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§1.5 Rates Of Change, Slope and Derivatives

The student will learn about:

average rate of change,

instantaneous rate of change, instantaneous rate of change, and all of the other names for these two processes.

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Rate of ChangeWe are going to define two rates of change. First will be the average rate of change.

Then through a study of limits we will examine the instantaneous rate of change.

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Definition Def: The average rate of change for a function y = f (x), where x changes from x to x + h is:

This is called the difference quotient. Note that it is the change in y over the change in x, or the slope of the secant line from point P ( x, f (x) ) to point Q at ( x + h, f (x + h) ).

f (x h) f (x) f (x h) f (x)

(x h) xx

y

h

Secant and Tangent Lines• A secant line to a curve is a line that passes

through two points of the curve. • A tangent line is a line that passes through a

point of the curve and matches exactly the steepness of the curve at that point.

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h = Δx

f (x + h) – f (x) = Δy

secant

Visual Interpretation

P

Q

h

)x(f)hx(f

x

ym

STATIC

-1 1 2

4

3

2

1

O (x) (x + h)

f (x)

f (x + h)

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ExampleThe profit (in dollars) from the sale of x car seats for infants is given by,

P (x) = 45 x – 0.025 x2 – 5,000,

where 0 ≤ x ≤ 2,400.

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Example continuedP (x) = 45 x – 0.025 x2 – 5,000

a. Find the change in profit if production is changed from 1,000 to 1,400 car seats.

ΔP = P (1,400) – P (1,000) = 9,000 – 15,000

= - 6,000

NOTE: We are finding the change in the profit.

What does this mean?

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Example continuedP (x) = 45 x – 0.025 x2 – 5,000

b. Find the average change in profit if production is changed from 1,000 to 1,400 car seats. i.e. the difference quotient.

,6 00015

400

f (x h) f (x) P

h h

This was the 4 step procedure!

What is the meaning of this number?

x = 1000, h = 400, x + h = 1400

From previous slide.

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This May Help!

The following terms all have the same meaning and mathematical process.

• The average rate of change.

• The difference quotient.

• The slope of the secant line.

h

)x(f)hx(f •

• The 4 step procedure.

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Rate of Change This is used to find the average change in revenue, average change in velocity, average change in profit, etc.

We will now go on to find the instantaneous rate of change of these functions. That occurs when h approaches 0.

The idea of instantaneous rate of change is fundamental to our studies this semester. It is basically one of the two operations that we do in the mathematics called calculus.

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Definition Def: The instantaneous rate of change for a function, y = f (x), at x = x is:

This is sometimes called the rate of change. Note that it is the change in y over the change in x, as the change in x approaches 0.

h 0

f (x h) f (x) ylim

h x

This is the limit as h approaches 0 of the difference quotient. It is the slope of the line (tangent) at point P ( x, f (x) ).

12h = Δx

f (x + h) – f (x) = Δy

Tangent

Let h approach 0

Visual Interpretation

P h

)x(f)hx(f

0h

lim

x

ym

DYNAMIC

-1 1 2 3

4

3

2

1

O (x) (x + h)

f (x)

f (x + h)

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Example

The profit (in dollars) from the sale of x car seats for infants is given by,

P (x) = 45 x – 0.025 x2 – 5,000,

where 0 ≤ x ≤ 2,400.

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Example continuedP (x) = 45 x – 0.025 x2 – 5,000

Find the instantaneous change in profit if production is 1,000 car seats.

We will use the previous definition:

h

)x(f)hx(flim

0h

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Example continuedP (x) = 45 x – 0.025 x2 – 5,000

Find the instantaneous change in profit if production is 1,000 car seats. I will use x = 1000 at the end.

h

)x(P)hx(Plim

0h

h

000,5x025.0x45000,5h025.0xh050.0x025.0h45x45lim

222

0h

h

)x(f)hx(flim

0h

Step 2

Step 3

Step 4

h

5000x025.0x45000,5)hx(025.0)hx(45lim

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0h

Step 1

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Example continuedP (x) = 45 x – 0.025 x2 – 5,000

Find the instantaneous change in profit if production is 1,000 car seats. By Definition continued.

h

h025.0xh050.0h45lim

2

0h

h025.0x050.045lim0h

45 – 0.050x

h

000,5x025.0x45000,5h025.0xh050.0x025.0h45x45lim

222

0h

And now we let x = 1000

= 45 – 50 = - 5 What does this mean?

h 0

(45 0.050x 0.025h) hlim

h

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Five Steps

1. f (x + h)

2. f ( x)

( - ) 3. f (x + h) – f (x)

quotientdifferencetheh

)x(f)hx(f 4.

h

)x(f)hx(flim

0h

5.

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Suggestion

1. f (x + h)

2. f ( x)

( - ) 3. f (x + h) – f (x)

h

)x(f)hx(f.4

h

)x(f)hx(flim

0h

5.

ALWAYS do it this way, with an x and an h even if you are given values for x and h. Thus you will always know how to start and if there are other parts to the question you will have an easier time.

This is the 5-step procedure

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This May Help!

The following terms all have the same meaning and mathematical process.

• The instantaneous rate of change of y with respect to x.

• The limit of the difference quotient.

• The slope of the tangent line.

• h

)x(f)hx(flim

0h

• The derivative.

• The 5 step procedure.

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Slope – tangent line

As mentioned earlier the slope of the tangent line is the limit of the difference quotient as h approaches zero and as defined above.

An example follows.

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Example

The profit (in dollars) from the sale of x car seats for infants is given by,

P (x) = 45 x – 0.025 x2 – 5,000,

where 0 ≤ x ≤ 2,400.

The slope of the tangent line is found as done previously by definition.

Now we will use a graphing calculator to get this value.

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Example continuedP (x) = 45 x – 0.025 x2 – 5,000

Find the instantaneous rate of change in profit if production is at 1,000 car seats.

tangent

0 ≤ x ≤ 2,400

0 ≤ y ≤ 20,000

Slope is the derivative as before.

I love my calculator!

Draw and Tangent

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Calculator Example

3. Do the above using a graphing calculator.

P (x) = 45 x – 0.025 x2 – 5,000

Using dy/dx under the “calc” menu.

Let x = 1000.

dy/dx

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Summary.

We have seen that the difference quotient occurs naturally and will be encountered in many different situations. This section has introduced the idea in an intuitive manner. We will work on a more precise and formal concept of limit in the next few sections. For now, become familiar with the two definitions of average rate of change and instantaneous rate of change.

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Summary.

We have seen two new ideas.1. The average rate of change.

h

)x(f)hx(f

2. The instantaneous rate of change.

h

)x(f)hx(flim

0h

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ASSIGNMENT

§1.5 On my website.

14, 15, 16.