The student will learn about: average rate of change, other names for these two processes.
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We 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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Def: The average rate of change for a function y = f (x), where x changes from x to x + h is: f (x h) f (x)
 x

  h


 y x
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) ).
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• 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.

4
3
Q f (x + h)
2 f (x + h) – f (x) = Δy
1
-1
P f (x)
1 O (x) (x + h) 2 m

 y
 x
 f ( x
 h )
 f ( x ) h h = Δx secant
T
I
C
S
T
A
5

The profit (in dollars) from the sale of x car seats for infants is given by,
P (x) = 45 x – 0.025 x 2 – 5,000, where 0 ≤ x ≤ 2,400.
6

P (x) = 45 x – 0.025 x 2 – 5,000 a. Find the change in profit if production is changed from 1,000 to 1,400 car seats.
NOTE: We are finding the change in the profit.
ΔP = P (1,400) – P (1,000) = 9,000 – 15,000
= - 6,000
What does this mean?
7

P (x) = 45 x – 0.025 x 2 – 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.
x = 1000, h = 400, x + h = 1400

 f (x
 h)
 f (x)


P h h

 
15
400
From previous slide.
This was the 4 step procedure!
What is the meaning of this number?
8

The following terms all have the same meaning and mathematical process.
•
The difference quotient.
f ( x
 h )
 f ( x )
• h
•
The average rate of change.
•
The slope of the secant line.
•
The 4 step procedure.
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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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Def: The instantaneous rate of change for a function, y = f (x), at x = x is: lim h

0 f (x
 h)
 f (x) h

 y
 x
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.
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) ).
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4
3
f (x + h)
2
Tangent
Let h approach 0 f (x + h) – f (x)
= Δy
-1
1
P m

 y
 x
 h lim

0 f ( x
 h )
 f ( x ) h f (x)
1 O (x) (x + h) 2 3 h = Δx
12

The profit (in dollars) from the sale of x car seats for infants is given by,
P (x) = 45 x – 0.025 x 2 – 5,000, where 0 ≤ x ≤ 2,400.
13

P (x) = 45 x – 0.025 x 2 – 5,000
Find the instantaneous change in profit if production is 1,000 car seats.
We will use the previous definition: h lim

0 f ( x
 h )
 f ( x ) h
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P (x) = 45 x – 0.025 x 2 – 5,000 h lim

0 f ( x
 h )
 f ( x ) h
Find the instantaneous change in profit if production is 1,000 car seats.
I will use x = 1000 at the end.
P ( x
 h )

P ( x ) h lim

0 h

Step 3
Step
1 h lim

0 h lim

0
45 ( x
45 x

 h )

0 .
025 ( x
 h )
2 
5 , 000
 h

45 x

0 .
025 x
2
Step 2

5000


Step 5
45 h

Step 4
0 .
025 x
2 h

0 .
050 xh

0 .
025 h
2 
5 , 000

45 x

0 .
025 x
2 
5 , 000

15

P (x) = 45 x – 0.025 x 2 – 5,000
Find the instantaneous change in profit if production is 1,000 car seats. By Definition continued.
h lim

0 h lim

0 h lim

0
45 x

45 h

0 .
025 x
2 
0 .
050 xh

0 .
025 h
2 
5 , 000

45 x

0 .
025 x
2 
5 , 000
 h
45 h

0 .
050 xh

0 .
025 h
2
45

0 .
050 x h

0 .
025 h

 lim h

0
(45
45 – 0.050x
 h

And now we let x = 1000
= 45 – 50 = - 5
What does this mean?
16

1. f (x + h)
2. f ( x)
( - ) 3. f (x + h) – f (x)
4.
5.
f ( x
 h )
 f ( x ) the h f ( x
 h )
 f ( x ) h lim

0 h difference quotient
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1. f (x + h) f ( x
 h )
 f ( x )
4 .
2. f ( x)
( - ) 3. f (x + h) – f (x) h f ( x
 h )
 f ( x )
5.
h lim

0 h
This is the 5-step procedure
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.
18

The following terms all have the same meaning and mathematical process.
•
•
The limit of the difference quotient.
h lim

0 f ( x
 h ) h
 f ( x )
•
The instantaneous rate of change of y with respect to x.
•
The slope of the tangent line.
•
The derivative.
•
The 5 step procedure.
19

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.
20

The profit (in dollars) from the sale of x car seats for infants is given by,
P (x) = 45 x – 0.025 x 2 – 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.
21

P (x) = 45 x – 0.025 x 2 – 5,000
Find the instantaneous rate of change in profit if production is at 1,000 car seats. tangent
Draw and
Tangent
0 ≤ x ≤ 2,400
0 ≤ y ≤ 20,000
      
I love my calculator!
      
22 Slope is the derivative as before.

P (x) = 45 x – 0.025 x 2 – 5,000
3. Do the above using a graphing calculator.
Let x = 1000.
Using dy/dx under the
“ calc
” menu.
dy/dx

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.
24

We have seen two new ideas.
1. The average rate of change.
f ( x
 h )
 f ( x ) h
2. The instantaneous rate of change.
h lim

0 f ( x
 h )
 f ( x ) h
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ASSIGNMENT
§1.5 On my website.
14, 15, 16.
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