notes on "applied calculus", flath/gleason/et.al

8/4/2019 Notes on "Applied Calculus", Flath/Gleason/et.al.

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Chapter 1: Functions

A function is like a free Snack Vending Machine.

L2

functioninput output

But in mathematics, a function can work with any kind of inputsand outputs.


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Like the Car Crusher Function.

In mathematics theres no need to be constrained by reality, so wecan even define the Car UnCrusher Function.

Function

UnCrusher


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Most mathematical functions deal with numbers: both the inputsand the outputs are usually numbers.

5 function 17

Functions are usually given single-letter nicknames, like f, H, or .

To represent the output of a function, the input is wrapped inparenthesis and placed to the right of the functions name.

f(5) = output of f when the input is 5

So in the case above,

f(5) = 17.


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There are different ways to explain what a function does, that is,to define it.

You can give examples.

f(1) = 3f(2) = 7f(3) = 13

For this tables are helpful.

input outputx f(x)

1 32 73 -13

But what happens on other inputs?


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There are different ways to explain what a function does, that is,to define it.

You can draw the graph.

Pretty, but not 100% accurate.


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There are different ways to explain what a function does, that is,to define it.You can give the mathematical formula.

f(x) =3

2(x 2) (x 3) 7 (x 1) (x 3) 13

2(x 1) (x 2)

Ugly, but precise.

For example, what happens if we input 2.7?

f(2.7) =3

2(2.7 2) (2.7 3)

7 (2.7 1) (2.7 3)13

2(2.7 1) (2.7 2)

= 4.48


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Another example;

f(x) = (x 1)

x +1

2

(x + 2)2

Can you tell what f(1), f(

1

2), and f(

2) are?

f(1) = 0 = f(12

) = f(2)

These inputs are called the zeros of the function f.


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Heres the graph of f.

The zeros are at 2, 0.5, and 1.Notice that theres one point on the curve where it crosses thevertical axis.

Thats where the input is 0.f(0) = (0 1) 0 + 1

2

(0 + 2)2 = 2

That point on the graph is called the vertical intercept.

The zeros can be called horizontal intercepts.


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The simplest kind of function is the constant function: whateverinputs you plug in, you always get the same answer.

f(1) = 37, f(2) = 37, f(3) = 37, etc.

In this case we can just say that f(x) = 37 for any x.

The second simplest function is the linear function.

Example:

f(1) = 5 f(2) = 8, f(3) = 11, f(5) = 17.

Still very predictable.

Its easy, for this function, to predict every value.

f(4) = ? f(1.5) = ? f(0) = ?14 6.5 2


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Because linear functions have nice predictable growth patterns, thegraph is a straight line.

Hence linear function.

All you need to know about this function:

It starts at 2;

it increases by 3s.


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More precisely:

1. The vertical intercept is 2, i.e. f(0) = 2;2. The slope is 3, i.e. whenever you move right 1, you go up 3.

Slope will be extremely important for us. In fact its thefoundation of all of Calculus.

Slope measures steepness when reading the graph from left toright.

(Note that steepness is reversed when reading from right to left.)

Numerically we define slope as the ratio of upward movement torightward movement.

J k d Jill h d hill


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Jack and Jill have a date on a hill.

Jack is climbing up the line from (1,5) to Jills picnic site at (4,14).

Jacks path;

1. Straight line; use Pythagorean Theorem to find distance;2. Horizontal and vertical components.

2.1 Horizontal: 4 1 = 32.2 Vertical: 14 5 = 9

Sl i th ti i /


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Slope is the ratio rise/run:

m =14 54 1 =

9

3= 3.

In general,

Slope = m =

y2

y1

x2 x1

Wh t th f l f thi f ti ?


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What s the formula for this function?

1. The vertical intercept is 2, i.e. f(0) = 2;

2. The slope is 3.

Fact 2 means that f grows like 3x. So

f(x) = 3x + ?.

To find the value of ?, use Fact 1:

2 = f(0) = 3 0 + ?.

So ? has a value of 2.

f(x) = 3x + 2

In general, the equation of a linear function is

f(x) = mx + b (Slope-Intercept Form)

where m is the slope and b is the vertical intercept.


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Heres another form of the linear equation:

Start with the definition of slope:

Slope = m = y2 y1x2 x1 .

Now rearrange terms to get rid of the fraction:

m(x2 x1) = y2

y1.

Since this holds for any (x2, y2) on the line, we can think of it asthe defining equation of the line:

m(x

x1) = y

y1.

This is called point-slope form.

Both forms have their strengths and weaknesses for problemsolving.

T f bl


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Types of problems:

1. Given information about a line, figure out its equation.

Point-Slope form is usually best.

2. Given the equation of a line, determine facts about it.

Slope-Intercept form is usually best.

