MAT 1234Calculus I

Section 3.3How Derivatives Affect the

Shape of a Graph (II)

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HW and …. WebAssign HW Take time to study for exam 2

The 1st Derv. Test Find the critical numbers Find the intervals of increasing and

decreasing Determine the local max./min.

The 1st Derv. Test Find the critical numbers Find the intervals of increasing and

decreasing Determine the local max./min.

Note that intervals of increasing and decreasing are part of the 1st test.

The 2nd Derv. Test We will talk about intervals of concave

up and down But they are not part of the 2nd test.

Preview We know the critical numbers give the

potential local max/min. How to determine which one is local

max/min?

Preview We know the critical numbers give the

potential local max/min. How to determine which one is local

max/min? 30 second summary!

Preview

Concave Up

𝑓 ’ (𝑐)=0

Concave Down

𝑓 ’ (𝑐)=0

Preview We know the critical numbers give the

potential local max/min. How to determine which one is local

max/min? 30 second summary! We are going to develop the theory

carefully so that it works for all the functions that we are interested in.

PreviewPart I Increasing/Decreasing Test The First Derivative TestPart II Concavity Test The Second Derivative Test

Definition(a) A function is called concave upward

on an interval if the graph of lies above all of its tangents on .

(b) A function is called concave downward on an interval if the graph of lies below all of its tangents on .

Concavity is concave up on

Potential local min.

Concavity is concave down on

Potential local max.

Concavity

has no local max. or min. has an inflection point at

c

Concave down

Concave up

Definition An inflection point is a point where the

concavity changes (from up to down or from down to up)

Concavity Test

(a) If on an interval , then is concave upward on .

(b) If on an interval , then is concave downward on .

Concavity Test

(a) If on an interval , then is concave upward on .

(b) If on an interval , then is concave downward on .

Why?

Why? implies is increasing. i.e. the slope of tangent lines is increasing.

( ) ( )df x f xdx

Why? implies is decreasing. i.e. the slope of tangent lines is decreasing.

Example 3Find the intervals of concavity and the inflection points

1362)( 23 xxxxf

Example 31362)( 23 xxxxf

(a) Find , and the values of such that

)(xf )(xf

x 0)( xf

Example 31362)( 23 xxxxf

(b) Sketch a diagram of the subintervals formed by the values found in part (a). Make sure you label the subintervals.

Example 31362)( 23 xxxxf

(c) Find the intervals of concavity and inflection point(s).

has an inflection point at ( , )

Expectation Answer in full sentence. The inflection point should be given by

the point notation.

Example 3 Verification

The Second Derivative TestSuppose is continuous near .(a) If and , then has a local minimum at .(b) If and , then f has a local maximum at c.

(c) If , then no conclusion (use 1st derivative test)

Second Derivative TestSuppose

If then has a local min. at

0)( cf

0)( cf

𝑐

𝑓 ”(𝑐)>0

𝑓 ’ (𝑐)=0

Second Derivative TestSuppose

If then has a local max. at

0)( cf

0)( cf

𝑐

𝑓 ”(𝑐)<0

𝑓 ’ (𝑐)=0

Example 4 (Example 2 Revisit)Use the second derivative test to find the local max. and local min.

10249)( 23 xxxxf

Example 4 (Example 2 Revisit)(a) Find the critical numbers of

10249)( 23 xxxxf

Example 4 (Example 2 Revisit)(b) Use the Second Derivative Test to find the local max/min of

10249)( 23 xxxxf

The local max. value of isThe local min. value of is

Second Derivative Test Step 1: Find the critical points Step 2: For each critical point,

• determine the sign of the second derivative;• Find the function value• Make a formal conclusion

Note that no other steps are required such as finding intervals of inc/dec, concave up/down.

The Second Derivative Test(c) If , then no conclusion

The Second Derivative Test(c) If , then no conclusion

4

3

2

2

( )

( ) 4 00

( ) 12

(0) 12 0 0

f x x

f x xx

f x x

f

The Second Derivative Test(c) If , then no conclusion

4

3

2

2

( )

( ) 4 00

( ) 12

(0) 12 0 0

g x x

g x xx

g x x

g

The Second Derivative Test(c) If , then no conclusion

3

2

( )

( ) 3 00

( ) 6(0) 6 0 0

h x x

h x xx

h x xh

The Second Derivative TestSuppose is continuous near .(a) If and , then has a local minimum at .(b) If and , then f has a local maximum at c.

(c) If , then no conclusion (use 1st derivative test)

Which Test is Easier? First Derivative Test Second Derivative Test

Final Reminder You need intervals of

increasing/decreasing for the First Derivative Test.

You do not need intervals of concavity for the Second Derivative Test.

Classwork Do part (a), (d) and (e) only