© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e –...

© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 1 of 78

Applied Calculus


Functions Derivatives Applications of Derivative Techniques of Differentiation Logarithmic Functions and Applications The Definite Integrals The Trigonometric Functions

Course Contents

…


Chapter 0

Functions


Functions and Their Graphs Some Important Functions The Algebra of Functions Zeros of Functions The Quadratic Formula and Factoring Exponents and Power Functions Functions and Graphs in Applications

Chapter Outline


Definition Example

Rational Number: A number that may be written as a finite or infinite repeating decimal, in other words, a number that can be written in the form m/n such that m, n are integers

Irrational Number: A number that has an infinite decimal representation whose digits form no repeating pattern

73205.13

Rational & Irrational Numbers

2857140

142857142857072

.

....


The Number LineA geometric representation of the real numbers is shown below.

3

The Number Line

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

72


Open & Closed IntervalsDefinition Example

Infinite Interval: The set of numbers that lie between a given endpoint and the infinity

Closed Interval: The set of numbers that lie between two given endpoints, including the endpoints themselves

[−1, 4]

Open Interval: The set of numbers that lie between two given endpoints, not including the endpoints themselves

(−1, 4)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

,44x

41 x

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

41 x


Functions• A function f is a rule that assigns to each value of a real variable x exactly one

value of another real variable y. • The variable x is called the independent variable and the variable y is called the

dependent variable.• We usually write y = f (x) to express the fact that y is a function of x. Here f (x) is

the name of the function.

EXAMPLES:EXAMPLES:

ttetk

xxh

xxg

xxf

2sin21

31

4

2

2

y = f (x)x y


Functions in Application

EXAMPLEEXAMPLEWhen a solution of acetylcholine is introduced into the heart muscle of a frog, it diminishes the force with which the muscle contracts. The data from experiments of the biologist A. J. Clark are closely approximated by a function of the form

where x is the concentration of acetylcholine (in appropriate units), b is a positive constant that depends on the particular frog, and R(x) is the response of the muscle to the acetylcholine, expressed as a percentage of the maximum possible effect of the drug.

(a) Suppose that b = 20. Find the response of the muscle when x = 60.

(b) Determine the value of b if R(50) = 60 – that is, if a concentration of x = 50 units produces a R = 60% response.

xbxxR

100


Functions in Application

Replace b with 20 and x with 60. 7560206010060

R

Therefore, when b = 20 and x = 60, R (x) = 75%.

This is the given function. xbxxR

100(b)

Replace x with 50. 505010050

b

R

Replace R(50) with 60.505010060

b

Multiply both sides by b + 50. 5050

50006050

bb

b

Distribute on the left side.5000300060 b

Subtract 3000 from both sides.200060 b

Divide both sides by 60.333.b

Therefore, b = 33.3 when R (50) = 60.

SOLUTIONSOLUTION


FunctionsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

If f (x) = x2 + 4x + 3, find f (a − 2).

This is the given function. 342 xxxf

Replace each x with a – 2. 32422 2 aaaf

Evaluate (a – 2)2 = a2 – 4a + 4. 324442 2 aaaaf

Remove parentheses and distribute. 384442 2 aaaaf

Combine like terms. 12 2 aaf


Domain of a Function

Definition Example

Domain of a Function: The set of acceptable values for the variable x.

The domain of the function

is

2 xxf

02 x2x)2[ ,-



4

22

x

xxf

2x 2 \Rx

Definition Example



is 042 x



x

xxf

3

2

03 x3x)3( ,

Definition Example



is


Graphs of Functions

Definition ExampleGraph of a Function: The set of all points (x, f (x)) where x is the domain of f (x). Generally, this forms a curve in the xy-plane.


The Vertical Line Test

Definition ExampleVertical Line Test: A curve in the xy-plane is the graph of a function if and only if each vertical line cuts or touches the curve at no more than one point.

Although the red line intersects the graph no more than once (not at all in this case), there does exist a line (the yellow line) that intersects the graph more than once. Therefore, this is not the graph of a function.


Graphs of EquationsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Is the point (3, 12) on the graph of the function ? 221

xxxf

This is the given function. 221

xxxf

Replace x with 3. 232133

f

Replace f (3) with 12. 2321312

Simplify. 55212 .

Multiply.51212 . false

Since replacing x with 3 and f (x) with 12 did not yield a true statement in the original function, we conclude that the point (3, 12) is not on the graph of the function.


Linear Equations

Equation Example

y = mx + b(This is a linear function)

x = a(This is not the graph of a

function)


Linear Equations

Equation Example

y = b

CONTINUECONTINUEDD


Piece-Wise FunctionsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Sketch the graph of the following function .