Example: 1 2 #4


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Example: 1.2 #4

Given the equation 4y + 2x + 8 = 0, find the slope andy-intercept.

Slope-Intercept form is usually best.Solution Rewrite in Slope-Intercept form:

4y = 2x 8y =

1

2x + 2.

Now we can read off the info we need:

m =1

2b = 2

Example: 1 2 #8


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Example: 1.2 #8

A line passes through the points (4, 5) and (2,1). What is theequation of the line?

Point-Slope form is usually best.Solution:First find the slope.

m =y2 y1x2 x1

=1 52 4

=62

= 3

Now apply the point-slope form.

y 5 = 3(x 4)

There are exceptions


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There are exceptions.

Example: 1.2 #12

The population was 30,700 and grew at a rate of 850 per year.

1. Give the formula for P as a function of t (years since 2000).2. f(2010) = ?

3. f(?) = 45, 000

Solution

1. P = 850t + 30, 700.Note that P = f(t) where f(t) = 850t + 30,700.Ill usually just write P(t) = 850t + 30,700.

2. Plug in t = 10.

P(2010) = 850

10 + 30,700 = 39,200

3. Plug in P = 45,000.

45,000 = 850t + 30,70014,300 = 850t

t 16.8


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Example: 1.2 #24

Latitude # Species11 3444 26

(a) First find the slope:

m =N2 N1

l2 l1 =34 2611 44 =

833

.

Now apply point-slope form:

N 26 = 833

(l 44).


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N 26 = 833

(l 44).(b) Lets first rewrite the equation in slope-intercept form:

N = 833

l + 833

44 + 26 = 833

l + 1103

.

Now we can say:

m =

8

33species per degree;

b = 1103

species at the equator.

(c)


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What if a function isnt linear?

This function is sometimes steep, sometimes shallow, sometimesrising, sometimes falling.

Even in this case we can talk about average slope


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Even in this case we can talk about average slope.

average slope = 3 10.5 (1) = 83Note that

increasing = slope is +decreasing = slope is

Lets play Jeopardy


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Let s play Jeopardy...

The answer is:

The average slope from

3 to

1 is

,

the average slope from 1 to 1 is 0,the average slope from 1 to 3 is +.

The question is...

What is the shape of the function y = x2?


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This shape is called concave up, and occurs whenever the slopes

are consistently increasing.If the slopes are consistently decreasing, the shape is calledconcave down.

This even applies to any piece of the functions.


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What shapes?


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Remember I said that slope is the foundation of calculus?

We havent really started calculus yet but...

(careful, this is confusing)

Concave Up Slope is Sloping UpConcave Down Slope is Sloping Down


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(a) Concave Up (b) Concave Down

(c)Slope is Sloping Up

(d)Slope is Sloping Down


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Problem: A coffee shop sells coffee to go in three sizes.

Assuming the relationship between price and size is linear, whatdoes the 12 oz. coffee cost?

Size in oz. Price in $


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x y

8 1.0012 ?

20 2.08

(Its OK to reverse x and y, just be consistent.)

Solution: First find the slope:

m = 2.08 1.0020 8 = 1.0812 = 0.09.

Thereforey 1.00 = 0.09(x 8)

and y = 0.09x + 0.28.

Finally, when x = 12,

y = 0.09 12 + 0.28 = $1.36.


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y = 0.09x + 0.28.

Problem: What do m = 0.09 and b = 0.28 (the y-intercept)mean in this problem?

Solution: The slope is rise/run, in this case $/oz.

So evidently coffee costs 9 per ounce.

The value of b is the value that y takes when x = 0.

So if you buy 0 ounces of coffee it still costs you 28.

Evidently that is the price of renting a mug.

b = fixed costsmx = variable costs

1.4 #12fi d t $650 000


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fixed costs = $650,000variable costs = $20 per shoeselling price = $70 per shoe

Find the total cost in terms of q, the quantity of pairs of shoes.Also find the total revenue and total profit.

Solution:We have m = 20 and b =650,000.

C(q) = 20q+ 650,000

Revenue, the total amount of money taken in, is even easier:

R(

q) = 70

q.

Profit is just the difference between the two:

(q) = R(q) C(q) = 70q (20q+ 650,000) = 50q 650,000.

In business problems, the term margin means one more unit.


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So cost at the margin means the cost of producing one morepair of shoes.

Instead of saying cost at the margin youll often see the phrasemarginal cost, which means the same thing.

Ifthe cost function is linear, then the marginal cost is just theslope of the cost function.

C(q) = 20q+ 650,000

So the marginal cost is $20.

R(q) = 70q.

The marginal revenue is $70.

(q) = 50q 650,000.

The marginal profit is $50.


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How does slope behave for the Demand function?

Consistently .


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How does slope behave for the Supply function?

Consistently +.

