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© 2010 Pearson Education Inc.Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 3 of 78 Chapter 0 Functions
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© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 1 of 78
Applied Calculus
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 2 of 78
Functions Derivatives Applications of Derivative Techniques of Differentiation Logarithmic Functions and Applications The Definite Integrals The Trigonometric Functions
Course Contents
…
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 3 of 78
Chapter 0
Functions
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 4 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 5 of 78
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
.
....
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 6 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 7 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 8 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 9 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 10 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 11 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 12 of 78
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[ ,-
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 13 of 78
Domain of a Function
4
22
x
xxf
2x 2 \Rx
Definition Example
Domain of a Function: The set of acceptable values for the variable x.
The domain of the function
is 042 x
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 14 of 78
Domain of a Function
x
xxf
3
2
03 x3x)3( ,
Definition Example
Domain of a Function: The set of acceptable values for the variable x.
The domain of the function
is
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 15 of 78
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 16 of 78
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 17 of 78
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 18 of 78
Linear Equations
Equation Example
y = mx + b(This is a linear function)
x = a(This is not the graph of a
function)
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 19 of 78
Linear Equations
Equation Example
y = b
CONTINUECONTINUEDD
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 20 of 78
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 21 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 22 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 23 of 78
Quadratic Functions
Definition Example
Quadratic Function: A function of the form
where a, b, and c are constants and a 0.
cbxaxxf 2
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 24 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 25 of 78
Rational Functions
Definition Example
Rational Function: A function expressed as the quotient of two polynomials.
15
32
4
xxxxxg
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 26 of 78
Power Functions
Definition Example
Power Function: A function of the form
.xxf r 25.xxf
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 27 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 28 of 78
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 29 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 30 of 78
Multiplying 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 the numerators and denominators. 23
12
xx
Evaluate.6
22
xx
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 31 of 78
Dividing FunctionsEXAMPLEEXAMPLE
SOLUTIONSOLUTION
Given and , express as a rational function. 3
2
x
xf
Replace f (x) and g(x) with the given functions.
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 32 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 33 of 78
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 34 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 35 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 36 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 37 of 78
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 38 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 39 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 40 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 41 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 42 of 78
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 43 of 78
Exponents
Definition Example
timesn
n bbbb 55553
nn bb 1
331
55
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 44 of 78
Exponents
Definition Example
344 343
555 mnn mnm
bbb
344 343
43
51
51
5
15
mnn mnm
nm
bbbb 111
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 45 of 78
Exponents
Definition Example
666666 133
32
31
32
31
srsr bbb
21
41
4
1421
21
rr
bb 1
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 46 of 78
Exponents
Definition Example
77777
7 133
31
34
31
34
srs
r
bbb
399999 21
84
85
548
5
54
rssr bb
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 47 of 78
Exponents
Definition Example
153527125
2712527125
33
313131
///
rrr baab
1625
105
10 44
4
4
r
rr
ba
ba
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 48 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 49 of 78
Applications of Exponents
31
31023/
/
xx 3273
CONTINUECONTINUEDD
31
3109/
/
xx
93 2
313109 //x srs
r
bbb
Subtract.399 /x
Divide.39x
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 50 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 51 of 78
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
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 52 of 78
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.
© 2010 Pearson Education Inc. Goldstein/Schneider/Lay/Asmar, CALCULUS AND ITS APPLICATIONS, 12e – Slide 53 of 78
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.
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