## Understanding Calculus

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 Preface 1. Why Study   Calculus 2. Numbers 3. Functions 4. The Derivative 5. Differentiation 6. Applications 7. Free Falling   Motion 8. Understanding  Derivative 9. Derivative  Approximations 10. Integration   Theory 11. Understanding   Integration 12. Differentials

 Inverse Functions Exponents Exponential  Functions Applications of   Exponential  Functions Sine and Cosine   Function Sine Function Sine Function -   Differentiation and   Integration Oscillatory Motion Mean Value   Theorem Taylor Series More Taylor Series Integration   Techniques

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CHAPTER 4

### Chapter 4 - The Derivative

#### Section 4.2- Average Rate of Change

We have learned that a change in the independent variable is defined as , and the corresponding change in the dependent variable over this interval is The question we now must ask ourselves is how can we measure the relative change of the dependent variable with respect to the independent variable? In other words, how do we calculate how much more or less changed compared to

To calculate how much more changed over an interval from , we simply divide the change in f over the change in x for the interval. Thus we divide, by the interval over which we are evaluating it, which is equal to . Thus the relative change of f with respect to x over an interval is defined as:

While this expression may seem rather simple, it does require some explanation. By dividing the change in f by the change in x what we are doing is calculating how much more f changed for a given change in x. For example in the function, , when x changed from 3 to 5, f changed from 81 to 375. Over this interval of from x=3 to x=5, the was 294. Thus the relative change in f with respect to a change in the independent variable x is:

The value of 147 tells us that f changes 147 times more than x over that interval of from x=3 to x=5 only. Thus for each unit change in x, , the corresponding change in f is 147. We can therefore define the rate of change of a function with respect to its independent variable to be:

The value, called the rate of change of the function,  refers to how much more or less changes for a unit change in x. It is only valid over the interval under consideration,

Another way of understanding what rate of change of a function means is to look at the steepness of the line connecting the two endpoints of the interval under consideration.

As increases, the steepness of the line connecting the two endpoints will increase. Thus, the greater the rate of change of the function, the greater  its slope or steepness over the interval under consideration. Since slope and rate of change are synonymous, then how is rate of change  defined for functions whose graphs do not have constant slopes? For example, from x=9 to x=12 of , the change in f is 2997. Thus the rate of change of the function over the interval is:

This value is significantly higher than the rate of change calculated for the previous interval from x = 3 to x = 5. We can only  conclude that the rate of change or slope of the graph must be increasing and is not constant over an interval . Look at the graph of the function to understand how this might be so:

Since the rate of change of a function can change, then we have to come up with a more refined definition of rate of change. We can define the average rate of change of a function over an interval , to be equal to

In the next section we will take a closer look at how we can define the exact or instantaneous rate of change of a function.

Next section -> Section 4.3 - Instantaneous Rate of Change

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