Unit 1A Library of Functions

Unit 1A Library of Functions

The building blocks for Calculus

The building blocks for Calculus

Functions and GraphsFunctions and Graphs

Function: A rule that takes a set of input numbers and assigns to each a definite output number.

The set of all input numbers is called the domain of the function and the set of all resulting output numbers is called the range of the function.

Function: A rule that takes a set of input numbers and assigns to each a definite output number.

The set of all input numbers is called the domain of the function and the set of all resulting output numbers is called the range of the function.

The Rule of Four (Multiple Representations)

The Rule of Four (Multiple Representations)

Representing mathematical functions with:

tables graphs formulas words

Representing mathematical functions with:

tables graphs formulas words

SituationThe value of a car decreases over time.

SituationThe value of a car decreases over time.

• Can this be represented using a:

• Graph?

• Table?

• Formula?

• Can this be represented using a:

• Graph?

• Table?

• Formula?

Linear Functions Linear Functions

Are fundamental to all mathematical patterns.

“the value of the function changes by equal amounts over equal intervals”.

In other words, as the independent variable changes by a fixed amount, a constant amount is added to the value of the function.

Are fundamental to all mathematical patterns.

“the value of the function changes by equal amounts over equal intervals”.

In other words, as the independent variable changes by a fixed amount, a constant amount is added to the value of the function.

A different look at the slope formula:A different look at the slope formula:

€

m =Δy

Δx=

f (x2 ) − f (x1 )

x2 − x1

€

m =Δy

Δx=

f (x2 ) − f (x1 )

x2 − x1

This expression is called the

difference quotient.

Exponential FunctionsExponential Functions

are characterized by a simple property: the value of the function changes by equal ratios over equal intervals.

In other words, as the independent variable changes by a fixed amount, the value of the function changes by a constant factor.

are characterized by a simple property: the value of the function changes by equal ratios over equal intervals.

In other words, as the independent variable changes by a fixed amount, the value of the function changes by a constant factor.

General Exponential FunctionGeneral Exponential Function

€

P = Poa t

€

P = Poa t

where Po is the initial quantity when t = 0 and a is the factor by which P changes when t increases by 1.

€

Po

€

€

If the rate a is > 1, we have exponential growth.

If the rate a is < 1 we have exponential decay.

If the rate a is > 1, we have exponential growth.

If the rate a is < 1 we have exponential decay.€

P = Poa t

€

P = Poa t

Population growth can often be modeled with an exponential function:

Ratio:

5023 4936 1.0176÷ ≈5111 5023 1.0175÷ ≈

1.01761.0246

1.0175

World Population:

1986 4936 million1987 50231988 51111989 52011990 53291991 5422

The world population in any year is about 1.018 times the previous year.

in 2010: ( )195422 1.018P = ⋅ 7609.7≈

About 7.6 billion people. →

Nineteen years past 1991.

Radioactive decay can also be modeled with an exponential function:

Suppose you start with 5 grams of a radioactive substance that has a half-life of 20 days. When will there be only one gram left?

After 20 days:1 5

52 2⋅ =

40 days:2 51

542

⎛ ⎞⋅ =⎜ ⎟⎝ ⎠

t days:201

52

t

y ⎛ ⎞= ⋅⎜ ⎟⎝ ⎠

In Pre-Calc you solved this using logs. →

ConcavityConcavity The graph of a function is concave up if

it bends upward as we move left to right.

It is concave down if it bends downward.

A line is neither concave up nor concave down.

The graph of a function is concave up if it bends upward as we move left to right.

It is concave down if it bends downward.

A line is neither concave up nor concave down.

Exponential function with base e

Exponential function with base e

When the growth or decay is “continuous” the function can be written:

When the growth or decay is “continuous” the function can be written:

€

P = Poekt

to represent continuous growth

Exponential function with base e

Exponential function with base e

When the growth or decay is “continuous” the function can be written as:

When the growth or decay is “continuous” the function can be written as:

to represent continuous decay

€

Q = Qoe−kt

New Functions from OldNew Functions from Old

• Understanding how certain parent functions can be transformed into new functions is essential in using functions to model real world situations.

• Understanding how certain parent functions can be transformed into new functions is essential in using functions to model real world situations.

Key PointsKey Pointso Graphical interpretation of linear combinations

of functions,

o Modeling interpretation of composition of functions,

o Basic manipulations of graphs, including shifts, flips, stretches

o Odd and even functions,

o The concept of an inverse function.

o Graphical interpretation of linear combinations of functions,

o Modeling interpretation of composition of functions,

o Basic manipulations of graphs, including shifts, flips, stretches

o Odd and even functions,

o The concept of an inverse function.

Inverse functions:

( ) 11

2f x x= + Given an x value, we can find a y value.

-5

-4

-3

-2

-10

1

2

3

4

5

-5 -4 -3 -2 -1 1 2 3 4 5

11

2y x= +

11

2y x− =

2 2y x− =

2 2x y= −

Switch x and y: 2 2y x= − ( )1 2 2f x x− = −

(inverse of x)

Inverse functions are reflections about y = x.

Solve for x:

→

Logarithms

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2004

Golden Gate BridgeSan Francisco, CA

Logarithmic FunctionsLogarithmic Functions

The logarithm (log) is the inverse of an exponential function.

• Base e is just another number. By varying the constant k in ekt, any exponential can be expressed with base e, with an inverse of the natural logarithm ln.

The logarithm (log) is the inverse of an exponential function.

• Base e is just another number. By varying the constant k in ekt, any exponential can be expressed with base e, with an inverse of the natural logarithm ln.

Consider ( ) xf x a=

This is a one-to-one function, therefore it has an inverse.

