Unit 4

Applications of Derivatives

Slide 2

4.1 Critical Number Test

Types of Maximums and Minimums

Absolute/Global Extreme Values• Maximum – • Minimum –

Local/Relative Extreme Values• Maximum –

• Minimum –

Slide 3

4.1 Critical Number Test

Critical Number/Value –

Slide 4

4.1 Critical Number Test1) For the following picture, find the absolute extrema, local extrema and critical numbers.

Slide 5

4.1 Critical Number TestExtreme Value Theorem – If f is continuous on a closed interval [a, b], then f has on that interval.

Slide 6

4.1 Critical Number Test

Critical Numbers Test for Finding Extrema

If f is continuous on a closed interval [a, b], then it’s absolute extreme values are paired with critical numbers or endpoints.

Steps for finding the absolute extrema.1.

2.

3.

4.

Slide 7

4.1 Critical Number TestFind the critical numbers of f. Then use the critical number test to determine the absolute extreme values of f.

1) 3 212 5 on 2, 2

2f x x x x

Slide 8

4.1 Critical Number TestFind the critical numbers of f. Then use the critical number test to determine the absolute extreme values of f.

2) 3 1 on 1, 1f x x x

Slide 9

4.2 Mean Value Theorem

If ( ) is continuous at every point of the closed interval , and

differentiable at every point of its interior , , then there is at least

( ) - ( )one point in , at which '( ) .

-

y f x a b

a b

fb f ac a b f c

b a

Slide 10

4.2 Mean Value TheoremDetermine if f satisfies the conditions of the mean value theorem. If so, find all possible values of c.

1) 3 3 5 on 3, 2f x x x

Slide 11

4.3 First Derivative Test

1 2If is defined for all in some interval and let and be in , then 1. is increasing if and

f x I x x I

f

2. is decreasing if and 3.

f

is constant if and

f

Slide 12

4.3 First Derivative Test

Suppose is continuous on , and differentiable on ,

1. If ' 0 for all in , , then

2. If ' 0 for all in , , then

f a b a b

f x x a b

f x x a b

Slide 13

4.3 First Derivative Test

First Derivative Test (for finding local extrema)

Let be continuous on , and

1. If ' 0 to the left of and ' 0 to the right of ,

then is a

f a b a c b

f x c f x c

f c

2. If ' 0 to the left of and ' 0 to the right of ,

then is a

f x c f x c

f c

Slide 14

4.3 First Derivative Test

2 2

Find the intervals over which is increasing and decreasing. Give the values of for which has local extreme values.1) 5 5 2

fx f

f x x x

Slide 15

4.3 First Derivative Test

2

Find the intervals over which is increasing and decreasing. Give the values of for which has local extreme values.1) sin sin on 0, 2

fx f

f x x x

Slide 16

4.4 Second Derivative Test• If the second derivative is positive, a curve looks like

• If the second derivative is negative, a curve looks like

Slide 17

4.4 Second Derivative Test• f is concave up on (a, b) if and only if the slopes of the tangents

to f are increasing• which means

• which means

• f is concave down on (a, b) if and only if the slopes of the tangents

to f are decreasing• which means

• which means

Slide 18

4.4 Second Derivative Test(c, f(c)) is an inflection point if and only if

1.

2.

Slide 19

4.4 Second Derivative Test

Second Derivative Test (for finding some local extrema)

1. If ' 0 and '' 0, then

2. If ' 0 and '' 0, then

f c f c

f c f c

3. If ' 0 and '' 0, then f c f c

Slide 20

4.4 Second Derivative Test

5 3Find the local extreme values using the second derivative test.1) 3 5 1f x x x

Slide 21

Relationship of f’ and f’’ to ff

4.4 Second Derivative Test

Slide 22

4.4 Second Derivative Test

2

3

Find the intervals for which is increasing and decreasing, local extreme values, intervals for which is concave up and concave down, inflection points and provide a sketch of the graph.1)

ff

f x x 1 x