The Equilibrium point is where the supply and demand curves


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The Equilibrium point is where the supply and demand curvesmeet: that is where the price and quantity are stable.

On the horizontal line, producers are depressed: not selling enough.On the vertical line, producers are manic: not producing enoughAt the equilibrium point, producers are perfectly content.

1.4 #26Demand: q = 120 000 500p


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Demand: q = 120,000 500pSupply: q = 1000p

(a) What happens at a price of$100 per unit?

(b) Find the equilibrium point.

Solution:

(a) Demand: q(100) = 50,000Supply: q(100) = 100, 000

Price is too high.

(b)120,000 500p = q = 1000p

120,000 = 1500p

p = 80At that price,

q = 1000 80 = 80,000.


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Suppose the company is selling pipes,which the government wants to discourage.So they tax the pipe company $5 per pipe.(This is called a specific tax.)

How do things change?

Old equations:


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Demand: q = 120,000 500pSupply: q = 1000p

Equilibrium: q = 80,000 and p = 80

New equations:

Demand: q = 120,000 500pSupply: q = 1000(p 5) = 1000p 5000

Equilibrium:

120,000 500p = q = 1000p 5000125,000 = 1500p

p = $83.33

At that price,

q = 1000 83.33 5000 = 78,333.

Youve probably heard about quantities that grow exponentially.For example debt


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For example, debt.Suppose you owe $100 to a loan shark who charges 30% interestper month.

If you dont pay anything back, next month youll owe $130.The month after that, youll owe the $130 plus an extra 30%:

130 + 0.30 130 = 1.30 130 = $169.

Every month you owe 1.3 times what you owed the month before.month debt

x y

0 $100

1 $100 1.3 = $1302 $100 1.32 = $1693 $100 1.33 = $219.70...

...x $100

1.3x


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So the formula for debt y as a function of time in months x is

y = 100

1.3x.

This called an exponential function because the input is pluggedinto an exponent.

Exponentially functions grow extremely quickly:

because the slope is continually getting steeper.When x = 24, you will owe 100 1.324 =$54,280.08.Other things that grow exponentially:

1. Population: people beget people;

2. Technology: inventions beget new inventions;

3. Mistakes: errors beget further errors.


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Some things decay exponentially:

The more you have, the more there is to be lost.

1. Radioactive elements;

2. Chemicals;

3. Endangered species.

Constantly decreasing, but at a slowing rate:slope is continually getting less steep.

It can take a very long time for the level to reach 0, and may neverquite happen.

1.5 #20Deforestation occurs at a rate of about 2.9% per year.


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p y

How much existing forest will remain in 35 years?

Solution:

Every year the amount of forest remaining is 2.9% less than it wasthe year before:

this year = (last year) .029(last year)= .971(last year)

So the base of the exponential will be .971.

Let x be the time in years from the start, and let y be the amountof forest remaining.And suppose the amount of forest at the start is F.

Then y = 0.971xF

so that after 35 years

y = 0.97135F = .357F.

So, about 36% of the original forest remains after 35 years.


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In general exponential functions have formulas like this:

P = P0at.

Here P is an exponential function of t, which is usually time.

And P0 is the initial value of P, that is, the value P takes when

time t = 0:P0 = P(0).

The constant a represents the ratio between values of P atsuccessive moments of time.

So a represents the growth or decay rate.

Growth Example


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pA towns population is currently 8000 and is growing at a rate of1% per year.

Express the population P as a function of time t from the present.

Solution:P = 8000 1.01t

Decay ExampleA towns population is currently 8000 and is falling at a rate of 2%per year.

Express the population P as a function of time t from the present.

Solution:P = 8000 0.98t


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A funny thing about bases...

M h h f b l


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Mathematicians have a favorite base, namely 2.718281828459045.

They call this base e.

So rather than write 0.98t, they prefer to write

e.02020270731751947t.

This is actually equivalent, because

e.02020270731751947 = 0.98,

so

e.02020270731751947t

= (e.02020270731751947

)t

= 0.98t.

This number .02020270731751947 is usually referred to as thedecay rate.

If k had been +, we would call it the growth rate.


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Remember the CarCrusher and Uncrusher functions?

Car Crusher

Function

Function

UnCrusher

Replace CarCrusher by exponential function and UnCrusher


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Replace CarCrusher by exponential function and UnCrusherby logarithmic:

So for example, if e3 = 20.08553692318767, that is


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3 exponential 20.08553692318767,

then20.08553692318767 logarithmic 3,

that is ln(20.08553692318767) = 3.

So the exponential and logarithmic functions simply undo each

other.

For any x and a > 0,

a = ex ln(a) = x.

For any x and a > 0,


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a = ex ln(a) = x.

Properties of ln:

1. Because e1 = e,ln(e) = 1.

2. Because e0 = 1,ln(1) = 0.

3. Because (ex)y = exy,

ln(ay) = yln(a).

