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OUTLINE:

1. Review of functions and graphs

2. Review of limits

3. Review of derivatives - the idea of velocity, tangent, and normal lines to curves

4. Review of related rates and max/min problems

1. Review of functions and graphs

2. Review of limits

3. Review of derivatives - the idea of velocity, tangent, and normal lines to curves

4. Review of related rates and max/min problems

1.
to MATH 104-002:

Calculus I

Calculus I

2.
Welcome to the Course

1. Math 104 – Calculus I

2. Topics: quick review of Math 103 topics, methods and

applications of integration, infinite series and applications.

3. Pace and workload:

Moves very fast

Demanding workload, but help is available!

YOU ARE ADULTS - how much do you need to practice

each topic?

Emphasis on applications - what is this stuff good for?

4. Opportunities to interact with professor, TA, and other

1. Math 104 – Calculus I

2. Topics: quick review of Math 103 topics, methods and

applications of integration, infinite series and applications.

3. Pace and workload:

Moves very fast

Demanding workload, but help is available!

YOU ARE ADULTS - how much do you need to practice

each topic?

Emphasis on applications - what is this stuff good for?

4. Opportunities to interact with professor, TA, and other

3.
Outline for Week 1

(a)Review of functions and graphs

(b)Review of limits

(c)Review of derivatives - idea of velocity,

tangent and normal lines to curves

(d)Review of related rates and max/min

problems

(a)Review of functions and graphs

(b)Review of limits

(c)Review of derivatives - idea of velocity,

tangent and normal lines to curves

(d)Review of related rates and max/min

problems

4.
Functions and Graphs

The idea of a function and of the graph of a function should be very familiar

4 3 2

f (x) = 6x − x − 9x + 4 x

The idea of a function and of the graph of a function should be very familiar

4 3 2

f (x) = 6x − x − 9x + 4 x

5.
Questions for discussion...

1. Describe the graph of the function f(x) (use calculus vocabulary as

appropriate).

2. The graph intersects the y-axis at one point. What is it (how do

you find it)?

3. How do you know there are no other points where the graph

intersects the y-axis?

4. The graph intersects the x-axis at four points. What are they (how

do you find them)?

5. How do you know there are no other points where the graph

intersects the x-axis?

6. The graph has a low point around x=4, y=-100. What is it exactly?

How do you find it?

7. Where might this function come from?

1. Describe the graph of the function f(x) (use calculus vocabulary as

appropriate).

2. The graph intersects the y-axis at one point. What is it (how do

you find it)?

3. How do you know there are no other points where the graph

intersects the y-axis?

4. The graph intersects the x-axis at four points. What are they (how

do you find them)?

5. How do you know there are no other points where the graph

intersects the x-axis?

6. The graph has a low point around x=4, y=-100. What is it exactly?

How do you find it?

7. Where might this function come from?

6.
Kinds of functions that

should be familiar:

Linear, quadratic

Polynomials, quotients of polynomials

Powers and roots

Exponential, logarithmic

Trigonometric functions (sine, cosine, tangent, secant,

cotangent, cosecant)

Hyperbolic functions (sinh, cosh, tanh, sech, coth, csch)

should be familiar:

Linear, quadratic

Polynomials, quotients of polynomials

Powers and roots

Exponential, logarithmic

Trigonometric functions (sine, cosine, tangent, secant,

cotangent, cosecant)

Hyperbolic functions (sinh, cosh, tanh, sech, coth, csch)

7.
Quick Question

The domain of the function

1 x is...

f(x)

2

x 2x

A. All x except x=0, x=2

B. All x < 1 except x=0.

C. All x > 1 except x=2.

D. All x < 1.

E. All x > 1.

The domain of the function

1 x is...

f(x)

2

x 2x

A. All x except x=0, x=2

B. All x < 1 except x=0.

C. All x > 1 except x=2.

D. All x < 1.

E. All x > 1.

8.
Quick Question

Which of the following has a graph that

is symmetric with respect to the y-axis?