The inverse is called a logarithm function.

Example:416 2= 24 log 16= Two raised to what power

is 16?

The most commonly used bases for logs are 10: 10log logx x=

and e: log lne x x=

lny x= is called the natural log function.

logy x= is called the common log function.

→

lny x=

logy x=

is called the natural log function.

is called the common log function.

In calculus we will use natural logs exclusively.

We have to use natural logs:

Common logs will not work.

→

Properties of Logarithms

loga xa x= log xa a x= ( )0 , 1 , 0a a x> ≠ >

Since logs and exponentiation are inverse functions, they “un-do” each other.

Product rule: log log loga a axy x y= +

Quotient rule: log log loga a a

xx y

y= −

Power rule: log logya ax y x=

Change of base formula:ln

loglna

xx

a=

→

Example

$1000 is invested at 5.25 % interest compounded annually.How long will it take to reach $2500?

( )1000 1.0525 2500t

=

( )1.0525 2.5t

= We use logs when we have an unknown exponent.

( )ln 1.0525 ln 2.5t

=

( )ln 1.0525 ln 2.5t =

( )ln 2.5

ln 1.0525t = 17.9≈ 17.9 years

In real life you would have to wait 18 years. →

Trig Functions

The Mean Streak, Cedar Point Amusement Park, Sandusky, OH

Trigonometric functions are used extensively in calculus.

→

When you use trig functions in calculus, you must use radian measure for the angles.

Trigonometric FunctionsTrigonometric Functions

Trigonometry is organized into 3 main categories:

1. Triangle trig (right triangle and non-right triangle relationships)

2. The Unit Circle and the graphs of trig functions.

3. Trig Identities and their properties

Trigonometry is organized into 3 main categories:

1. Triangle trig (right triangle and non-right triangle relationships)

2. The Unit Circle and the graphs of trig functions.

3. Trig Identities and their properties

Even and Odd Trig Functions:

“Even” functions behave like polynomials with even exponents, in that when you change the sign of x, the y value doesn’t change.

Cosine is an even function because: ( ) ( )cos cosθ θ− =

Secant is also an even function, because it is the reciprocal of cosine.

Even functions are symmetric about the y - axis.

→

Even and Odd Trig Functions:

“Odd” functions behave like polynomials with odd exponents, in that when you change the sign of x, the

sign of the y value also changes.

Sine is an odd function because: ( ) ( )sin sinθ θ− = −

Cosecant, tangent and cotangent are also odd, because their formulas contain the sine function.

Odd functions have origin symmetry.

→

The rules for shifting, stretching, shrinking, and reflecting the graph of a function apply to trigonometric functions.

( )( ) y a f b x c d= + +

Vertical stretch or shrink;reflection about x-axis

Horizontal stretch or shrink;reflection about y-axis

Horizontal shift

Vertical shift

Positive c moves left.

Positive d moves up.

The horizontal changes happen in the opposite direction to what you might expect. →

is a stretch.1a >

is a shrink.1b >

-1

0

1

2

3

4

-1 1 2 3 4 5x

When we apply these rules to sine and cosine, we use some different terms.

( ) ( )2sinf x A x C D

B

π⎡ ⎤= − +⎢ ⎥⎣ ⎦

Horizontal shift

Vertical shift

→

is the amplitude.A

is the period.B

A

B

C

D ( )21.5sin 1 2

4y x

π⎡ ⎤= − +⎢ ⎥⎣ ⎦

2π− 3

2

π− π−

2

π−

2

π π

3

2

π

2π

Trig functions are not one-to-one.

However, the domain can be restricted for trig functions to make them one-to-one.

These restricted trig functions have inversescalled “arc”.

π

siny x=

Power FunctionsPower Functions

€

f (x) = kx p

where k and p are any constants with k≠0

Polynomial FunctionsPolynomial Functions

Flexibility of polynomial graphs on a small scale.

Basic resemblance to power functions on a large scale.

Varying the coefficient to produce particular graphs.

Flexibility of polynomial graphs on a small scale.

Basic resemblance to power functions on a large scale.

Varying the coefficient to produce particular graphs.

Polynomials are the sums of power functions with nonnegative integer

exponents

Polynomials are the sums of power functions with nonnegative integer

exponents

€

y = p(x) = an x n + an−1 x n−1 +...a1 x1 + a0

n is a nonnegative integer called the degree of the polynomial.

Rational FunctionsRational Functions

Identifying horizontal and vertical asymptotes of rational functions.

Rational Functions are ratios of polynomial functions

Identifying horizontal and vertical asymptotes of rational functions.

Rational Functions are ratios of polynomial functions

€

f (x) =p(x)

q(x)

Rational FunctionsRational Functions

Vertical asymptotes of a rational function occur at every root of the denominator that are not also roots of the numerator.

Holes can occur in a rational function when a root of the denominator is also a root of the numerator.

Vertical asymptotes of a rational function occur at every root of the denominator that are not also roots of the numerator.

Holes can occur in a rational function when a root of the denominator is also a root of the numerator.

3 cases to identify a horizontal asymptote3 cases to identify a

horizontal asymptote Case 1: The degree of the numerator and

denominator are the same. Case 1: The degree of the numerator and

denominator are the same.

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y =a

bis a horizontal asymptote: a is the leading coefficient of the numerator and b is the leading coefficient of the denominator.

Case 2: The degree of the numerator is less than the degree of the denominator, the x-axis (y = 0) is a horizontal asymptote.

Case 3: The degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Case 2: The degree of the numerator is less than the degree of the denominator, the x-axis (y = 0) is a horizontal asymptote.

Case 3: The degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.