(a = ex)4. Because exey = ex+y,

ln(ab) = ln(a) + ln(b).


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Example:

Find the number k such that

.98t = ekt.

Solution:ln(.98t) = ln(ekt)tln(.98) = ktln(e) .0202t = kt

.0202 = k

ExampleSuppose a population is initially 35,000 and grows 12% in 10 years.


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Express the population as a function of time, using base e.

Solution: We know the general form of the exponential function:

P(t) = P0ekt.

We also know that P0 = 35,000.Now use the known growth data to find the growth rate k:

P(10) = 1.12 35,000= 35,000e10k

1.12 35,000 = 35,000e10k1.12 = e10k

ln(1.12) = 10kln(e)0.1133 = 10k 1

k = 0.01133

So P(t) = 35,000e

0.01133t

.

In an exponential growth problem, the doubling time is the timerequired for the population to double.


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q p p

This does not depend on the initial population!

ExampleSuppose a population is initially P0 and has a growth constant of0.02.How long does the population take to double in size?

Solution: P = P0e0.02t

When is P = 2P0?

2P0 = P0e0.02t

2 = e0.02tln(2) = 0.02t

t =ln(2)

0.02= 34.66

The corresponding concept for exponential decay is half-life, thetime it takes for the quantity to be reduced to half the initial size.


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Example: An isotope of Cadmium has a half-life of 13.6 years.

How long will it take for 99% of the original quantity to decay?Solution:We know the equation looks like

Q = Q0ekt

First we need to figure out k.We know that Q = 1

2Q0 when t = 13.6 years.

1

2Q0 = Q0e

13.6k

12 = e

13.6k

ln(12

) = 13.6k

k =1

13.6ln(1

2) = 0.051

Example: An isotope of Cadmium has a half-life of 13.6 years.


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How long will it take for 99% of the original quantity to decay?

Solution:

So now we have this:

Q = Q0e0.051t.

We want to know at what time t the quantity Q will have been

reduced to 0.01Q0.

0.01Q0 = Q0e0.051t

0.01 = e0.051t

ln(0.01) =

0.051t

t = 10.051

ln(0.01) = 90.3

So it will take 90.3 years.

Suppose you deposit $1000 in a savings account at an interest rate


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Suppose you deposit $1000 in a savings account at an interest rateof 2% APY (annual percentage yield).

Every year your balance grows by a factor of 1 + 0.02 = 1.02.So after t years your balance has grown to 1000(1.02)t.

In general, if

the interest rate = r% APYthe initial deposit = B0

the number of years elapsed = tand the balance after t years = B

thenB = B0(1 + r)

t.


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Now well look at interest from a slightly different perspective.

We saw in the previous problem that $1000 invested at 3% APY


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for 20 years yields $1806.11.

We say that the future value of$1000 now is $1806.11.

We can also say that the present value of$1806.11 in the future is$1000.

These are just two different ways of saying the same thing.

If we let P be the present value, i.e. B = P0,then

B = P(1 + r)t.

This gives us B in terms of P.

If we want P in terms of B we can rearrange the equation:

P =B

(1 + r)t.


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Present and Future Value

B = P(1 + r)t

P =B

(1 + r)t= B(1 + r)t.

Where

t = timer = interest rate (APY)

P = present value

B = future value

1.7 #34Assume an interest rate of 7.75% APY.


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Buy: $12,000 nowTaxes: $580 in 1 year

$464 in 2 years$290 in 3 years

Sell: $5000 in 3 yearsNominal Cost: $8334

Versus:

Lease: $2650 now$2650 in 1 year$2650 in 2 years$2650 in 3 years

Nominal Cost: $10,600

Which deal is better?

Solution: Compute the present value of each to compare.

Lease: $2650 now


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Lease: $2650 now$2650 in 1 year$

2650 in 2 years$2650 in 3 yearsNominal Cost: $10,600

Compute present value of the costs with r = 7.75% using

P =B

(1 + r)t= B(1 + r)t.

Total Cost =2650

(1.0775)0

+2650

(1.0775)1

+2650

(1.0775)2

+2650

(1.0775)3

= $9510.23

Buy: $12,000 nowT $580 i 1


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Taxes: $580 in 1 year$464 in 2 years

$290 in 3 yearsSell: $5000 in 3 years

Nominal Cost: $8334

Compute present value of the cost with r = 7.75%.

Total Cost =12,000

(1.0775)0+

580

(1.0775)1+

464

(1.0775)2+

290

(1.0775)3

5000

(1.0775)3

= $9149.68

Conclusion: Buying is still cheaper than leasing.

Suppose you deposit $1000 in the bank at 2% interestcompounded quarterly.That means you are paid 2 = 0 5% interest every quarter (3


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That means you are paid4

= 0.5% interest every quarter (3months).