3for 23for 1

xxx

xf

We graph the function f (x) = 1 + x only for those values of x that are less than or equal to 3.

-6

-4

-2

0

2

4

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Notice that for all values of x greater than 3, there is no line.


Piece-Wise Functions

Now we graph the function f (x) = 2 only for those values of x that are greater than 3.

Notice that for all values of x less than or equal to 3, there is no line.

CONTINUECONTINUEDD

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


Piece-Wise Functions

Now we graph both functions on the same set of axes.

CONTINUECONTINUEDD

-6-5-4-3-2-10123456

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


Quadratic Functions

Definition Example

Quadratic Function: A function of the form

where a, b, and c are constants and a 0.

cbxaxxf 2


Polynomial Functions

Definition Example

Polynomial Function: A function of the form

where n is a nonnegative integer and a0, a1, ..., an are given numbers.

01

1 axaxaxf nn

nn

517 23 xxxf


Rational Functions

Definition Example

Rational Function: A function expressed as the quotient of two polynomials.

15

32

4

xxxxxg


Power Functions

Definition Example

Power Function: A function of the form

.xxf r 25.xxf


Absolute Value Function

Definition Example

Absolute Value Function: The function defined for all numbers x by

such that |x| is understood to be x if x is positive and –x if x is negative

,xxf xxf

212121 f


Adding FunctionsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Given and , express f (x) + g(x) as a rational function. 3

2

x

xf

Replace f (x) and g(x) with the given functions.

2

1

x

xg

21

32

xx

f (x) + g(x) =

Multiply to get common denominators.3

32

13

222

xx

xxxx

Evaluate. 323

3242

xxx

xxx

Add and simplify the numerator. 32342

xxxx

3213

xxx

613

2

xx

xEvaluate the denominator.


Subtracting FunctionsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Given and , express f (x) − g(x) as a rational function. 3

2

x

xf

Replace f (x) and g(x) with the given functions.2

13

2

xx

f (x) − g(x) =

Multiply to get common denominators.3

32

13

222

xx

xxxx

Evaluate. 323

3242

xxx

xxx

Subtract. 32

342

xx

xx

Simplify the numerator and denominator. 32

7

xxx

2

1

x

xg

67

2

xx

x


Multiplying FunctionsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Given and , express f (x)g(x) as a rational function. 3

2

x

xf


2

1

x

xg

21

32

xx

f (x)g(x) =

Multiply the numerators and denominators. 23

12

xx

Evaluate.6

22

xx


Dividing FunctionsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Given and , express as a rational function. 3

2

x

xf


2

1

x

xg

21

32

x

x

Rewrite as a product (multiply by reciprocal of denominator).1

23

2

xx

Multiply the numerators and denominators.

13

22

xx

Evaluate.342

xx

xgxf

xgxf


Composition of FunctionsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Table 1 shows a conversion table for men’s hat sizes for three countries. The function

converts from British sizes to French sizes, and the function converts from French sizes to U.S. sizes. Determine the function h (x) = f (g (x)) and give its interpretation.

xxf81

18 xxg

This is what we will determine.h (x) = f (g (x))

In the function f, replace each occurrence of x with g (x).

xg81

Replace g (x) with 8x + 1. 1881

x


Composition of Functions

Distribute.1818

81

x

CONTINUECONTINUEDD

Multiply.81

x

Therefore, h (x) = f (g (x)) = x + 1/8. Now to determine what this function h (x) means, we must recognize that if we plug a number into the function, we may first evaluate that number plugged into the function g (x). Upon evaluating this, we move on and evaluate that result in the function f (x). This is illustrated as follows.

g (x) f (x)British French French U.S.

h (x)Therefore, the function h (x) converts a men’s British hat size to a men’s U.S. hat size.


Composition of FunctionsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Given and , find f (g (x)). 3

2

x

xf

Replace x by g(x) in the function f (x)

2

1

x

xg

32

xg

f (g (x)) =

Substitute.3

21

2

x

Multiply the numerators and denominators by x + 2.2

2

32

12

xx

x

Simplify.5342

x

x


Zeros of Functions

Definition Example

Zero of a Function: For a function f (x), all values of x such that f (x) = 0.

12 xxf

10 2 x1x


Quadratic FormulaDefinition Example

Quadratic Formula: A formula for solving any quadratic equation of the form .The solution is:

There is no solution if

These are the solutions/zeros of the quadratic function

02 cbxax

0232 xx

.2

42

aacbbx

2;3;1 cba

12

21433 2 x

2173

x

.232 xxxf.042 acb


Graphs of Intersecting FunctionsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Find the points of intersection of the pair of curves.;xxy 9102 9xy

-40

-20

0

20

40

60

80

100

-5 0 5 10 15

The graphs of the two equations can be seen to intersect in the following graph. We can use this graph to help us to know whether our final answer is correct.


