edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is 10 centimeters? ........................................................................... 10 6.2

The rate of changes of the volume for a=10cm is equal to

 cm3  dV  900   . How fast is the dt  s 

volume changing when a is equal to 10 centimeters? ........................................................................... 12 6.3

Sand is falling of a conveyor and form a prism at the rate 5 m^3/min. Side of the base if

a  10m and the height is H  10m . At what rate is the level of the sand changing for h  5m ? 21 6.4

Volume of the balloon (page 150) .............................................................................................. 13

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6.5 7

Volume of the balloon (page 150) .............................................................................................. 13

Concavity and the Second Derivative Test........................................................................................ 146 7.1

Concave functions ..................................................................................................................... 146

7.1.1

(*) Proof ............................................................................................................................ 147

7.1.2

Consider the function f x  x 3  9x 2  7x  8 . a. Find the points of inflection(s), if

 

any. b. Determine the intervals on which the function is concave up or concave down................. 149

7.1.3

Consider the function

x3 x4 f x  1  2x  x   2 12



2

a. Find the points of inflection(s),

if any. b. Determine the intervals on which the function is concave up or concave down.............. 151

7.1.4

Consider the function

x2 x 4 f x  1  2x   2 12



a. Find the points of inflection(s), if

any. b. Determine the intervals on which the function is concave up or concave down. ................ 153

7.1.5

Consider the function

x4 x5 f x  1  2x  2x   4 20



2

a. Find the points of

inflection(s), if any. b. Determine the intervals on which the function is concave up or concave down. 155

7.1.6

Consider the function

25x 2 x 4 f x  7  5x   12 2



a. Find the points of inflection(s), if

any. b. Determine the intervals on which the function is concave up or concave down. ................ 157

7.1.7 7.2

Consider the function

x3 x4 f x 7 x x   2 12



2

.................................................. 159

Second Derivative Test.............................................................................................................. 161

7.2.1

Theorem ............................................................................................................................ 161

7.2.2

*Proof (1) .......................................................................................................................... 163

7.2.3

Example f x  x2 .......................................................................................................... 170

7.2.4

Example

5x 2 x 3 f x  1  6x   ................................................................................ 171 2 3

7.2.5

Example f x  1  9x  3x 2  x 3 ................................................................................ 172

 



 

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Example

x3 f x  2  5x  2x  3

7.2.7

Example

f x  1  12x 

7.2.8

Example f x  2  45x  3x 2  x 3 ............................................................................. 177

7.2.6

8



2

................................................................................ 173

x2 x3  ................................................................................ 175 2 3



 

A summary of Curve Sketching ......................................................................................................... 179 8.1

 

Example f x  x 3  9x 2  7x  8 .................................................................................... 179



f x 

x2 1 .......................................................................................................... 204 x2  4

8.2

Example

8.3

Example f x  x 2  2x ....................................................................................................... 211

8.4

Example f x  1  9x 

8.5

2 3 Example f  x   3  4x  3x  x .................................................................................. 226

8.6

Example f x  1  4x 2  4x 3  x 4 .................................................................................. 229

8.7

Example f x  1  2x 

8.8

Example

 



x3 ............................................................................................... 216 3 2

3

 



3x 2 x 3  ..................................................................................... 235 2 3



(critical points) .................................................................... 237

f x  sin2 x  cos x



2 3

f x  3x  2x .................................................................................................... 240

8.9

Example

8.10

Example f x  8 x  7 x 

8.11

Example f x  2  9x  3x 2  x 3 ........................................................................................ 245

8.12

Example f x  1  12x  9x2  2x 3 .................................................................................. 248

8.14

x 1 x 1 x  2  x  5  (*) Vertical asymptote y  x  1 x  1  x  2 x  3 x  4 



2

7x 3 x 4  ............................................................................. 243 3 4

 

 

2

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3

2

.................................. 252

8.15 9

(*) Nonlinear inequality 

x

2



1 x 2 1



x  1 x  2 x  3 

 0 .......................................................... 252

Global Optimization .......................................................................................................................... 117

10

Optimization Problems and the Applications of the Derivatives.................................................. 255

10.1 A manufacturer wants to design a box having a square base and volume of 108 cubic inches. What dimension will produce a box with minimum surface area? Round your answer to 3 decimal places. 255 10.1.1

Method I (closed box) ...................................................................................................... 255

10.1.2

Method II (closed box) ...................................................................................................... 258

10.1.3

Method III (open box) ....................................................................................................... 261

10.1.4

Method IV (open box) ....................................................................................................... 262

10.2 A manufacturer wants to design an open box having a square base and surface area of 108 cubic inches. What dimension will produce a box with maximum volume? Round your answer to 3 decimal places. (Example 1 Section 3.7) ............................................................................................... 263 10.3 A rectangular solid with a square base has a surface area of 433.5 square centimeters. Find the dimensions that will result in a solid with maximum volume. ............................................................. 265 10.3.1

Method 1........................................................................................................................... 265

10.3.2

Method 2........................................................................................................................... 268

10.4

 

Which points on the graph f x  4  x 2 are the closest points to the point (0,2)?........... 271

10.4.1

Method 1..............................................................................................................................