Download Handout for Derivative and more Exercises Mathematics in PDF only on Docsity!
Section 2.2- Average Rate of Change & Derivative
In this section, you learn how to find the average rate of change, the instantaneous rate of change, and the
limit definition of derivatives and their applications in real-life situations.
Learning Objectives:
At the end of this unit, you will be able to:
- Find the average rate of change for given functions or situations
- Find the instantaneous rate of change
- Find the derivative of a function (using the definition of the derivative)
- Find the equation of the tangent line to the graph of a function at a given point.
- Find the point(s), if any, at which the derivative is nonexistent.
- Solve application problems
In this section we first work on Average rate of change and then we discuss about Instantaneous rate of
change as well as the definition of derivative.
Average Rate of Change
We can find the average rate of change of a function y=(๐ฅ) over an interval [๐,๐] by treating the function as
though it were linear and using the slope between ๐ and ๐.
Average rate of change =
ฮ๐
ฮ๐ฅ
ฮ๐ฆ
ฮ๐ฅ
๐(๐)โ ๐(๐)
๐โ๐
[ Note: This is similar to the slope of a line (๐) =
๐ฆ 2 โ๐ฆ 1
๐ฅ 2 โ๐ฅ 1
]
Example 1 : Suppose we take a trip from Vancouver driving south to Seattle. Every half-hour we note how
far we have traveled, with the following results for the first three hours. Find the average speed over the
time interval from t = 0 to t = 3
Times (t) in hrs 0 0.5 1 1.5 2 2.5 3
Distance travelled in Miles f(t) 0 30 55 80 104 124 138
a) Find the average speed over the time interval from t = 0 to t = 3
b) What is the average speed for the first 1 hours?
Example 2 : Find the average rate of change for the function over the given interval:
a) ๐(๐ฅ) = โ 4 ๐ฅ
2
โ 6 between x=2 and x=
b) ๐
2
- 3 ๐ฅ + 4 between x= 2 and x=2+h
Example 3 : Figure 23 below shows the total amount appropriated annually (in billions of dollars) for the
U.S. Department of Education in recent years. Find the average rate of change per year in Department of
Education appropriation from 2009 to 2013. [ Source: U.S. Department of Education ]
Average Rate of Change as Slope of Secant Line
Average rate of change can be interpreted as the slope of a secant line.
From the figure above, for y = f ( x ), the average rate of change from x to x + h is
๐(๐ฅ+h)โ๐(๐ฅ)
๐ฅ+hโ๐ฅ
, h โ 0.
That is, Average rate of change =
๐(๐ฅ+h)โ๐(๐ฅ)
h
, h โ 0
The above expression is also called a difference quotient. It is in fact, the slope of a secant line PQ.
Slope of Secant Line and Slope of Tangent Line
As we have seen before, we need two points to determine the slope of a line. How can we find the slope of
a curve, at just one point? The answer is by finding the slope of the tangent line to the curve at that point.
Definition:
A secant line is a line between two points on a curve.
A tangent line is a line that touches a curve at one point.
Instantaneous Rate of Change:
Let y = f ( x ), the instantaneous rate of change is given by: ๐ฅ๐ข๐ฆ
hโ 0
๐(๐ฅ+h)โ๐(๐ฅ)
h
, (or lim
๐โ๐
๐(๐)โ๐(๐)
๐โ๐
) provided
that the limit exists. It can be interpreted as the slope of the tangent at the point ( x , f ( x )).
Who invented Calculus? Watch the following video just for fun!
https://youtu.be/PIR5_3G2naw
3 - Steps to find the derivative using limit definition of derivative
Find ๐(๐ฅ + h) and simplify
Write the definition of derivative
hโ 0
๐(๐ฅ+h)โ๐(๐ฅ)
h
- Substitute the values of ๐(๐ฅ + h) & ๐(๐ฅ) in step 3 and evaluate the limit if it exists.
Example1 : For each of the following functions find the derivative fโ(x) using the definition of derivative.
2
b) ๐
๐ฅ c) ๐
3
Example 2 : Consider the function ๐(๐ฅ) = 6 โ ๐ฅ
2
a) Find the equation of the secant line through the points where x has the values ๐ฅ = โ 1 ๐๐๐ ๐ฅ = 3
b) Find the equation of the tangent line when x=-1.
Where a Function is Not Differentiable:
A function f ( x ) is not differentiable at a point x = a , if there is a โcornerโ at a.
A function f ( x ) is not differentiable at a point x = a , if there is a vertical tangent at a.
A function f ( x ) is not differentiable at a point x = a , if it is not continuous at a. In the following figure, f(x)
is not differentiable at x=-2.
Example 3 : In the following figure, the function does not have derivative at
๐ฅ = ๐ฅ 1 , ๐ฅ 2 , ๐ฅ 3 , ๐ฅ 4 , ๐ฅ 5 , ๐ฅ 6 for different reasons as mentioned below.
Example 4 : Let ๐
๐ฅ 2 โ 1
๐ฅ+ 2
. Find the point(s) where f is not differentiable and why?
Example 5: Suppose the temperature T in degrees Fahrenheit at a height x in feet above the ground is
given by ๐ฆ= ๐
(
)
2
a) Find the derivative T'(x) of this function and give an interpretation of it.
b) Find ๐
'(100) and explain its meaning.
The Derivative as a Function
We now know how to find (or at least approximate) the derivative of a function for any x-value; this means
we can think of the derivative as a function, too. The inputs are the same xโs; the output is the value of the
derivative at the point.
Example 6: The Fig. below is the graph of a function y=f(x). Using the information in the graph, fill in a
table showing values
Note: At various values of x, draw your best guess at the tangent line and measure its slope. You might have
to extend your lines so you can read some points. In general, your estimate of the slope will be better if you
choose points that are easy to read and far away from each other. Here are my estimates for a few values
of x (parts of the tangent lines I used are shown)
x y=f(x) f '(x)= estimated
slope of tangent
line to the curve
at the point (x,y)
0
1
2
3
4
5
