2 2 The First and Second Derivative Rules

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2. 2 The First and Second Derivative Rules

2. 2 The First and Second Derivative Rules

We can get information about the graph of a function f(x) by examining the

We can get information about the graph of a function f(x) by examining the first and second derivatives, f’(x) and f’’(x). The first derivative of a function f(x) can indicate if the graph of f(x) is increasing or decreasing.

First Derivative Rule If f’(a) > 0 then f(x) is increasing at x =

First Derivative Rule If f’(a) > 0 then f(x) is increasing at x = a. If f’(a) < 0 then f(x) is decreasing at x = a. Note: Relative extreme points can occur when f’(a) = 0.

The second derivative of a function f(x) gives useful information about the concavity of

The second derivative of a function f(x) gives useful information about the concavity of the graph of f(x).

Second Derivative Rule If f’’(a) > 0, then f(x) is concave up at x

Second Derivative Rule If f’’(a) > 0, then f(x) is concave up at x = a. If f(‘’(a) < 0, then f(x) is concave down at x = a. Note: A point of inflection might exist at a if f’’(a) = 0.

Sketch the graph of a function f(x) with all the following properties. a. (2,

Sketch the graph of a function f(x) with all the following properties. a. (2, 3), (4, 5), and (6, 7) are on the graph. b. f’(6) = 0 and f’(2) = 0 c. f’’(x) > 0 for x < 4, f’’(4) = 0, and f’’(x) < 0 for x > 4.

Connections Between the Graphs of f(x) and f’(x)

Connections Between the Graphs of f(x) and f’(x)

We can think of the derivative of f(x) as a “slope-function” for f(x). The

We can think of the derivative of f(x) as a “slope-function” for f(x). The y-values on the graph of y = f’(x) are the slopes of the corresponding points on the original graph y = f(x).

s The slope is given at several points. How is the slope changing on

s The slope is given at several points. How is the slope changing on the graph?

Compare the slopes on the graph of f(x) with the y-coordinates of the points

Compare the slopes on the graph of f(x) with the y-coordinates of the points on the graph of f’(x) below. What is the correspondence?

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