MATH 150 SUPPLEMENTAL NOTES 1

RATES OF CHANGE AND LIMITS
SPEED

Suppose we are tracking Barney as he falls from the Sears Tower. To find the average speed Barney is falling over any particular time interval, divide the amount of distance covered during the time interval by the length of the time interval. Since Barney is falling freely near the surface of the earth, then his position can be calculated with the equation y = 16t² ft. To find his average speed for the first 10 seconds, we will find D y/D t.

If we want to find Barney's average speed during a 1-second interval between 12 and 13 seconds, then the average speed will have the following form.

Suppose we want to find Barney's generic speed as he travels from t₀ to t₀ + h. Barney's average speed will be

So for any initial time t₀ and increment h, we can find his speed. This is an example of the average rate of change.

AVERAGE RATE OF CHANGE AND SECANT LINES

Given any arbitrary function y = f (x), we can calculate the average rate of change of y with respect to x over the interval [x₁, x₂] by

DEFINITION:

The average rate of change of y = f (x) with respect to x over the interval [x₁, x₂] is

NOTE:

The average rate of change of f over [x₁, x₂] is the slope of the secant line from P (x₁, y₁) to Q (x₂, y₂). Also, if you had pre-calculus from me, then you would also recognize this formula as the difference quotient.

EXAMPLE 1:

Find the average rate of change of f (x) = x⁴ - 2x² on the interval [-2, 1].

SOLUTION: First of all, evaluate the f (x) at the endpoints of the interval.

Now plug the values into the formula.

LIMITS OF FUNCTIONS

QUESTION:

SOLUTION:

So x = -4 is a removable point of discontinuity (and it is not in the domain of this function). When you look at the graph of the original function, there will be a hole in the line y = x - 4 (See figure 1) at x = -4. (I know that this graph does not show

figure 1

that fact.) So let us look at what happens as x gets "close" to -4.

x	-4.1	-4.01	-4.001	-3.9	-3.99	-3.999
	-8.1	-8.01	-8.001	-7.9	-7.99	-7.999

This example leads us to the informal definition of the limit.

DEFINITION:

INFORMAL DEFINITION OF LIMIT

Let f (x) be defined on an open interval about x₀, except possibly at x₀, itself. If f (x) gets arbitrarily close to L for all x sufficiently close to x₀, we say that f approaches the limit L as x approaches x₀, and we write

EXAMPLE 2:

Consider the following graph of the piecewise function.

QUESTION 1:

SOLUTION: Yes, the limit does exist.

Since the right-hand and the left-hand limits exist and are equal, then the

QUESTION 2:

SOLUTION: No, the limit does not exist.

The right-hand and left-hand limits do exist, but they are not equal to each other. Therefore, the limit as x approaches 1 does not exist.

Remember this - for a limit to exist both the right-hand and left-hand limits must exist and be equal to each other. If they do not equal each other, then the limit does not exist.

FACT:

If f is the identity function, f (x) = x, then for any value of x₀,

FACT:

If f is the constant function, f (x) = k, then for any value of x₀,

LIMITS BY SUBSTITUTION

EXAMPLE 3:

Find the limit by substitution.

SOLUTION: To evaluate this limit, I will plug in 2 for x.

EXAMPLE 4:

Find the limit by substitution.

SOLUTION: To find this limit, I will plug -1 in for x.

EXAMPLE 5:

Find the limit by substitution.

SOLUTION:

EXAMPLE 6:

Find the limit by substitution.

SOLUTION:

In this set of supplemental notes we covered two important topics. The first one was the average rate of change. This topic will help us bridge the way from limits to the derivative of a function, which we will find out is the instantaneous rate of change. The second topic was the concept of the limit. We first looked at an example of a function that had a removable point of discontinuity. We looked at the graph and a table of values, from which we determine that the limit did exist and we found the value. You need to remember that a limit means we approach the value, but never quite get there. In the next example, we learned that when we approach the limit from both sides, both limits must exist and be equal for the limit to exist. We will talk more about this topic later. Finally, we evaluated limits by substituting the limiting value directly into the function. This is the easiest way of evaluating a limit, but it does not always work for every limit. We will see some limits later in this course, which do not allow us to just plug in and evaluate. Work through these examples, and make sure that you understand what is being discussed in these notes. If you have any problems or questions, please feel free to contact me.

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