Unit 11 – Derivative GraphsSection 11.1 – First Derivative Graphs

First DerivativeSlope of the Tangent Line

This is the graph of f(x)

Given the graph of FUNCTION f(x):

If f x is increasing, then f ' x 0

This is the graph of f(x)

Slope of tangent line positive

This is the graph of f(x)

Given the graph of FUNCTION f(x):

If f x is decreasing, then f ' x 0

This is the graph of f(x)

Slope of tangent line negative

Given the function f ' x :

This is the graph of f ' x

If f ‘ (x) is positive:

•Slopes of f(x) are positive•f(x) is increasing

If f ‘ (x) is negative:

•Slopes of f(x) are negative•f(x) is decreasing

This is the graph of f(x)

2f x x

This is the graph of f ‘ (x)

f ' x 2x

This is the graph of f ‘ (x)

At x = 3…..

f ' 3 6The graph of f(x) is increasing.

At x = -1…..

f ' 1 2 The graph of f(x) is decreasing.

This is the graph of f ‘ (x)

On what intervals is the graph of f(x) increasing?

2 , 0,1 , 2,

On what intervals is the graph of f(x) decreasing?

2, 0 , 1, 2

For what values of x is f ‘ (x) = 0?

X X X X

-2, 0, 1, 2

BONUS QUESTION:

For what values of x is f “ (x) = 0?

X

X

X

-1.2, 0.4, 1.5

The First Derivative Test For Maximum/Minimum

The solutions to f ‘ (x) = 0 are CRITICAL POINTS.

If f ‘ (x) changes fromnegative to postive, a

RELATIVE MINIMUM exists.

If f ‘ (x) changes frompositive to negative, a

RELATIVE MAXIMUM exists.

This is the graph of f ‘ (x)

For what values of x is f ‘ (x) = 0?

-2, 0, 1, 2

The critical points of f(x) are-2, 0, 1, 2

The relative maxima of f(x) are at-2 and 1

because f ‘ (x) changes frompositive to negative

The relative minima of f(x) are at0 and 2

because f ‘ (x) changes fromnegative to positive

Copy the graph ->

If the graph represents f(x), mark with an x the criticalnumbers

X

XX

XIf the graph represents f ‘ (x), mark with an x the criticalnumbers

X X X

If the graph represents f(x), estimate to one decimal place thevalue(s) of x at which there is a relative maximum.

-1.4, 0.4

If the graph represents f ‘ (x), estimate to one decimal place thevalue(s) of x at which there is a relative minimum.

-1.9, 1.8

3

2

x 1dyGiven

dx x 3

a) For what value(s) of x will there be a horizontal tangent on f(x) ?1

b) For what value(s) of x will the graph of f(x) be increasing?

1,

c) For what value(s) of x will there be a relative minimum on f(x)?1

d) For what value(s) of x will there be a relative maximum on f(x)?none

CALCULATOR REQUIRED

If the graph represented is f(x), for what values of x is the first derivative equal to zero? If the graph represented if f ‘ (x), for what values of x would the local max(s) and local min(s) be? If the graph represented is f(x), write using interval notation the interval(s) on which the graph is increasing. If the graph represented is f ‘ (x), write using interval notation the interval(s) on which the graph is decreasing.

-1 and 2

-2, 1, 3

(-3, -1), (2, 4)

[-3, -2) U (1, 3]

The graph is on the interval [-3, 4]

The graph is on the interval [-2, 2]

If the graph represented is f(x), for what value(s) of x if f ‘ (x) = 0? 

If the graph represented is f ‘ (x), for what values of x is there a relative minimum? If the graph represented is f(x), write using interval notation the interval(s) on which f ‘ (x) is positive. If the graph represented if f ‘ (x), what at value(s) of x is there a relative maximum?

-1, 1

(-2, -1.5), (-0.5, 0.5), (1.5, 2)

0

-1.5, -0.5, 0.5, 1.5

This is the graph of f ‘ (x)on the interval [-5, 3]

Where are the critical point(s) of f(x)?

Where is the ABSOLUTEmaximum of f(x) on [-5, 3]?

What is f ‘ (1)?

Where is the ABSOLUTEminimum of f(x) on [-5, 3]?

x = 1

0

x = 3

x = -5