Assignment: P. 62: 16, 18, 22-24 P. 63: 49, 51, 53, 54,

Assignment: P. 62: 16, 18, 22-24 P. 63: 49, 51, 53, 54,

Objectives:

1.

To calculate average rate of change

2.

To determine whether a function is even, odd, or neither

3.

To use a graphing utility to find minima and maxima

4.

To find the zeros of a function algebraically

Assignment:

P. 62: 16, 18, 22-24

P. 63: 49, 51, 53, 54,

63-69 odd

P. 63: 71-76 some

P. 64: 88

P. 65: 91-95 odd

Rate of Change

Tangent Line

Maximum Point

Relative Maximum

Secant Line

Minimum Point

Relative Minimum

Slope is often referred to as

rate of change

. Why is the rate of change for any given line always constant?

4

C

B

2

-10 5 10

A

-5

-2

Would the rate of change be constant for other graphs, like circles or parabolas?

6 6

4

2

4

2

-5 5 -5 5

For these graphs, the rate of change is different at every point along the curve.

6 6

4

2

4

2

-5 5 -5 5

It was the search for this rate of change, called the

instantaneous rate of change

, that eventually lead to the discovery of differential calculus.

6 6

4

2

4

2

-5 5 -5 5

Calculus tells us that the rate of change at any given point on a graph is equal to the slope of the

tangent line

at that point.

6 6

4

2

4

2

-5 5 -5 5

A line is a

tangent

if and only if it intersects a curve in one point.

Finding the slope of a tangent line to a circle is fairly easy, even though you only have one point on the line.

You simply find the slope of the radius, and then take the negative reciprocal.

-2

6

4

2

5

But what about other curves? For example, shown is the graph of

y

= 6 – x

2

. How would we find the slope of the tangent line at, say, (1, 5)?

The problem is that a parabola, or most other curves, do not have a radius that is perpendicular to the tangent line at any given point, and we only have one point on the line.

It was the resolution of this problem, by

Fermat, Newton, and Leibniz that led to the discovery of differential calculus.

It begins with another line, called the

secant line

.

-5

6

4

2

-2

5

A line is a

secant

if and only if it intersects a curve in two points.

Watch how a series of secants can get closer and closer to the tangent line.

Watch how a series of secants can get closer and closer to the tangent line.

Finding the slope of a secant line gives us the

average rate of change

.

The

average rate of change

between any two points is the slope of the secant line through the two points:

Average rate of change =

=

(

2

)

( )

1

x

2

x

1

y

x

=m sec

Find the average rate of change of f (x) = x

2

– 2x for x

1

= −1 and x

2

= 3.

for x

1

= 3 and x

2

= 8.

In physics, the position of something (i.e., a thrown rock) is given by the equation

s

 

16

t

2

v t

0

s

0 where s = position, t = time, v

0 and s

0

= initial position.

= initial velocity,

If we knew calculus, we could calculate the instantaneous rate of change at any point; but since we don’t, we’ll have to settle for the average rate of change.

In physics, the position of something (i.e., a thrown rock) is given by the equation

s

 

16

t

2

v t

0

s

0

Write a function that represents the situation, then find the average rate of change.

1.

An object (rock) is thrown upward from a height of 6.5 feet at a velocity of 72 fps.

t

1

= 0, t

2

= 4

In physics, the position of something (i.e., a thrown rock) is given by the equation

s

 

16

t

2

v t

0

s

0

Write a function that represents the situation, then find the average rate of change.

2.

An object (rock) is dropped from a height of

80 feet.

t

1

= 1, t

2

= 2

You will be able to determine whether a function is even, odd, or neither

Shown below are two types of symmetry that the graph of a function can have.

If the graph has y-axis symmetry, the function is said to be

even

.

If the graph has origin symmetry, the function is said to be

odd

.

You don’t always want to look at a graph to see if a function is even or odd, so you can perform these simple algebraic tests instead.

1.

A function f (x) is

even

if, for each x in the domain, f (x) = f (−x).