A. y=

x 1 5 2

D. y= x 2x

x

x

4 E. y= 3

B. y= 2x 1 x 3

3

C. y= x 2x

Which of the following has a graph that

is symmetric with respect to the y-axis?

A. y=

x 1 5 2

D. y= x 2x

x

x

4 E. y= 3

B. y= 2x 1 x 3

3

C. y= x 2x

9.
Quick Question

The period of the function

3 x

f ( x) sin is...

5

A. 3

B. 3/5

C. 10/3

D. 6/5

E. 5

The period of the function

3 x

f ( x) sin is...

5

A. 3

B. 3/5

C. 10/3

D. 6/5

E. 5

10.
Quick Question

a

a

If log a 5 , then a=

3

A. 5

B. 15

C. 25

D. 125

E. None of these

a

a

If log a 5 , then a=

3

A. 5

B. 15

C. 25

D. 125

E. None of these

11.
Limits

Basic facts about limits

The concept of limit underlies all of calculus.

Derivatives, integrals and series are all different kinds of limits.

Limits are one way that mathematicians deal with the infinite.

Basic facts about limits

The concept of limit underlies all of calculus.

Derivatives, integrals and series are all different kinds of limits.

Limits are one way that mathematicians deal with the infinite.

12.
First things first...

First some notation and a few basic facts.

Let f be a function, and let a and L be fixed numbers.

Then lim f( x) L is read

x a

"the limit of f(x) as x approaches a is L"

You probably have an intuitive idea of what this means.

And we can do examples:

First some notation and a few basic facts.

Let f be a function, and let a and L be fixed numbers.

Then lim f( x) L is read

x a

"the limit of f(x) as x approaches a is L"

You probably have an intuitive idea of what this means.

And we can do examples:

13.
For many functions...

...and many values of a , it is true that

lim f ( x) f (a )

x a

And it is usually apparent when this is not true.

"Interesting" things happen when f(a) is not

well-defined, or there is something "singular"

about f at a .

...and many values of a , it is true that

lim f ( x) f (a )

x a

And it is usually apparent when this is not true.

"Interesting" things happen when f(a) is not

well-defined, or there is something "singular"

about f at a .

14.
Definition of Limit

So it is generally pretty clear what we

mean by

lim f ( x) L

x a

But what is the formal mathematical

So it is generally pretty clear what we

mean by

lim f ( x) L

x a

But what is the formal mathematical

15.
Properties of real numbers

One of the reasons that limits are so difficult to

define is that a limit, if it exists, is a real number.

And it is hard to define precisely what is meant

by the system of real numbers.

Besides algebraic and order properties (which

also pertain to the system of rational numbers),

the real numbers have a continuity property.

One of the reasons that limits are so difficult to

define is that a limit, if it exists, is a real number.

And it is hard to define precisely what is meant

by the system of real numbers.

Besides algebraic and order properties (which

also pertain to the system of rational numbers),

the real numbers have a continuity property.

16.
Least upper bound property

If a set of real numbers has

an upper bound, then it has

a least upper bound.

If a set of real numbers has

an upper bound, then it has

a least upper bound.