So after k quarters your account has grown to

1000(1 + 0.005)k.

If t is number of years, then k = 4t.

So we get the formula

P = P0(1 + .005)4t.

In general, if r is the nominal annual interest rate, and n is thenumber of compounding periods,then the formula is

B = P(1 + rn

)nt,

where t = time in years.

Compound Interest

B P(1 r )nt


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B = P(1 + rn

)nt

t = time in yearsn = # compounding periods per year

Example: Suppose you deposit $100 in a savings account at 2%nominal interest.

Compute the future balance at 1 and 5 years under yearly,quarterly, and daily compounding.

Solution:Yearly Compounding, n = 1:

B = 100(1 + .021

)11 = 100 1.021 = 102

B = 100(1 + .021

)15 = 100 1.025 = 110.41

Compound Interest


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B = P(1 + rn

)t

t = time in yearsn = # compounding periods per year

Quarterly Compounding, n = 4:

B = 100(1 + .024 )41 = 100 1.0054 = 102.015

B = 100(1 + .024

)45 = 100 1.00520 = 110.49

Daily Compounding, n = 365:

B = 100(1 + .02365

)3651 = 100 1.00005479365 = 102.02

B = 100(1 + .02365

)3655 = 100 1.000054791825 = 110.52

Compound Interest

B = P(1 + rn

)t


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(n

)

t = time in years

n = # compounding periods per year

What happens as the number of compounding periods getsgreater?

1. Daily Compounding, n = 365:

B = 100(1 + .02365

)3651 = 100 1.00005479365 = 102.02

2. Hourly Compounding, n = 8760:

B = 100(1 +.028760)

87601

= 100 1.0000023418760

= 102.02

3. Per Minute Compounding, n = 525600:

B = 100(1 + .02525600

)5256001 = 100 1.00000003805175525600 = 102.02

Moral: For large n the results are all nearly the same.

So we can define continuous compounding to mean compounding


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p g p gwhen n is large.

Since the particular value of n doesnt matter, well just say that

n = .

So if you invest $100 in a savings account at 2% nominal interestcompounded continuously, after one year your balance has grownto

100

1 +

0.02

= $102.02.


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Another big number is

.

Suppose that

=r.

then

= r .


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Substituting

= r

into the growth factor

1 +

r

gives 1 +

1r


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So the growth factor can also be written

1 +

1 r

.

Notice that the expression in [ ]s is a constant.Evaluate it on your calculator:

1 +

1

= 2.71828...

Remember that number?

Its mathemticians favorite base, e.


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So the growth factor is er,

and the one year continuous interest formula is

B = Per.

What happens if you wait longer than one year?

You multiply by the growth factor once for each year.

B = Pererer...

So if the money is invested for t years,

B = P(er)t = Pert.

Continuous Compounding Formula


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B = Pert

where

t is time in years;r is the nominal interest rate;

P is the initial investment;B is the balance after t years.

Example: Suppose $1500 is invested at 3% interest compounded

continuously.What is the balance after 20 years?

Solution: B = 1500e0.0320 = $2733.18

Recycling functions:


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Functions can be modified to get the behavior you want.

Example:Suppose you have a function f that has the shape you want, but...

f(2) = 4 while you want it to be 5.

Solution:Define a new function g(x) = f(x) + 1.

Same shape, but g(2) = 5.

Because every y-value has been increased by 1,

we call g a vertical shift of f.

This might be f:


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Then this would be g:

Theres another way to get almost the same effect.Define

h(x) =5

f (x).


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h(x) 4

f(x).

Thenh(2) =

54

f(2) =54

4 = 5.

The graph of h is almost the same shape as the graph of f, itsjust been stretched vertically.

So know we know how to vertically shift and stretch functions.

Vertical Shift g = f + C


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Vertical Shift g = f+ C

Vertical Stretch g = Cf

In both forumulas C is a constant, like 1 or 54

.

Vertical Shift: if C is + then the shift is up;

if C is then the shift is down.

Vertical Stretch: if |C| > 1 then the stretch is out (expands);if

|C

|< 1 then the stretch is in (shrinks);

if C is + then the stretch preserves orientation;if C is then the shift reverses orientation.

VerticalShift

VerticalStretch

formula: g = f+ C g = Cf

C > 0: up rightside up


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C > 0: up rightside-up

C < 0: down upside-down

|C| > 1: expand|C| < 1: shrink

Examples:

f(x) f(x) + 2

f(x) f(x) 2

Sexual Reproduction of Functions

Wh t f ti l t th


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When two functions are alone together...

f(x) g(x)

...sometimes they get intimate.

f(g(x))

This produces a new function well call h.

h(x) = f(g(x))

Mathematicians call this composition of functions.

The functions f and g have been composed to form h.

Composition is what geeks mean by daisy-chaining.