2.

A function f (x) is

odd

if, for each x in the domain, f (x) = −f (x).

Plugging in −x for x in f (x) doesn’t change the function.

Plugging in −x for x in f (x) gives you −f (x).

Let n be an even number:

 

n

x n

For example:

 

2

x

2

 

4 

x

4

Let m be an odd number:

 

m

 

x m

For example:

 

3  

x

3

 

5  

x

5

Determine whether each of the following functions are even, odd, or neither.

( )

x

2

x

4

( )

x

5

2

x

3

x

( )

x

5

2

x

4

x

2

( )

2

x

3

1 ( )

 

x

6

x

8

( )

x

9

3

x

5

 

x

What do you notice about all of even functions?

What do you notice about all of the odd functions? Can you come up with a shortcut?

( )

x

2

x

4

( )

x

5

2

x

3

x

( )

x

5

2

x

4

x

2

( )

2

x

3

1 ( )

 

x

6

x

8

( )

x

9

3

x

5

 

x

You don’t look at the graph or perform an algebraic test to see if a function is even or odd. Just look at the powers.

1.

A function f (x) is

even

if, for each x in the domain, f (x) = f (−x).

2.

A function f (x) is

odd

if, for each x in the domain, f (x) = −f (x).

This happens when all the powers of x are even.

This happens when all the powers of x are odd.

Caveats:

1.

The previous shortcuts only work on polynomial functions.

2.

You have to think of a constant as kx

0

(a number times x

0

) which is an even power of

x

.

The

vertex

of a parabola marks the

turning point

of the graph of a quadratic function.

A

turning point

is a point at which the function values “turn” from increasing to decreasing or vice versa.

The y-coordinate of a parabola’s turning point marks the

absolute minimum

or

maximum

of the function since there are no other points above or below it.

Other polynomial functions also have various turning points that mark minima and maxima; however, they may not be absolute.

Extrema

(min/max values) come in two varieties:

1.

Absolute

2. Relative (Local)

• Relative (Local) Minimum

:

The y-coordinate of a turning point that is lower on a graph than its surrounding points

Relative (Local) Maximum:

The y-coordinate of a turning point that is higher on a graph than its surrounding points

It’s a fairly easy exercise to approximate the location of relative extrema using your graphing utility.

1.

Press

Y= and enter the function.

2.

Choose your favorite

ZOOM setting.

3.

Press

2 nd

TRACE for the

CALC menu.

4.

Choose minimum or maximum

.

5.

Set the left bound, right bound, and a guess.

6.

Magic!

Use a graphing utility to find the extrema of each of the following functions.

1.

h

(x) = 0.5x

3

+x

2

x + 2

2.

j

(x) = x

4

+ 3x

3

x

2

– 4x – 5

A square piece of sheet metal is 10 inches by 10 inches. Squares of side length x are cut from the corners and the remaining piece is folded to make an open box.

What size square(s) can be cut from the corners to create a box with a maximum volume?

You will be able to find the zeros of a function using algebra

A

zero

of a function is the x-value that makes the function equal zero.

Zeros = Roots Roots = 𝑥-intercepts

To find the find the zeros of a function, set it equal to zero and solve for x.

How many zeros should you expect the function below to have?

f x

5

x

7

3

x

4

2

x

9

How many zeros should you expect the function below to have?

f x

5

x

7

3

x

4

2

x

9

For polynomial functions, the

degree

(highest power) determines the number of zeros.

However, some may be repeated

Find the zeros of each function algebraically.

1.

( )

3

x

2

 

10

2.

( )

81

x

2

3.

3

x

7

x

5

Objectives:

1.

To calculate average rate of change

2.

To determine whether a function is even, odd, or neither

3.

To use a graphing utility to find minima and maxima

4.

To find the zeros of a function algebraically

Assignment:

P. 62: 16, 18, 22-24

P. 63: 49, 51, 53, 54,

63-69 odd

P. 63: 71-76 some

P. 64: 88

P. 65: 91-95 odd

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