Graphs of Intersecting Functions

To determine the intersection points, set the equations equal to each other, since they both equal the same thing: y.

99102 xxx

This is the equation to solve.

Now we solve the equation for x using the quadratic formula.

CONTINUECONTINUEDD

99102 xxx

Subtract x from both sides.99112 xx

Add 9 to both sides.018112 xx

Use the quadratic formula.

1218141111 2

x

Here, a = 1, b = −11, and c =18.

Simplify.2

7212111 x


Graphs of Intersecting FunctionsCONTINUECONTINUE

DD

We now find the corresponding y-coordinates for x = 9 and x = 2. We can use either of the original equations. Let’s use y = x – 9.

Simplify.2

4911x

Simplify.2

711x

Rewrite.2

7112

711 ,x

Simplify.29,x

9x 2x9xy 9xy99 y 92 y

0y 7y


Graphs of Intersecting FunctionsCONTINUECONTINUE

DDTherefore the solutions are (9, 0) and (2, −7). This seems consistent with the two intersection points on the graph. A zoomed in version of the graph follows.

-20

-15

-10

-5

0

5

10

0 2 4 6 8 10


FactoringEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Factor the following quadratic polynomial.326 xx

This is the given polynomial.326 xx

Factor 2x out of each term. 232 xx

Rewrite 3 as 2232 xx .3

2

Now I can use the factorization pattern: a2 – b2 = (a – b)(a + b).

Rewrite . 33 as 3 22xxx xxx 332


FactoringEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Solve the equation for x.

2

651xx

This is the given equation.2

651xx

Multiply everything by the LCD: x2. 22

2 651 xxx

x

Distribute.22

22 65 xx

xx

x

Multiply.652 xx

Subtract 5x + 6 from both sides.0652 xx

Factor. 061 xx

0601 xx

1x 6x

Set each factor equal to zero.

Solve.


Exponents

Definition Example

timesn

n bbbb 55553

nn bb 1

331

55


Exponents

Definition Example

344 343

555 mnn mnm

bbb

344 343

43

51

51

5

15

mnn mnm

nm

bbbb 111


Exponents

Definition Example

666666 133

32

31

32

31

srsr bbb

21

41

4

1421

21

rr

bb 1


Exponents

Definition Example

77777

7 133

31

34

31

34

srs

r

bbb

399999 21

84

85

548

5

54

rssr bb


Exponents

Definition Example

153527125

2712527125

33

313131

///

rrr baab

1625

105

10 44

4

4

r

rr

ba

ba


Applications of ExponentsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Use the laws of exponents to simplify the algebraic expression.

3

32527xx /

This is the given expression. 3

32527xx /

3

3253227x

x // rrr baab

31

3253227/

//

xx

nn bb 1

rssr bb

31

3103227/

//

xx

31

31023 27

/

/

xx mnn mn

m

bbb


Applications of Exponents

31

31023/

/

xx 3273

CONTINUECONTINUEDD

31

3109/

/

xx

93 2

313109 //x srs

r

bbb

Subtract.399 /x

Divide.39x


Geometric ProblemsEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Consider a rectangular corral with two partitions, as shown below. Assign letters to the outside dimensions of the corral. Write an equation expressing the fact that the corral has a total area of 2500 square feet. Write an expression for the amount of fencing needed to construct the corral (including both partitions).

First we will assign letters to represent the dimensions of the corral.

x x x x

y

y


Geometric Problems

Now we write an equation expressing the fact that the corral has a total area of 2500 square feet. Since the corral is a rectangle with outside dimensions x and y, the area of the corral is represented by:

CONTINUECONTINUEDD

xyA

Now we write an expression for the amount of fencing needed to construct the corral (including both partitions). To determine how much fencing will be needed, we add together the lengths of all the sides of the corral (including the partitions). This is represented by:

yyxxxxF yxF 24


Surface AreaEXAMPLEEXAMPLE

SOLUTIONSOLUTION

Assign letters to the dimensions of the geometric box and then determine an expression representing the volume and the surface area of the box.

First we assign letters to represent the dimensions of the box.

xy

z

Therefore, an expression that represents the volume is:

V = xyz.


Surface Area

Now we determine an expression for the surface area of the box. Note, the box has 5 sides which we will call Left (L), Right (R), Front (F), Back (B), and Bottom (Bo). We will find the area of each side, one at a time, and then add them all up.

L: yz

xy

z

CONTINUECONTINUEDD

R: yz

F: xz B: xz

Bo: xy

Therefore, an expression that represents the surface area of the box is:

S = yz + yz + xz + xz + xy = 2yz + 2xz + xy.

© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e –...

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Transcript of © 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e –...