17.
Important example

2

The set of real numbers x such that x 2. The

corresponding set of rational numbers has no least

upper bound. But the set of reals has the number 2

In an Advanced Calculus course, you learn how to start

from this property and construct the system of real

numbers, and how the definition of limit works from

2

The set of real numbers x such that x 2. The

corresponding set of rational numbers has no least

upper bound. But the set of reals has the number 2

In an Advanced Calculus course, you learn how to start

from this property and construct the system of real

numbers, and how the definition of limit works from

18.
Official definition

lim f(x) = L means that for any ε > 0,

x→ a

no matter how small, you can find a δ > 0

such that if x is within δ of a, i.e., if x-a < δ,

then f(x)-L < ε

lim f(x) = L means that for any ε > 0,

x→ a

no matter how small, you can find a δ > 0

such that if x is within δ of a, i.e., if x-a < δ,

then f(x)-L < ε

19.
For example….

2

lim x 25

x 5

because if 1 and we choose 111

Then for all x such that x 5 we have

5 x 5 and so

25 10 2 x 2 25 10 2

which implies

2

2

2

x 25 10 10

11 121

2

lim x 25

x 5

because if 1 and we choose 111

Then for all x such that x 5 we have

5 x 5 and so

25 10 2 x 2 25 10 2

which implies

2

2

2

x 25 10 10

11 121

20.
Top ten famous limits:

1 1

1. lim lim

x 0 x x 0 x

1

2. lim 0

x x

1 1

1. lim lim

x 0 x x 0 x

1

2. lim 0

x x

21.
n

3. (A) If 0 < x < 1 then lim x 0

n

n

(B) If x > 1, then lim x

n

sin x 1 cos x

4. lim 1 and lim 0

x 0 x x 0 x

x x

5. lim e 0 and lim e

x x

3. (A) If 0 < x < 1 then lim x 0

n

n

(B) If x > 1, then lim x

n

sin x 1 cos x

4. lim 1 and lim 0

x 0 x x 0 x

x x

5. lim e 0 and lim e

x x

22.
n

x

6. For any value of n, lim x 0

x e

ln x

and for any positive value of n, lim n 0

x x

1

lim sin 6-10

7. x 0

x

does not exist!

x

6. For any value of n, lim x 0

x e

ln x

and for any positive value of n, lim n 0

x x

1

lim sin 6-10

7. x 0

x

does not exist!

23.
8. lim x ln( x) 0

x 0

x

1

9. lim 1 e

x

x

10. If f is differentiable at a, then

f ( x) f (a)

lim f ' (a)

x a x a

x 0

x

1

9. lim 1 e

x

x

10. If f is differentiable at a, then

f ( x) f (a)

lim f ' (a)

x a x a

24.
Basic properties of

limits

I. Arithmetic of limits:

If both lim f ( x) and lim g ( x) exist, then

x a x a

lim f ( x) g ( x) lim f ( x) lim g ( x)

x a x a x a

lim f ( x) g ( x) lim f ( x) lim g ( x)

x a x a x a

and if f ( x) lim f ( x)

lim g ( x) 0 , then lim x a

x a x a g ( x) lim g ( x)

x a

limits

I. Arithmetic of limits:

If both lim f ( x) and lim g ( x) exist, then

x a x a

lim f ( x) g ( x) lim f ( x) lim g ( x)

x a x a x a

lim f ( x) g ( x) lim f ( x) lim g ( x)

x a x a x a

and if f ( x) lim f ( x)

lim g ( x) 0 , then lim x a

x a x a g ( x) lim g ( x)

x a

25.
II. Two-sided and one-sided

limits:

lim f ( x) L if and only if

x a

BOTH lim f ( x) L and lim f ( x) L

x a x a

III. Monotonicity:

If f(x) g(x) for all x near a,

then lim f( x) lim g( x)

x a x a

limits:

lim f ( x) L if and only if

x a

BOTH lim f ( x) L and lim f ( x) L

x a x a

III. Monotonicity:

If f(x) g(x) for all x near a,

then lim f( x) lim g( x)

x a x a

26.
IV. Squeeze theorem:

If f(x) g(x) h(x) for all x near a, and if

lim f ( x) lim h( x), then lim g( x) exists and

x a x a x a

is equal to the common value of the other

two limits.

1

−x ≤x sin ≤x

x

1

lim x sin = 0

x→ 0 x

If f(x) g(x) h(x) for all x near a, and if

lim f ( x) lim h( x), then lim g( x) exists and

x a x a x a

is equal to the common value of the other

two limits.

1

−x ≤x sin ≤x

x

1

lim x sin = 0

x→ 0 x

27.
Let’s work through a few:

x 5 x 5

lim lim

x 2 x 2 x 2 x 2

2

x 4

lim

x 2 x 2

x 5 x 5

lim lim

x 2 x 2 x 2 x 2

2

x 4

lim

x 2 x 2

28.
Now you try this one...

2 t 2

lim

t 0 t

A. 0 E. -1

B. F. 2

C. -1/2 G. -2

1 1

D. H.

2 2 2 2

2 t 2

lim

t 0 t

A. 0 E. -1

B. F. 2

C. -1/2 G. -2

1 1

D. H.

2 2 2 2

29.
Continuity

A function f is continuous at x = a if it is true

that lim

x→ a

f(x) = f(a)

(The existence of both the limit and of f(a) is

implicit here).

