12 Rates of Change Average Rates of Change. Concepts: Average Rates of Change

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1 12 Rates of Change Concepts: Average Rates of Change Calculating the Average Rate of Change of a Function on an Interval Secant Lines Difference Quotients Approximating Instantaneous Rates of Change (Section 2.7) 12.1 Average Rates of Change We briefly introduced rates of change when we discussed the slope of a line. Recall that rates of change arise when two different quantities are changing simultaneously. The word per is very often associated with a rate of change. Think about the speed of a car. The position of the car is changing as the time changes. Both position and time are changing. We measure the speed (a.k.a. the rate of change of the distance traveled by the car with respect to time) in miles per hour. We can think of this as d. This looks a lot like the t formula for the slope of a line. To find the Average Rate of Change of one quantity (quantity 1) with respect to another quantity (quantity 2), you simply divide the change in quantity 1 by the change in quantity 2. quantity 1 quantity 2 One of the most common examples used to introduce average rates of change is average speed. The average speed of an object is the average rate of change of the distance traveled by the object with respect to time. 1

2 Example 12.1 (Average Speed) Joni leaves the house at 9:00AM. She travels 5 miles to the grocery store, stops the car, shops, and drives back home. She returns to her home at 11:30AM. What is the average speed of Joni s car from 9:00AM to 11:30AM? Example 12.2 (Average Speed) Bogey leaves the house at 8:00AM. He travels on a road that is due east from his front door. The total distance Bogey has traveled t hours after 8:00AM is given by where d(t) is measured in miles. d(t) = 5t 3 (a) What was Bogey s average speed between 8:00AM and 9:00AM? (b) What was Bogey s average speed between 10:00AM and 11:00AM? (c) If Bogey reaches his destination at 11:00AM, what was his average speed for the whole trip? 2

3 Definition 12.3 (Average Rate of Change of a Function) If f is a function, then the average rate of change of f(x) with respect to x as x changes from a to b is given by f(b) f(a) b a = f(a) f(b) a b = f(x) x Example 12.4 (Average Rate of Change of a Function) Let g(x) = x 2 + 3x + 5. Find the average rate of change of g(x) with respect to x as x changes from 5 to 3. Example 12.5 (Average Rate of Change of a Function) Let g(x) = x 2 +3x+5. How can you interpret the average rate of change of g(x) with respect to x as x changes from 5 to 3 graphically? We can now see that the average rate of change of a function with respect to x is simply the slope of the line that connects two points on the graph of the function. The line that connects these two points is called a secant line. 3

4 Example 12.6 (Average Rate of Change of a Function) Air is being pumped into a tire. When the outer radius of the tire is R feet, the volume of the air in the tire is given by V (R) = π(r 2 1) where V is measured in ft 3. What is the average rate of change of the volume of air in the tire with respect to the outer radius of the tire as the outer radius changes from 18 inches to 2 feet? Difference Quotients It is common to compute the average rate of change of a function for small x values. For example, you might need to compute the average rate of change for x between 4 and 4.1, or between 5 and 5.01, or between 7 and Sometimes it is even necessary to compute several of these values. If this is the case, it is often helpful to find a general formula for the average rate of change. If you are computing the average rate of change from x to x + h, what is x? Example 12.7 (Difference Quotient) Let g(x) = x 2 + 3x + 5. Find the average rate of change of g(x) on the interval from x to x + h. (You should assume that h 0. Nothing changed if h = 0.) 4

5 Example 12.8 Use your result from the previous example to find the average rate of change of g(x) on the interval from a to b. What is x for each problem? What is h? a b x h Ave ROC Definition 12.9 Let f be a function. The difference quotient of f is the average rate of change of f on the interval from x to x + h. f(x + h) f(x) (x + h) x = f(x + h) f(x) h Instantaneous Rates of Change In this class, we do not yet have the tools to formally define an instantaneous rate of change. These tools will not be developed until Calculus. Nevertheless, we can begin to develop an intuitive understanding which can help you to understand the definition when it is finally introduced in Calculus. When you look at the speedometer in your car, it is not measuring the average speed for your entire trip. It is attempting to measure your speed right now. In other words, your speedometer is measuring an instantaneous speed. How do you measure speed at an instant since there is no change in time ( t = 0). We cannot divide by zero. We can begin to understand this by doing the following. Suppose you want to measure the speed at the instant when the time is t. Look at really small intervals that contain t. ( Really small intervals means that t is small.) For example, what if we want to know our (instantaneous) speed 4 hours into our trip. Perhaps we could measure our average speed from 3.9 hours to 4 hours, from 3.99 hours to 4 hours, from 4 hours to 4.1 hours, from 4 hours to If the average speeds over shorter and shorter intervals seem 5

6 to be approaching the same value, then we could make good guess about our instantaneous speed at 4 hours. Example The table below contains some average speeds for our trip. Based on this information, what would you guess our instantaneous speed was 4 hours into the trip? Example (Difference Quotient) Let g(x) = x 2 + 3x + 5. Average Speed from Time 1 Time 1 Time 2 to Time hours 4 hours 65.6 mph 3. 7 hours 4 hours 65.4 mph 3. 9 hours 4 hours mph hours 4 hours mph hours 4 hours mph 4 hours 4.7 hours 64.2 mph 4 hours 4.5 hours 65.2 mph 4 hours 4.3 hours 64.9 mph 4 hours 4.1 hours mph 4 hours hours mph 1. Make an educated guess about the instantaneous rate of change of g(x) at x = Make an educated guess about the instantaneous rate of change of g(x) at x = Make an educated guess about the instantaneous rate of change of g(x) at x = a. 6