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Composition of Functions


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p

input g f outputoutput = f(g(input))

y = f(g(x))

Example:f(x) = 2x + 3, g(x) = 5x 1, and h(x) = f(g(x)).

Find h(4).Solution:h(4) = f(g(4)) = f(19) = 41



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p

input g f outputoutput = f(g(input))

y = f(g(x))

Example:f(x) = 2x + 3, g(x) = 5x 1, and h(x) = f(g(x)).

Find h(x).Solution:h(x) = f(g(x)) = f(5x1) = 2(5x1)+3 = 10x2+3 = 10x+1



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p

input f g outputoutput = g(f(input))

y = g(f(x))

Example:f(x) = 2x + 3, g(x) = 5x 1, and h(x) = g(f(x)).

Find h(4).Solution:h(x) = g(f(4)) = g(11) = 54



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input f g outputoutput = g(f(input))

y = g(f(x))

Example:f(x) = 2x + 3, g(x) = 5x 1, and h(x) = g(f(x)).

Findh

(x

).Solution:h(x) = g(f(x)) = g(2x + 3) = 5(2x + 3) 1 = 10x + 14

The shape of the graph of f(g(x)) can be hard to predict.Example:f(x): g(x):


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f(g(x)): g(f(x)):

The only time its easy to predict the shape of f(g(x)) is when gis a linear function.

Example: f = g (x) = x + 1


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Example: f = , g(x) = x + 1.

What is the graph of f(g(x))?

Notice that g is a shift applied to the simple function y = x.

But the output of g is the input of f.

So its the x values of f that are changed, not the y values.

Note the shift is to the left


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Note the shift is to the left.

To remember the direction of the shift, think:f(x + 1) is 1 unit ahead of f(x)

(when reading from left to right).

f(x 1) is 1 unit behind f(x), so its graph is

You can also do horizontal stretches: f(Cx).As with shifts, directions are reversed in comparison with verticalstretches:

f (x) f (2x)


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f(x) f(2x)

f(x) f(12

x)


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HorizontalShift

HorizontalStretch

formula: g(x) = f(x + C) g(x) = f(Cx)

C > 0: left usual orientation

C < 0: right mirror orientation

|C| > 1: shrink|C| < 1: expand

In calculus it will useful to able to decompose complicatedfunctions into their simpler constituent parts.

Example: h(x) = ex + x2

If d fi f ( ) x d ( ) 2 h


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If we define f(x) = ex and g(x) = x2 then

h(x) = f(x) + g(x).

Example: h(x) = x2ex

Solution: h(x) = g(x)f(x)

Example: h(x) = ex2

Solution: h(x) = f(g(x))

Example: h(x) = e2x

Solution: h(x) = g(f(x))

g(f(x)) = g(ex) = (ex)2 = e2x

ProportionalityRecall the predictability of linear functions:

Example:S f i li d f (3) 7 d f (5) 19


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Suppose f is linear and f(3) = 7 and f(5) = 19.

What is f(3.4)?

Solution:The input 3.4 is 20% of the way from 3 to 5.

Therefore f(3.4) is 20% of the way from 7 to 19.

f(3.4) = 7 + 0.20(19 7) = 7 + 0.20 12 = 7 + 2.4 = 9.4The rise and the run are proportional:they are each the same % of something.

Run is a % of 5

3 = 2;

Rise is a % of 19 7 = 12.So rise is always 6 times bigger than run.

rise = 6 run

Whenever variables x and y are related by

y = kx


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for some constant k,we say that y is directly proportional to xwith constant of proportionality k.

Examples:

1. Gas consumed and miles driven. (At least for constant speed.)2. Temperature and power consumption. (AC)

3. Population and food consumption.

4. Force of a spring and its displacement (stretch).


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Example: A spring in free equilibrium is 3 inches long.When stretched with a 2 pound weight it stretches to 7 inches.

1. How much would it stretch with a 1.5 pound weight?

2. How much weight would be necessary to stretch it to 8inches?

Solution: Let x be displacement and y be the weight.The y = kx for some constant k.

We know that y = 2 when x = 4.

2 = k4k = 2

4= 0.5

1. How much would it stretch with a 1.5 pound weight?

2. How much weight would be necessary to stretch it to 8


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2. How much weight would be necessary to stretch it to 8

inches?

1. When y = 1.5,

1.5 = 0.5x x = 1.50.5 = 3.

So the spring would stretch to 3 + 3 = 6 inches.

2. When x = 8 3 = 5,

y = 0.5 5 = 2.5.

So you would need to use 2.5 pounds.

StocksSuppose a company is worth $50,000,000 and has issued 1,000,000shares of stock.


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Then each share is worth (nominally) $50.

value per share =$50,000,000

1,000,000 = $50

Let x be the number of shares issued,

let y be the value per share,and let k be the total value of the company. Then

y =k

x= k

1

x.