€ Functions that are continuous at every point

of an interval are called "continuous on the

interval".

A function f is continuous at x = a if it is true

that lim

x→ a

f(x) = f(a)

(The existence of both the limit and of f(a) is

implicit here).

€ Functions that are continuous at every point

of an interval are called "continuous on the

interval".

30.
Intermediate value theorem

The most important property of continuous functions is the

"common sense" Intermediate Value Theorem:

Suppose f is continuous on the interval [a,b], and f(a) = m, and

f(b) = M, with m < M. Then for any number p between m and

M, there is a solution in [a,b] of the equation f(x) = p.

The most important property of continuous functions is the

"common sense" Intermediate Value Theorem:

Suppose f is continuous on the interval [a,b], and f(a) = m, and

f(b) = M, with m < M. Then for any number p between m and

M, there is a solution in [a,b] of the equation f(x) = p.

31.
Application of the intermediate-value theorem

3

f ( x) x 2 x 2

Maple graph

Since f(0)=-2 and f(2)=+2, there must be a root of f(x)=0 in

between x=0 and x=2. A naive way to look for it is the

"bisection method" -- try the number halfway between the

two closest places you know of where f has opposite signs.

3

f ( x) x 2 x 2

Maple graph

Since f(0)=-2 and f(2)=+2, there must be a root of f(x)=0 in

between x=0 and x=2. A naive way to look for it is the

"bisection method" -- try the number halfway between the

two closest places you know of where f has opposite signs.

32.
3

f ( x) x 2 x 2

We know that f(0) = -2 and f(2) = 2, so there is a root

in between. Choose the halfway point, x = 1.

Since f(1) = -3 < 0, we now know (of course, we already

knew from the graph) that there is a root between 1 and 2.

So try halfway between again:

f(1.5) = -1.625

So the root is between 1.5 and 2. Try 1.75:

f(1.75) = -.140625

f ( x) x 2 x 2

We know that f(0) = -2 and f(2) = 2, so there is a root

in between. Choose the halfway point, x = 1.

Since f(1) = -3 < 0, we now know (of course, we already

knew from the graph) that there is a root between 1 and 2.

So try halfway between again:

f(1.5) = -1.625

So the root is between 1.5 and 2. Try 1.75:

f(1.75) = -.140625

33.
3

f ( x) x 2 x 2

We had f(1.75) < 0 and f(2) > 0. So the root is

between 1.75 and 2. Try the average, x = 1.875

f(1.875) = .841796875

f is positive here, so the root is between 1.75 and 1.875.

Try their average (x=1.8125):

f(1.8125) = .329345703

So the root is between 1.75 and 1.8125. One more:

f (1.78125) = .089141846

So now we know the root is between 1.75 and 1.8125.

You could write a computer program to continue this to

any desired accuracy.

f ( x) x 2 x 2

We had f(1.75) < 0 and f(2) > 0. So the root is

between 1.75 and 2. Try the average, x = 1.875

f(1.875) = .841796875

f is positive here, so the root is between 1.75 and 1.875.

Try their average (x=1.8125):

f(1.8125) = .329345703

So the root is between 1.75 and 1.8125. One more:

f (1.78125) = .089141846

So now we know the root is between 1.75 and 1.8125.

You could write a computer program to continue this to

any desired accuracy.

34.
Derivatives

Let’s discuss it:

1. What, in a few words, is the derivative of a function?

2. What are some things you learn about the graph of a

function from its derivative?

3. What are some applications of the derivative?

4. What is a differential? What does dy = f '(x) dx mean?

Let’s discuss it:

1. What, in a few words, is the derivative of a function?

2. What are some things you learn about the graph of a

function from its derivative?

