1.3 Average Rates Of Change
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Math 135 Class Notes
1.3
Business Calculus
Spring 2009
Average Rates of Change
Consider a function y = f (x) and two input values x1 and x2 . The change in input, or the change in x, is x2 − x1 . The corresponding change in output, or the change in y, is y2 − y1 = f (x2 ) − f (x1 ). DEFINITION OF AVERAGE RATE OF CHANGE The average rate of change of y = f (x) with respect to x, as x changes from x1 to x2 , is the ratio of the change in output to the change in input:
where x1 6= x2 .
y2 − y1 f (x2 ) − f (x1 ) = x2 − x1 x2 − x1
If we look at the graph of the function, then the average rate of change will equal the slope of the line passing through the points P (x1 , y1 ) and Q(x2 , y2 ). The line passing through P and Q is called a secant line.
EXAMPLE The graph in the figure shows a typical response to adversing. Aftr an amount a is spent on advertising, the company sells N (a) units of a product. Find the average rate of change of N as a changes N a) from 0 to 1 700 600 500
b) from 1 to 2
400 300 200
c) from 2 to 3 100 0
11
0
1
2
3
4
a
12
Chapter 1
Differentiation
DIFFERENCE QUOTIENTS AS AVERAGE RATES OF CHANGE We can rewrite the average rate of change of a function in a different notation as follows. Instead of using x1 for an initial input, use x. From x, we move h units to the second input x2 = x + h. Then the average rate of change is f (x2 ) − f (x1 ) f (x + h) − f (x) = x2 − x1 (x + h) − x f (x + h) − f (x) = h DEFINITION OF DIFFERENCE QUOTIENT The average rate of change of a function f with respect to x is also called the difference quotient. It is given by f (x + h) − f (x) h ° ¢ ° ¢ The difference quotient is equal to the slope of the line from x, f (x) to x + h, f (x + h) .
EXAMPLE Let f (x) = x2 . Find the difference quotient when: a) x = 5 and h = 3
b) x = 5 and h = 0.1
EXAMPLE Let f (x) = x2 . Find a simplified form of the difference quotient. Then find the value of the difference quotient when x = 5 and h = 0.1.
1.3
Average Rates of Change
EXAMPLE
Let f (x) = x3 . Find a simplified form of the difference quotient.
EXAMPLE
Let f (x) = 3/x. Find a simplified form of the difference quotient.
13
1.3
Business Calculus
Spring 2009
Average Rates of Change
Consider a function y = f (x) and two input values x1 and x2 . The change in input, or the change in x, is x2 − x1 . The corresponding change in output, or the change in y, is y2 − y1 = f (x2 ) − f (x1 ). DEFINITION OF AVERAGE RATE OF CHANGE The average rate of change of y = f (x) with respect to x, as x changes from x1 to x2 , is the ratio of the change in output to the change in input:
where x1 6= x2 .
y2 − y1 f (x2 ) − f (x1 ) = x2 − x1 x2 − x1
If we look at the graph of the function, then the average rate of change will equal the slope of the line passing through the points P (x1 , y1 ) and Q(x2 , y2 ). The line passing through P and Q is called a secant line.
EXAMPLE The graph in the figure shows a typical response to adversing. Aftr an amount a is spent on advertising, the company sells N (a) units of a product. Find the average rate of change of N as a changes N a) from 0 to 1 700 600 500
b) from 1 to 2
400 300 200
c) from 2 to 3 100 0
11
0
1
2
3
4
a
12
Chapter 1
Differentiation
DIFFERENCE QUOTIENTS AS AVERAGE RATES OF CHANGE We can rewrite the average rate of change of a function in a different notation as follows. Instead of using x1 for an initial input, use x. From x, we move h units to the second input x2 = x + h. Then the average rate of change is f (x2 ) − f (x1 ) f (x + h) − f (x) = x2 − x1 (x + h) − x f (x + h) − f (x) = h DEFINITION OF DIFFERENCE QUOTIENT The average rate of change of a function f with respect to x is also called the difference quotient. It is given by f (x + h) − f (x) h ° ¢ ° ¢ The difference quotient is equal to the slope of the line from x, f (x) to x + h, f (x + h) .
EXAMPLE Let f (x) = x2 . Find the difference quotient when: a) x = 5 and h = 3
b) x = 5 and h = 0.1
EXAMPLE Let f (x) = x2 . Find a simplified form of the difference quotient. Then find the value of the difference quotient when x = 5 and h = 0.1.
1.3
Average Rates of Change
EXAMPLE
Let f (x) = x3 . Find a simplified form of the difference quotient.
EXAMPLE
Let f (x) = 3/x. Find a simplified form of the difference quotient.
13
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