We can say that the value per share is directly proportional to theinverse of the number of shares,

or we can say that the value per share is inversely proportional tothe number of shares.

Example: Suppose that y is inversely proportional to x,and y = 20 when x = 12.

What would y be when x = 16? When x = 8?


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What would y be when x 16? When x 8?

Solution: First find k:

20 =k

12

240 = k

Then when x = 16,

y =240

16= 15,

and when x = 8,

y =240

8= 30.

GravityIf F is the gravitational force between two bodies,and r is the distance between them, then

G


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F =

G

r2

where G is the gravitational constant.

We can describe this in three different ways.

1. Force is inversely proportional to the square of the distance;2. Force is directly proportional to the inverse of the square of

the distance;

F = G

1

r2

3. Force is a power function of the inverse of the distance.

F = G

1

r

2

Power FunctionsWe say that y is a power function of x if

y = kxp


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where k is the constant of proportionalityand p is another constant.

If

F =G

r2

then we can say:F is a power function of

1

r

F = G1r2

orF is a power function of r

F = Gr2.

If y = 5x then y is a power function of x:y = 5x

1

2 .

Power functions are the third simplest kinds of functions.


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Power functions are the third simplest kinds of functions.

But theres quite a variety of shapes.y = x2

-4

-2

0

2

4

-4 -3 -2 -1 0 1 2 3 4

x2

x

y = x2

0

2

4

1/x

2

Polynomials

A monomial is a power function whose power is a whole number.

E l s 2 3 17 12 2


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Examples: 2x3, 17x12, ex2.

A polynomial is a sum of monomials.

Examples:3x2 4x + 1

5 4x + 3x2 9x3 + 2x4

(x 3)(2x + 5) = 2x2 x 15

3x2

e1/2

x + The highest power of the variable is called the degree of the

polynomial.

Graphs of PolynomialsThe number of bumps can be as large as the degree.

The graphs of polynomials can look like virtually anything.


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y = 13

(x3 3x)

-4

-2

0

2

4

-4 -3 -2 -1 0 1 2 3 4

(x

3-3*x)

/3

x

y = 112

(3 x4 + 4 x3 12 x2)

2

4

1.9 #38:

C = 10,000 + 35qR = pqq = 3000 20p (Demand)


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(a)

C = 10,000 + 35(3000 20p) = 115,000 700p

R = p(3000 20p) = 3000p 20p2

(b)C = 115,000 700pR = 3000p 20p2

60000

80000

100000

120000

$

Revenue

Cost

411

Remember these equations?


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Remember these equations?

e =

1 +

1

er =

1 +

r

Unprofessional looking...

Lets go back to using n for the gigantic number.

1 n


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e =

1 +

1

n

But we need a reminder this is valid only ifthe value n is gigantic.

e = Ifn

1 + 1nn

Read the as approaches.The answer is more reliable for bigger values of n.

Theres just one hitch...

e = Ifn

1 +

1

n

nThis works fine for the present expression


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f(n) =

1 +

1

n

n

0

0.5

1

1.5

2

2.5

3

10 20 30 40 50 60 70 80 90 100

(1+1/n)n

n

But what about other functions?

Suppose that f is this function:

1

1.5


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-1.5

-1

-0.5

0

0.5

10 20 30 40 50 60 70 80 90 100

(1+1/n)n

n

What isIf

nf(n) = ?

We say in this case that f(n) has no limit as n .For the previous function the limit was e.

Since we need the limits to exist, we will write


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limn

f(n).

Thus

limn1 +

1

n

n

= e

limn

1 +

r

n

n= er.

Consider some other limits.

limx

1

x= 0

1


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limx

3 + x

= 3

limx

1

x + 3

= 0

limx

x

x + 3

= 1

limx x + 5

x + 3 = 1

limx

x

x2 + 3

= 0

limx

x =

lim

x =


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x

limx

x2 =

limx

x2 + x =

limx

x2 x =

limx

x2

100x =

limx

100x x2 =

We get the following rule:

If f is a polynomial, then


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limx f(x) = where the sign is the same asthat of the leading term in f.

By leading term we mean the term of the highest degree (power).For example,

limx

3x5 100x4 90x3 + 7x2 + 3x 1 =

limx

3x5 + 100x4 90x3 + 7x2 + 3x 1 = .

These two are easy:limx

ex =

lim

ln(x) = .


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x

But this one is tricky:

limx

ex x5

= ?

Key: ex more than doubles on every step to the right.

ex+1

ex= e = 2.718.

By comparison, x5

hardly seems to grow at all:

(x + 1)5

x5 1.

So ex consistently outgrows x5, and therefore

limx

ex x5 = 5


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limx

x5

ex = limx

x5

ex

= 0

limx

ex

x5

=

The same is true when you replace x5 by any polynomial:

limx

ex

100x7 3x4 + 25x3 2

= .

