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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.

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introduction to functions and graphs

1

Introduction to Functions and Graphs

1.5 Functions and Their Rates of Change

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

increasing and decreasing functions
Increasing and Decreasing Functions

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).

graphs of increasing and decreasing functions
Graphs of Increasing and Decreasing Functions

when x1 < x2, then f(x1) < f(x2) and f is increasing

when x1 < x2, then f(x1) > f(x2) and f is decreasing

graphs of increasing and decreasing functions1
Graphs of Increasing and Decreasing Functions

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.

interval notation
Interval Notation

A convenient notation for number line graphs is called interval notation.

Instead of drawing number lines . . .

closed and open intervals
Closed and Open Intervals

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.

half open intervals
Half-open Intervals

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

union symbol
Union Symbol

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

increasing decreasing and endpoints
Increasing, Decreasing, and Endpoints

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).

example 2 determining where a function is increasing or decreasing
Example 2: Determining where a function is increasing or decreasing

and

Use the graph of

interval notation to identify where f is increasing or decreasing.

Solution

Decreasing:

Increasing:

average rate of change
Average Rate of Change

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

average rate of change1
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).

example 4 modeling braking distance for a car
Example 4: Modeling braking distance for a car

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.

example 4 modeling braking distance for a car1

Braking distance increases, on average, by 10 feet for

each 1-mile-per-hour increasein speed between 30 and 60 miles per hour.

Example 4: Modeling braking distance for a car

Solution

(a)

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
The Difference Quotient

The difference quotient of a function f is an expression of the form

  • where h ≠ 0.
example 6 calculating a difference quotient
Example 6: Calculating a difference quotient

Let the distance d in feet that a racehorse runs in t seconds be d(t) = 2t2 for 0 ≤ t ≤ 10.

(a)Find d(t+h).

(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.

example 6 calculating a difference quotient2
Example 6: Calculating a difference quotient

Solution

(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.

example 6 calculating a difference quotient3
Example 6: Calculating a difference quotient

Solution

(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.

key ideas for this section
Key Ideas for this section:
  • How do we know a function is increasing or decreasing?
  • What notation do we use to describe increasing and decreasing functions?
  • How do we use interval notation?
    • Half open, closed, infinite
  • What is the average rate of change?
  • How do we calculate average rate of change?
key ideas for this section1
Key Ideas for this section:
  • What is the secant line?
  • What is the difference quotient?
  • How can we calculate the difference quotient?