MAT 1234 Calculus I Section 3.3 How Derivatives Affect the Shape of a Graph (II)

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Presentation transcript:

MAT 1234 Calculus I Section 3.3 How Derivatives Affect the Shape of a Graph (II)

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

The 1 st Derv. Test Find the critical numbers Find the intervals of increasing and decreasing Determine the local max./min.

The 1 st 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 1 st test.

The 2 nd Derv. Test We will talk about intervals of concave up and down But they are not part of the 2 nd 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

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.

Preview Part I Increasing/Decreasing Test The First Derivative Test Part II Concavity Test The Second Derivative Test

Definition

Concavity

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

Why?

Example 3 Find the intervals of concavity and the inflection points

Example 3 (a) Find, and the values of such that

Example 3 (b) Sketch a diagram of the subintervals formed by the values found in part (a). Make sure you label the subintervals.

Example 3 (c) Find the intervals of concavity and inflection point(s).

Expectation

Example 3 Verification

The Second Derivative Test

Second Derivative Test

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

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

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

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

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