3. What are some applications of the derivative?

4. What is a differential? What does dy = f '(x) dx mean?

35.
Derivatives (continued)

Derivatives give a comparison between the rates of

change of two variables:

When x changes by so much, then y changes by so much.

Derivatives are like "exchange rates".

6/03/10 1 US Dollar = 0.83 Euro

1 Euro = 1.204 US Dollar (USD)

6/04/10 1 US Dollar = 0.85 Euro

1 Euro = 1.176 US Dollar (USD)

Definition of derivative:

dy f ( x h) f ( x )

lim

dx h 0 h

Derivatives give a comparison between the rates of

change of two variables:

When x changes by so much, then y changes by so much.

Derivatives are like "exchange rates".

6/03/10 1 US Dollar = 0.83 Euro

1 Euro = 1.204 US Dollar (USD)

6/04/10 1 US Dollar = 0.85 Euro

1 Euro = 1.176 US Dollar (USD)

Definition of derivative:

dy f ( x h) f ( x )

lim

dx h 0 h

36.
Common derivative formulas:

d

d p

x px p 1 f ( x) g ( x) f ( x) dg df g ( x)

dx dx dx dx

d x

dx

e e x d f ( x) g ( x) f ' ( x) f ( x) g ' ( x)

dx g ( x) g ( x) 2

d 1

ln x d

dx x f ( g ( x) f ' ( g ( x)) g ' ( x)

dx

d

sin x cos x

dx

d Let’s do some examples…..

cos x sin x

d

d p

x px p 1 f ( x) g ( x) f ( x) dg df g ( x)

dx dx dx dx

d x

dx

e e x d f ( x) g ( x) f ' ( x) f ( x) g ' ( x)

dx g ( x) g ( x) 2

d 1

ln x d

dx x f ( g ( x) f ' ( g ( x)) g ' ( x)

dx

d

sin x cos x

dx

d Let’s do some examples…..

cos x sin x

37.
Derivative question #1

5 1

Find f '(1) if f ( x) x

x9 / 5

A. 1/5 E. -1/5

B. 2/5 F. 4/5

C. -8/5 G. 8/5

D. -2/5 H. -4/5

5 1

Find f '(1) if f ( x) x

x9 / 5

A. 1/5 E. -1/5

B. 2/5 F. 4/5

C. -8/5 G. 8/5

D. -2/5 H. -4/5

38.
Derivative question #2

8

Find the equation of a line tangent to y

4 3x

at the point (4,2).

A. 6x+y=26 E. 5x+21y=62

B. 4x+2y=20 F. 4x+15y=46

C. 3x-4y=4 G. 3x+16y=44

D. 7x+18y=64 H. 2x-y=6

8

Find the equation of a line tangent to y

4 3x

at the point (4,2).

A. 6x+y=26 E. 5x+21y=62

B. 4x+2y=20 F. 4x+15y=46

C. 3x-4y=4 G. 3x+16y=44

D. 7x+18y=64 H. 2x-y=6

39.
Derivative question #3

x

2 e

Calculate d f if f ( x)

dx 2 x

x

e x 4 4 e x x 2

A. E.

x x3

ex x2 1 4 F.

e x x 2 5x

x x 3

x

e x x 2

G.

ex x2 2x 2 3

x 4

x

x

e x 3 2

e x x3 4 x 2 3 3

D. 4 H. x

x

x

2 e

Calculate d f if f ( x)

dx 2 x

x

e x 4 4 e x x 2

A. E.

x x3

ex x2 1 4 F.

e x x 2 5x

x x 3

x

e x x 2

G.

ex x2 2x 2 3

x 4

x

x

e x 3 2

e x x3 4 x 2 3 3

D. 4 H. x

x

40.
Derivative question #4

What is the largest interval on which the

x

function f ( x) 2 is concave upward?

x 1

A. (0,1) E. (1, 3 )

B. (1,2) F. ( 3 , )

C. (1, ) G. ( 2 , )

D. (0, ) H. (1/2, )

What is the largest interval on which the

x

function f ( x) 2 is concave upward?

x 1

A. (0,1) E. (1, 3 )

B. (1,2) F. ( 3 , )

C. (1, ) G. ( 2 , )

D. (0, ) H. (1/2, )

41.
Discussion

Here is the graph of a function.

