Average rate of change

[Pages:33]1.4 Rates of Change (Minimum Homework: 1, 3, 5, 7, 9, 11, 15, 19, 21, 25)

Average rate of change: The average rate of change of function over the interval (, ) is given by this equation: = ()-()

-

? Average rate of change is a measure of how much a function changes per unit, on average, over that interval.

? Average rate of change is just the slope of the straight line connecting the interval's endpoints on the graph of the function.

Average rate of example 1:

Find the average rate of change for each function over the given interval. Sketch a graph to model your answer. (You may use your calculator obtain the graph, be sure to label the necessary points.) () = -22 + 4 - 5 between = 1 = 3

Create 2 points using the given x values.

= 1; (1) = -2(1)2 + 4(1) - 5 = -3

Creates the point: (1, -3)

= 3; (3) = -2(3)2 + 4(3) - 5 = -11

Creates the point: (3, -11)

Average rate of change is just the slope of the line that connects the points.

= -11-(-3) = -8 = -4

3-1

2

Answer: Average Rate of Change = -

Average rate of change example 2:

At 10 AM a car's odometer read 10,300 miles. At noon, the car's odometer read 10,420 miles. What is the car's average rate of change measured in miles per hour?

We need to create two points.

Since we are asked to find the average rate of change in miles per hour

- of the points must be time in hours (hours are mentioned second)

- of the points must be distance in miles (miles are mentioned first)

(, ) These are the points needed: (10, 10300) (12, 10420)

= 10420-10300 () = 120 =

12-10 ()

2

60

Answer: Average Rate of Change (velocity):

? Average rate of change tells us the average rate at which a function changes over an interval.

? The average rate of change only tells us an average change between two values of x. It gives no specific information inbetween the two values of .

That is, we have no idea how the function behaves at any specific instant in the interval between the given values of x.

? In the car example we computed the average rate of change (velocity) was 60 over the 2-hour trip. This is an average speed for the entire trip. This does not mean the car traveled at precisely 60 for the entire trip. This is just the average speed. In fact, it likely went faster than 60 at times and slower than 60 at other times. The car could have been stopped for a chunk of time.

We need to calculate the car's instantaneous rate of change to know its speed (velocity) at a specific moment.

Instantaneous Rate of Change (Derivative):

? Instantaneous rate of change at = is the average rate of change over the interval (, )

? Instantaneous rate of change cannot be calculated with Algebra alone because the of the equal x-coordinates.

? The denominator of the fraction in the computation would be

-. ? This gives a fraction with 0 in the denominator. This is undefined.

Calculus is needed to compute instantaneous rate of change.

Instantaneous rate of change of a function at = : Is the average rate of change of the function at = .

? The instantaneous rate is essentially the average rate of change over the interval (, + ).

? To find the instantaneous rate of change, we take the limit as 0.

? The instantaneous rate of change is the average rate of change at a single point, since h is changed to zero and we get the interval (, + 0) = (, )

This is the formula to compute instantaneous rate of change of a function = .

=

= lim (+)-() = lim (+)-()

0 (+)-

0

Instantaneous rate of change is a measure of the slope of the line connecting the points: (, ()) ( + , ( + ))

We will use the word DERIVATIVE very often this semester.

? The terms instantaneous rate of change and derivative have the same definition and they are interchangeable words.

? The value of a derivative of function when = is just the instantaneous rate of change of the function at =

This is the formula to compute derivative of a function = . =

Derivative

=

()

=

lim

0

(+)-() (+)-

=

lim

0

(+)-()

? We use the symbol () =

? A derivative is a measure of the slope of the line connecting the points: (, ()) ( + , ( + ))

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