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1.2. RATE OF CHANGE. Example:. Rate of Change of a Function. The average rate of change , or rate of change , of Q with respect to t over an interval is. Increasing and Decreasing Functions. If Q = f ( t ) for t in the interval a ≤ t ≤ b,

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1.2

RATE OF CHANGE


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Example:


Rate of change of a function

Rate of Change of a Function

The average rate of change, or rate of change, of Q with respect to t over an interval is


Increasing and decreasing functions

Increasing and Decreasing Functions

If Q = f(t) for t in the interval a ≤ t ≤ b,

• f is an increasing function if the values of f increase as t increases in this interval.

• f is a decreasing function if the values of f decrease as t increases in this interval.


Rates of change for increasing and decreasing functions

Rates of Change for Increasing and Decreasing Functions

If Q = f(t),

• If f is an increasing function, then the average rate of change of Q with respect to t is positive on every interval.

• If f is a decreasing function, then the average rate of change of Q with respect to t is negative on every interval.


Graphs of increasing and decreasing functions

Graphs ofIncreasing and Decreasing Functions

  • The graph of an increasing function rises when read from left to right.

  • The graph of a decreasing function falls when read from left to right.


Graphs of increasing and decreasing functions1

Graphs ofIncreasing and Decreasing Functions

Many functions have some intervals on which they are increasing and other intervals on which they are decreasing. These intervals can often be identified from the graph.

Decreasing for -2 < x < 3

Increasing for x < -2

Increasing for x > 3


Function notation for the average rate of change

Function Notation for the Average Rate of Change


1 2

y = x2

(3, 9)

Slope = 4

(-2, 4)

Slope = -1

(1, 1)

Average rate of change of f(x) on an interval is the slope of the dashed line (secant line) on that interval


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