The opposite is true for ln.

60

80

100%e

x

log(x)


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0

20

40

0 20 40 60 80 100

y

x

limx

ln(x) x5 =

limx

x

5

ln(x) = limx

x5

ln(x)

=

You can take limits as x

anything, not just

.

limx

100x7 + 3x4 + 25x3 2 =

limx0

100x7 + 3x4 + 25x3 2 = 2


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x0 limx0

1

x=

-4

-2

0

2

4

-4 -2 0 2 4

1/x

x

limx0

1

x + 2=

1

2

lim 1


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limx2 x + 2 =

limx2

1

x + 2=

1

4

limx2

x + 1x + 2

= 34

limx2

x + 1

x + 2=

limx2

x + 2

x + 2= 1

limx2

x + 2

x + 2= 1

(x + 2)2


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limx2 x + 2 = 0

Because(x + 2)2

x + 2= x + 2.

limx2

(x

+ 2)(x

+ 1)x + 2 = 1

Because(x + 2)(x + 1)

x + 2= x + 1.

limx2

x2 + 3x + 2

x + 2= 1

limx0

x2 + 3x

x= 3

Becausex2 + 3x

= x + 3.


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x

limx0

5x2 4xx

= 4

limx0

5x3

4x2

x = 0

limx0

5x3 4x2 + 7xx

= 7

limx0

5x3 4x2 + 7x + 1x

=

Instantaneous Rate of ChangeOr, Instantaneous Slope.

Problem: Find the slope of the function f(x) = x2 at the point(1, 1).


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0

0.5

1

1.5

2

2.5

-0.5 0 0.5 1 1.5

x

Not the average

slope between 0and 2 or between.5 and 1.5.

The slope at the

point x = 1.

What we want is the slope of the tangent line.

2

2.5


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0

0.5

1

1.5

2

-1 -0.5 0 0.5 1 1.5

y

x

How do we calculate that?

We can use average slope to get a pretty good approximation of

instantaneous slope.

4

5


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0

1

2

3

0.5 1 1.5 2 2.5 3

y

x

4

5

Instantaneous Slope = lim (Average Slope)

Lets use this to compute the slope of the tangent line.

5


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0

1

2

3

4

0.5 1 1.5 2 2.5 3

y

x

Here the two pointsused for the averageslope are (1, 1) and(a, a2),

and a 1.

The average slope is

Average Slope =a2 1a 1 .

Average Slope =a2 1a 1 .


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We want the limit as a 1:lima1

a2 1a 1

= lima1

(a

1)(a + 1)

a 1

= lima1

(a + 1) = 2

You can use this technique to calculate the instantaneous slopeanywhere on the parabola.

Heres another way to do the algebra.Lets use h to represent the distance of the second point from thefirst.

Then a = 1 + h.

A d k h l h


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And we take the limit as h 0.

limh0

(1 + h)2 1(1 + h) 1

= limh0

1 + 2h + h2 1h

= limh0

2h + h2

h

= limh0

(2 + h) = 2

This style is used more generally than the previous one.

The derivative of a function is the value of the instantaneous slope(or equivalently, the slope of the tangent line).

Note that the derivative is also a function: its values depend on

h i


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the input x.

In other words, the value of the derivative varies from point topoint.

Shorthand notation: the derivative of f can be denoted f.

So for the previous function, f(1) = 2.

f(5) is probably something completely different.

For a linear function g with slope m, we get g(x) = m for all x.

For example, if g(x) = 4x 6 then g(1) = 4 = g(2) = g(19).

Lets find a general formula for the derivative of f(x) = x2

.We know that f(1) = 2, now wed like to know f(x) =?.

To warm up lets compute f(3).

The two points used in the average slope are (3, 9) andh ( h)2


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3 + h, (3 + h)2

.

f(3) = limh0

(3 + h)2 9(3 + h) 3

= limh0

9 + 6h + h2 9h

= lim

h0

6h + h2

h

= limh0

(6 + h) = 6

So now we know that f(1) = 2 and f(3) = 6.

Lets repeat the calculation for x instead of 1 or 3.

The two points used in the average slope are (x, x2) and

x + h, (x + h)2.


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f(x) = lim

h0

(x + h)2 x2(x + h) x

= limh0

x2 + 2xh + h2

x2

h

= limh0

2xh + h2

h

= limh0

(2x + h) = 2x

So f(x) = 2x.

2

3

4

ff

I f h d i i f


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-4

-3

-2

-1

0

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

y

x

In fact the derivative of anyquadratic function is linear.

More generally, the derivative ofa polynomial is always simpler

(lower degree) than the original.

Note that f slopes up because fis concave up.

notes on "applied calculus", flath/gleason/et.al

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