MAT 1234 Calculus I Section 3 3 Derivatives

MAT 1234 Calculus I Section 3. 3 Derivatives and Graphs Part 2 http: //myhome. spu. edu/lauw

HW and …. l l Web. Assign HW Quiz 8 (3. 3 ONLY) Quiz 9 (2. 9, 3. 2) Take time to study for exam 2

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

The 1 st Derv. Test l l l 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 l l l We will talk about intervals of concave up and down They are related, but not part of the 2 nd test. The second test is relatively “fit” in terms of the “size” of the solution.

Preview l l We know the critical numbers give the potential local max/min. How to determine which one is local max/min?

Preview l l l We know the critical numbers give the potential local max/min. How to determine which one is local max/min? 30 second summary!

Preview

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

Preview 3. 3 Part I l Increasing/Decreasing Test l The First Derivative Test 3. 3 Part II l Concavity Test l The Second Derivative Test

Definition l

Concavity l

Concavity l

Concavity Concave down Concave up c

Definition l An inflection point is a point where the concavity changes (from up to down or from down to up)

Concavity Test l

Concavity Test l

Why? l

Why? l

Example 1 Find the intervals of concavity and the inflection points

Example 1 l

Example 1 l

Example 1 l

Example 1 (a) Find , and the values of such that

Example 1 (b) Sketch a diagram of the subintervals formed by the values found in part (a). Make sure you label the subintervals.

Example 1 (c) Find the intervals of concavity and inflection point(s).

Expectations l

Example 1 Verification

The Second Derivative Test l

Second Derivative Test l

Second Derivative Test l

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

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

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

Second Derivative Test l l l 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 l

The Second Derivative Test l

The Second Derivative Test l

The Second Derivative Test l

The Second Derivative Test l

Which Test is Easier? l l First Derivative Test Second Derivative Test

Final Reminder l l You need intervals of increasing/decreasing for the First Derivative Test. You do not need intervals of concavity for the Second Derivative Test.