1. Introduction to Functions and Graphs. 1.5 Functions and Their Rates of Change. Functions and Their Rates of Change. 1.5. Identify where a function is increasing or decreasing Use interval notation Use and interpret average rate of change Calculate the difference quotient.
Identify where a function is increasing or decreasing
Use interval notation
Use and interpret average rate of change
Calculate the difference quotient
Suppose that a function f is defined over an interval I on the number line. If x1 and x2 are in I,
(a) fincreases on I if, whenever x1 < x2, f(x1) < f(x2);
(b) fdecreases on I if, whenever x1 < x2, f(x1) > f(x2).
when x1 < x2, then f(x1) < f(x2) and f is increasing
when x1 < x2, then f(x1) > f(x2) and f is decreasing
If could walk from left to right along the graph of an increasing function, it would be uphill.
For a decreasing function, we would walk downhill.
A convenient notation for number line graphs is called interval notation.
Instead of drawing number lines . . .
When a set includes the endpoints, the interval is a closedinterval and brackets are used.
When a set does not include the endpoints, the interval is an openinterval and parentheses are used.
When a set includes one endpoint and not the other, the interval is a half-open and 1 bracket and 1 parenthesis is used.
This represents the interval
An inequality in the form x < 1 orx > 3 indicates the set of real numbers that are either less than 1 or greater than 3.
The union symbol U can be used to write this inequality in interval notation as
The concepts of increasing and decreasing apply only to intervals of the real number line and NOT to individual points.
Decreasing: (–∞, 0]
Increasing: [0, ∞)
Do NOT say that the function f both increases and decreases at the point (0, 0).
Use the graph of
interval notation to identify where f is increasing or decreasing.
Graphs of nonlinear functions are not straight lines, so we speak of average rate of change.
The line L is referred to as the secant line, and the slope of Lrepresents the average rate of change fromx1 to x2.
Different values of x1 and x2 usually yield different secant lines and different average rates of change.
Animation: Average Rate of Change
Let (x1, y1) and (x2, y2) be distinct points on the graph of a function f. The average rate of change of f from x1 to x2 is
That is, the average rate of change from x1 to x2 equals the slope of the line passing through (x1, y1) and (x2, y2).
Highway engineers sometimes estimate the braking distance in feet for a car traveling atxmiles per hour on wet, level pavement by
using the formula
(a)Evaluate f(30) and f(60). Interpret these results.
(b)Calculate the average rate of change of f from 30 to 60. Interpret this result.
each 1-mile-per-hour increasein speed between 30 and 60 miles per hour.Example 4: Modeling braking distance for a car
At 30 miles per hour the braking distance is 100 feet, and at 60 mph it is 400 ft.
(b)Average rate of change of f from 30 to 60:
The difference quotient of a function f is an expression of the form
Let the distance d in feet that a racehorse runs in t seconds be d(t) = 2t2 for 0 ≤ t ≤ 10.
(b)Find the difference quotient of d and simplify the result.
(c)Evaluate the difference quotient for t=7and h=0.1. Interpret your results.
(d)Evaluate the difference quotient for t=4and h=1.Then sketch a graph that illustrates the result.
(c) If t = 7 and h = 0.1, then the difference quotient becomes
4t + 2h = 4(7) +2(0.1) = 28.2.
The average rate of change, or average velocity, of the horse from 7 seconds to 7 + 0.1 = 7.1 seconds is 28.2 feet per second.
(d) If t = 4 and h = 1, then4t + 2h = 4(4) + 2(1) = 18
If t = 4 and d = 2(4)2 = 32, the first point is (4, 32). If t = 4 and h = 1, it follows that t + h = 4 + 1 = 5 so d = 2(5)2 = 50, the second point is (5, 50).
Thus the slope of the line passing through (4, 32) and (5, 50) is m = 18.