Reaction Time + Braking Distance

Exponential and Logarithmic Functions

5.1Combining Functions

Stopping distance

Reaction time + Braking distance

x

/ 0 / 12 / 24 / 36 / 48 / 60
r(x) / 0 / 44 / 88 / 132 / 176 / 220
b(x) / 0 / 16 / 64 / 144 / 256 / 400
s(x) / 0 / 60 / 152 / 276 / 432 / 620

Operations on Functions

If and both exist, the sum, difference, product, and quotient of two functions f and g are defined by

Domains

x

/ -2 / 0 / 1 / 4
f(x) / -3 / 1 / 3 / 9
g(x) / 0 / 1 / 2

Let and .

Domains

Difference of Functions

Cost of producing DVDs

Master - $12,000

Mass producing - $5 each

Sell - $12 each (3000)

Let x=3000

Let

Composition of Functions

Miles Feet Inches

If f and g are functions, then the composition function , or the composition of g and f is defined by

The domain of is all x in the domain of f such that is in the domain of g.

Let and .

Domains

Let and .

x / 1 / 2 / 3 / 4
/ 2 / 3 / 4 / 1
x / 1 / 2 / 3 / 4
/ 4 / 3 / 2 / 1
x / 0 / 1 / 2 / 3 / 4 / 5 / 6
f(x) / 0 / 1.5 / 3.0 / 4.5 / 6.0 / 7.5 / 9.0
x / 0 / 1.5 / 3.0 / 4.5 / 6.0 / 7.5 / 9.0
g(x) / 0 / 5.25 / 10.5 / 15.75 / 21.0 / 26.25 / 31.5
x / 0 / 1 / 2 / 3 / 4 / 5 / 6

Find functions f and g such that

5.2Inverse Functions and their Representations

Open door, get in, close door, start engine

Shut off eng, open door, get out, close door

A B C … X Y Z

C D E … Z A BHELP JGNR

Reversible actions Gal to pints

Pts to gal

Notation

% time skies cloudy in Augusta GA

1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12
43 / 40 / 39 / 29 / 28 / 26 / 27 / 25 / 30 / 26 / 31 / 39

a x

b y

A function is a one-to-one function if, for elements c and d in the domain of f,

implies

(different inputs always result in different outputs)

Determine graphically

The Horizontal Line Test

If every horizontal line intersects the graph of a function f at most once, then f is a one-to-one function.

Inverse functions

Let f be a one-to-one function. Then is the inverse of f , if

, and

Let . Find/verify inverse function.

Finding an inverse function

1)Verify f is a one-to-one function

2)Solve for x in terms of y.

3)Interchange x and y to obtain .

4)To verify , show that:

AND

The function gives the percentage of China’s population that may live in urban areas x years after 2000, .

1-1?

Check

find

Restricted domain

Let

Inverse?

Let

Inverse?

US Population with 4+ years of college in yr x.

1940 / 1970 / 2015
5 / 11 / 34

Domain of f is the range of

Graphical

If lies on the graph of f, then lies on the graph of .

on the graph of f on the graph of

Let .

5.3 Exponential Functions and Models

Linear function:

Power function:

Exponential fcn:

x / 0 / 1 / 2 / 3 / 4 / 5
/ 3 / 5 / 7 / 9 / 11 / 13
x / 0 / 1 / 2 / 3 / 4 / 5
/ 3 / 6 / 12 / 24 / 48 / 96

A function f represented by where , is an exponential function with base a.

If , then .

Linear and Exponential data

x / 0 / 1 / 2 / 3
y / -3 / -1.5 / 0 / 1.5
x / 0 / 1 / 2 / 3 / 4
y / 16 / 4 / 1 / /
x / 0 / 1 / 2 / 3
y / 3 / 4.5 / 6.75 / 10.125
x / 0 / 1 / 2 / 3 / 4
y / 16 / 12 / 8 / 4 / 0

Find C and a such that:

and

The number of centenarians (age 100+) in India is projected to increase by a factor of 1.04 annually from 2010 to 2100.

Function?

If 50,000 in 2010, how many in 2050?

An exponential function has the following properties:

Domain is , range is .
f is conts, x-axis is horiz asymptote
No x-int, y-int is C.
If , f is increasing, , f is dec.
f is 1-1, therefore exists.

Behavior of

Reflections - is a reflection of

is a reflection of

Compound Interest

$2000 deposited at 8%

If dollars is deposited into an account paying an annual rate of interest r, compounded n times per year, then after t years, the account will contain dollars, where

Suppose a 20-year old deposits $1000 in an IRA at 12% interest. Compare at age 65:

Compounded annually:

Compounded quarterly:

Natural Exponentiation

For calculation purposes, assume one puts $1 in a savings account for 1 year at 100% interest.