Draw a graph of its derivative.

Here is the graph of a function.

Draw a graph of its derivative.

42.
The meaning and uses of

derivatives, in particular:

• (a) The idea of linear approximation

• (b) How second derivatives are related to

quadratic functions

• (c) Together, these two ideas help to solve

max/min problems

derivatives, in particular:

• (a) The idea of linear approximation

• (b) How second derivatives are related to

quadratic functions

• (c) Together, these two ideas help to solve

max/min problems

43.
Basic functions --linear and

quadratric.

• The derivative and second derivative

provide us with a way of comparing

other functions with (and approximating

them by) linear and quadratic functions.

• Before you can do that, though, you need

to understand linear and quadratic

functions.

quadratric.

• The derivative and second derivative

provide us with a way of comparing

other functions with (and approximating

them by) linear and quadratic functions.

• Before you can do that, though, you need

to understand linear and quadratic

functions.

44.
Let’s review

• Let's review: linear functions of

one variable in the plane are

determined by one point + slope

(one number):

• y = 4 + 3(x-2)

• Let's review: linear functions of

one variable in the plane are

determined by one point + slope

(one number):

• y = 4 + 3(x-2)

45.
Linear functions

• Linear functions occur in calculus as

differential approximations to more

complicated functions (or first-order

Taylor polynomials):

• f(x) = f(a) + f '(a) (x-a)

(approximately)

• Linear functions occur in calculus as

differential approximations to more

complicated functions (or first-order

Taylor polynomials):

• f(x) = f(a) + f '(a) (x-a)

(approximately)

46.
Quadratic functions

• Quadratic functions have parabolas as

their graphs:

2 2

x x

y x 2, y x 1

2 2

• Quadratic functions have parabolas as

their graphs:

2 2

x x

y x 2, y x 1

2 2

47.
Quadratic functions

• Quadratic functions occur as second-

order Taylor polynomials:

• f(x) = f(a) + f '(a)(x-a) + f "(a)(x-a)2/2!

(approximately)

• Quadratic functions occur as second-

order Taylor polynomials:

• f(x) = f(a) + f '(a)(x-a) + f "(a)(x-a)2/2!

(approximately)

48.
They also help us tell...

• … relative maximums from relative

minimums -- if f '(a) =0 the quadratic

approximation reduces to

• f(x) = f(a) + f "(a)(x-a)2/2! and the

sign of f "(a) tells us whether x=a is a

relative max (f "(a)<0) or a relative min (f

"(a)>0).

• … relative maximums from relative

minimums -- if f '(a) =0 the quadratic

approximation reduces to

• f(x) = f(a) + f "(a)(x-a)2/2! and the

sign of f "(a) tells us whether x=a is a

relative max (f "(a)<0) or a relative min (f

"(a)>0).

49.
by way of -review,

max recall

and that

minto problems

find the

maximum and minimum values of a function on any

interval, we should look at three kinds of points:

1. The critical points of the function. These are the points where

the derivative of the function is equal to zero.

2. The places where the derivative of the function fails to exist

(sometimes these are called critical points,too).

3. The endpoints of the interval. If the interval is unbounded,

this means paying attention to

lim f ( x) and/or lim f ( x).

x x ( )

max recall

and that

minto problems

find the

maximum and minimum values of a function on any

interval, we should look at three kinds of points:

1. The critical points of the function. These are the points where

the derivative of the function is equal to zero.

2. The places where the derivative of the function fails to exist

(sometimes these are called critical points,too).

3. The endpoints of the interval. If the interval is unbounded,

this means paying attention to

lim f ( x) and/or lim f ( x).

x x ( )

50.
Position, velocity, and acceleration:

You know that if y = f(t) represents the position of an object

moving along a line, the v = f '(t) is its velocity, and a = f "(t) is

its acceleration.

