1-4 Study Guide and Intervention - Weebly

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1-4 Study Guide and Intervention

Extrema and Average Rates of Change

Increasing and Decreasing Behavior Functions can increase, decrease, or remain constant over a given interval. The points at which a function changes its increasing or decreasing behavior are called critical points. A critical point can be a relative minimum, absolute minimum, relative maximum, or absolute maximum. The general term for minimum or maximum is extremum or extrema.

Example: Estimate to the nearest 0.5 unit and classify the extrema for the graph of f(x). Support the answers numerically.

Analyze Graphically

It appears that f(x) has a relative maximum of 0 at x = ?1.5, a relative minimum of

?3.5 at x = ?0.5, a relative maximum of ?2.5 at x = 0.5, and a relative minimum of

?6 at x = 1.5. It also appears that

= ? and

= , so

there appears to be no absolute extrema.

Exercises Use a graphing calculator to approximate to the nearest hundredth the relative or absolute extrema of each function. State the x-value(s) where they occur. Sketch the graphs and label the max's and min's.

1. f(x) = + ?

2. f(x) = +

Chapter 1

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Glencoe Precalculus

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1-4 Study Guide and Intervention (continued)

Extrema and Average Rates of Change

Average Rate of Change The average rate of change between any two points on the graph of f is the slope of the line through those points. The line through any two points on a curve is called a secant line. The average rate of change on the interval [ , ] is the slope of the secant line, .

=

Example: Find the average rate of change of f(x) = 0. a. [?3, ?1]

+ 2x on each interval.

=

Substitute ?3 for and ?1 for .

=

= ?

or

Evaluate f(?1) and f(?3). Simplify.

b. [?1, 1] =

=

or

Substitute ?1 for and 1 for . Evaluate and simplify.

Exercises Find the average rate of change of each function on the given interval.

1. f(x) = + ? x ? 1; [?3, ?2]

2. f(x) = + ? x ? 1; [?1, 0]

3. f(x) = + ? 7x ? 4; [?3, ?1]

4. f(x) = + ? 7x ? 4; [1, 3]

5. f(x) = + 8x ? 3; [?4, 0]

6. f(x) = ? + 8x ? 3; [0, 1]

Chapter 1

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Glencoe Precalculus

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