Compounding /

N

/ A
Annual / 1 / 2
Monthly / 12 / 2.613035
Daily / 365 / 2.714567
Hourly / 8760 / 2.718127
Minutely / 525,600 / 2.718279
Secondly / 31,536,000 / 2.718282
Continuous

Euler’s number – irrational, like

- natural exponentiation

Calculator exercise

Continuously compounded interest

$1000 deposited at 12%, continuously compounded

e-Coli bacteria proliferation

Population can double every 49.5 minutes

After x minutes,

Let x be 99 minutes.

When will the number reach 25 million?

Future atmospheric concentration

2000 / 2050 / 2100 / 2150 / 2200
364 / 467 / 600 / 769 / 987

Traffic flow

30 cars per hour, cars enter the intersection at in x minutes.

Half life of a Facebook link

When posted, no hits have occurred, 100% of hits yet to come. Assume half life of 3 hours, find percentage after 4 hours.

, C units, x is the age, k is the half life.

Radium-226 has a half life of 1600 years. After 9600 years, a 2 gram sample decays to:

Radioactive Carbon-14 has a half life of 5700 years. Suppose a fossil contains 5% of the Carbon-14 as when it was alive.

5.4 Logarithmic Functions and Models

Growth of Bacteria

0 / 1 / 2 / 3 / 4
1 / 10 / 100 / 1000 / 10,000
1 / 10 / 100 / 1000 / 10,000
0 / 1 / 2 / 3 / 4

Common Logarithms

The common logarithm of a positive number x, denoted , is defined by

if and only if

where k is a real number. The function given by

is called the common logarithm function.

-3 / -2 / -1 / 0 / 1 / / 2 / 3
-3 / -2 / -1 / 0 / 1 / / 2 / 3

The following inverse properties hold for the common logarithm

for any real x, and

for any positive x.

Malaria deaths in Africa

The number of malaria deaths (millions) in Africa, x years after 2000 can be modeled by

Logarithms with other bases

-3 / -2 / -1 / 0 / 1 / / 2 / 3
-3 / -2 / -1 / 0 / 1 / / 2 / 3

If for some k, then .

Natural logarithms

If for some k, then .

John Napier (1550-1617)

The logarithm with base a of a positive number x, denoted , is defined by

if and only if

where , and k is a real number. The function given by

is called the logarithmic function with base a.

(a logarithm is an exponent)

The following inverse properties hold for the logarithms with base a.

for any real x, and

for any positive x.

Domain?

Solving equations

Zeroes and Ones

Some types of data can be modeled by

Number of species of birds on islands

Area / 0.1 / 1 / 10 / 100 / 1000
Species / 26 / 39 / 52 / 65 / 78

Model data

If the size of the island increases tenfold, how many additional species are there?

What size island would have 50 species of birds?

5.5Properties of Logarithms

John Napier (1550-1617)

Johannes Kepler (1571-1630)

Isaac Newton (1642-1727)

Sum of logs is log of product

For positive numbers m, n, and ,

1. and

Sound levels in decibels can be computed by .

Simplify.

Ordinary conversation has an intensity of w/cm². Find decibel level.

Change of Base formula

Let x, , and be positive real numbers. Then,

5.6Exponential and Logarithmic Equations

Population of the world 7 billion in 2011, when will it reach 8 billion?

Thickness of a runway

Thickness of pavement t” vs. wt. of aircraft (000)

What thickness is required for 130,000 lb plane?

The population of bluefin tuna has declined exponentially from 1974 to 1991, and can be modeled by , where x is the year, x=0 corresponds to 1974. Find the number in 1974 and 1991. When was the population about 50,000?

Finding a modeling function

In 1945, computers could perform 1000 computations with 1 kilowatt-hour of electricity. This number doubled every 1.6 years.

Model this exponential function , x is years after 1945.

Find .

When did computers first perform 1M calculations per kilowatt-hour?

A pot of coffee with temp of 100ºC is set down in a room with temp of 20ºC. The coffee cools to 60ºC after 1 hour. Model this data using by finding values for T, D, and a, find the temperature of the coffee in ½ hour, and when the coffee reached 50ºC.

Solve graphically:

Solving logarithmic equations

1) Isolate log on one side of =

2) Exponentiate each side of equation with the same base.

3) Solve for variable.

In developing countries, there is a relationship between the amount of land owned and daily calories consumed, which can be modeled by , x is acres owned, . Find the amount of calories consumed by a person who owns no land. Suppose a person consumes 2000 calories. Estimate the amount of land owned.

Exponential and Logarithmic Inequalities