Functions 2

Increasing and Decreasing Functions; Average Rate of Change 2.3

Increasing & Decreasing Funcns.; Avg. Rate of Change Functions are often used to model changing quantities. In this section, we learn how to: Determine if a function is increasing or decreasing. Find the rate at which its values change as the variable changes.

Increasing and Decreasing Functions

It is very useful to know where the graph of a function rises and where it falls.

Increasing and Decreasing Functions The graph shown here rises, falls, then rises again as we move from left to right. It rises from A to B, falls from B to C, and rises again from C to D.

Increasing and Decreasing Functions The function f is said to be: Increasing when its graph rises. Decreasing when its graph falls.

Increasing Function—Definition f is increasing on an interval I if f(x 1 ) < f(x 2 ) whenever x 1 < x 2 in I.

Decreasing Function—Definition f is decreasing on an interval I if f(x 1 ) > f(x 2 ) whenever x 1 < x 2 in I.

E.g. 1—Intervals Where a Func. Increases & Decreases The graph gives the weight W of a person at age x. Determine the intervals on which the function W is increasing and on which it is decreasing.

E.g. 1—Intervals Where a Func. Increases & Decreases The function is: Increasing on [0, 25] and [35, 40]. Decreasing on [40, 50]. Constant (neither increasing nor decreasing) on [25, 35] and [50, 80].

E.g. 1—Intervals Where a Func. Increases & Decreases This means that: The person gained weight until age 25. He gained weight again between ages 35 and 40. He lost weight between ages 40 and 50.

E.g. 2—Graph to Find Where a Function Incrs. & Decrs. (a) Sketch the graph of the function f(x) = x 2/3. (b) Find the domain and range of the function. (c) Find the intervals on which f increases and decreases.

E.g. 2—Sketching the Graph We use a graphing calculator to sketch the graph here. Example (a)

E.g. 2—Domain & Range From the graph, we observe that: The domain of f is. The range is [0, ∞). Example (b)

E.g. 2—Increase & Decrease We see that f is: Decreasing on (-∞, 0]. Increasing on [0, ∞). Example (c)

Average Rate of Change

We are all familiar with the concept of speed. If you drive a distance of 120 miles in 2 hours, then your average speed, or rate of travel, is:

Average Rate of Change Now, suppose you take a car trip and record the distance that you travel every few minutes. The distance s you have traveled is a function of the time t: s(t) = total distance traveled at time t

Average Rate of Change We graph the function s as shown. The graph shows that you have traveled a total of: 50 miles after 1 hour 75 miles after 2 hours 140 miles after 3 hours and so on.

Average Rate of Change To find your average speed between any two points on the trip, we divide the distance traveled by the time elapsed. Let’s calculate your average speed between 1:00 P.M. and 4:00 P.M. The time elapsed is 4 – 1 = 3 hours.

Average Rate of Change To find the distance you traveled, we subtract the distance at 1:00 P.M. from the distance at 4:00 P.M., that is, 200 – 50 = 150 mi

Average Rate of Change Thus, your average speed is:

Average Rate of Change The average speed we have calculated can be expressed using function notation:

Average Rate of Change Note that the average speed is different over different time intervals.

Average Rate of Change For example, between 2:00 P.M. and 3:00 P.M., we find that:

Average Rate of Change—Significance Finding average rates of change is important in many contexts. For instance, we may be interested in knowing: How quickly the air temperature is dropping as a storm approaches. How fast revenues are increasing from the sale of a new product.

Average Rate of Change—Significance So, we need to know how to determine the average rate of change of the functions that model these quantities. In fact, the concept of average rate of change can be defined for any function.

Average Rate of Change—Definition The average rate of change of the function y = f(x) between x = a and x = b is:

Average Rate of Change—Definition The average rate of change is the slope of the secant line between x = a and x = b on the graph of f. This is the line that passes through (a, f(a)) and (b, f(b)).

E.g. 3—Calculating the Average Rate of Change For the function f(x) = (x – 3) 2, whose graph is shown, find the average rate of change between the following points: (a) x = 1 and x = 3 (b) x = 4 and x = 7

E.g. 3—Average Rate of Change Example (a)

E.g. 3—Average Rate of Change Example (b)

E.g. 4—Average Speed of a Falling Object If an object is dropped from a tall building, then the distance it has fallen after t seconds is given by the function d(t) = 16t 2. Find its average speed (average rate of change) over the following intervals: (a) Between 1 s and 5 s (b) Between t = a and t = a + h

E.g. 4—Avg. Spd. of Falling Object Example (a)

E.g. 4—Avg. Spd. of Falling Object Example (b)

E.g. 4—Avg. Spd. of Falling Object Example (b)

Difference Quotient & Instantaneous Rate of Change The average rate of change calculated in Example 4 (b) is known as a difference quotient. In calculus, we use difference quotients to calculate instantaneous rates of change.

Instantaneous Rate of Change An example of an instantaneous rate of change is the speed shown on the speedometer of your car. This changes from one instant to the next as your car’s speed changes.

E.g. 5—Average Rate of Temperature Change The table gives the outdoor temperatures observed by a science student on a spring day.

E.g. 5—Average Rate of Temperature Change Draw a graph of the data. Find the average rate of change of temperature between the following times: (a) 8:00 A.M. – 9:00 A.M. (b) 1:00 P.M. – 3:00 P.M. (c) 4:00 P.M. – 7:00 P.M.

E.g. 5—Average Rate of Temperature Change A graph of the temperature data is shown. Let t represent time, measured in hours since midnight. Thus, 2:00 P.M., for example, corresponds to t = 14.

E.g. 5—Average Rate of Temperature Change Define the function F by: F(t) = temperature at time t

E.g. 5—Avg. Rate of Temp. Change The average rate of change was 2°F per hour. Example (a)

E.g. 5—Avg. Rate of Temp. Change The average rate of change was 2.5°F per hour. Example (b)

E.g. 5—Avg. Rate of Temp. Change The average rate of change was about –4.3°F per hour during this time interval. The negative sign indicates the temperature was dropping. Example (c)

Average Rate of Change The graphs show that, if a function is: Increasing on an interval, then the average rate of change between any two points is positive. Decreasing on an interval, then the average rate of change between any two points is negative.

E.g. 6—Linear Functions Let f(x) = 3x – 5. Find the average rate of change of f between the following points. (a) x = 0 and x = 1 (b) x = 3 and x = 7 (c) x = a and x = a + h What conclusion can you draw from your answers?

E.g. 6—Linear Functions Example (a)

E.g. 6—Linear Functions Example (b)

E.g. 6—Linear Functions Example (c)

E.g. 6—Linear Functions Have Constant Rate of Change It appears that the average rate of change is always 3 for this function. In fact, part (c) proves that the rate of change between any two arbitrary points x = a and x = a + h is 3.

Linear Functions As Example 6 indicates, for a linear function f(x) = mx + b, the average rate of change between any two points is the slope m of the line. This agrees with what we learned in Section 1-10: The slope of a line represents the rate of change of y with respect to x.