2

Example: For falling objects, y = 0 y v0 t 16t

is the height of the object at time t, where y0 is the

initial height (at time t=0), and v0 is its initial velocity.

You know that if y = f(t) represents the position of an object

moving along a line, the v = f '(t) is its velocity, and a = f "(t) is

its acceleration.

2

Example: For falling objects, y = 0 y v0 t 16t

is the height of the object at time t, where y0 is the

initial height (at time t=0), and v0 is its initial velocity.

51.
Related Rates

Recall how related rates work. This is one of the big ideas that

makes calculus important:

If you know how z changes when y changes (dz/dy) and how y

changes when x changes (dy/dx), then you know how z changes

when x changes:

dz dz dy

=

dx dy dx

Remember the idea of implicit differentiation: The derivative of

f(y) with respect to x is f '(y)dy

dx

Recall how related rates work. This is one of the big ideas that

makes calculus important:

If you know how z changes when y changes (dz/dy) and how y

changes when x changes (dy/dx), then you know how z changes

when x changes:

dz dz dy

=

dx dy dx

Remember the idea of implicit differentiation: The derivative of

f(y) with respect to x is f '(y)dy

dx

52.
More on related rates

The idea is that "differentiating both

sides of an equation with respect to

x" [or any other variable] is a legal

(and useful!) operation.

This is best done by using examples...

The idea is that "differentiating both

sides of an equation with respect to

x" [or any other variable] is a legal

(and useful!) operation.

This is best done by using examples...

53.
Related Rates Greatest Hits

A light is at the top of a 16-ft pole. A boy 5 ft tall walks away from

the pole at a rate of 4 ft/sec. At what rate is the tip of his shadow

moving when he is 18 ft from the pole? At what rate is the length

of his shadow increasing?

A man on a dock is pulling in a boat by means of a rope attached

to the bow of the boat 1 ft above the water level and passing through

a simple pulley located on the dock 8 ft above water level. If he pulls

in the rope at a rate of 2 ft/sec, how fast is the boat approaching the

dock when the bow of the boat is 25 ft from a point on the water

directly below the pulley?

A light is at the top of a 16-ft pole. A boy 5 ft tall walks away from

the pole at a rate of 4 ft/sec. At what rate is the tip of his shadow

moving when he is 18 ft from the pole? At what rate is the length

of his shadow increasing?

A man on a dock is pulling in a boat by means of a rope attached

to the bow of the boat 1 ft above the water level and passing through

a simple pulley located on the dock 8 ft above water level. If he pulls

in the rope at a rate of 2 ft/sec, how fast is the boat approaching the

dock when the bow of the boat is 25 ft from a point on the water

directly below the pulley?

54.
Greatest Hits...

A weather balloon is rising vertically at a rate of 2 ft/sec. An

observer is situated 100 yds from a point on the ground directly

below the balloon. At what rate is the distance between the balloon

and the observer changing when the altitude of the balloon is 500 ft?

The ends of a water trough 8 ft long are equilateral triangles whose

sides are 2 ft long. If water is being pumped into the trough at a rate

of 5 cu ft/min, find the rate at which the water level is rising when the

depth is 8 in.

Gas is escaping from a spherical balloon at a rate of 10 cu ft/hr. At

what rate is the radius chaing when the volume is 400 cu ft?

A weather balloon is rising vertically at a rate of 2 ft/sec. An

observer is situated 100 yds from a point on the ground directly

below the balloon. At what rate is the distance between the balloon

and the observer changing when the altitude of the balloon is 500 ft?

The ends of a water trough 8 ft long are equilateral triangles whose

sides are 2 ft long. If water is being pumped into the trough at a rate

of 5 cu ft/min, find the rate at which the water level is rising when the

depth is 8 in.

Gas is escaping from a spherical balloon at a rate of 10 cu ft/hr. At

what rate is the radius chaing when the volume is 400 cu ft?

55.
Check the WEB for

assignments and other

course information!

EMAIL [email protected]

in case of difficulty!

Next week: INTEGRALS!

assignments and other

course information!

EMAIL [email protected]

in case of difficulty!

Next week: INTEGRALS!