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MAT 1221 Survey of Calculus

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MAT 1221 Survey of Calculus. Section 3.3 Concavity and the Second Derivative Test. http://myhome.spu.edu/lauw. Expectations. Check your algebra. Check your calculator works Formally answer the question with the expected information. 1 Minute….

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## MAT 1221 Survey of Calculus

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### MAT 1221Survey of Calculus

Section 3.3

Concavity and the Second Derivative Test

http://myhome.spu.edu/lauw

Expectations
• Check your algebra.
• Check your calculator works
• Formally answer the question with the expected information
1 Minute…
• You can learn all the important concepts in 1 minute.
1 Minute…
• Critical numbers – give the potential local max/mins
1 Minute…
• Critical numbers – give the potential local max/mins
• If the graph is“concave down”at a critical number, it has a local max
1 Minute…
• Critical numbers – give the potential local max/mins
• If the graph is“concave up”at a critical number, it has a local min
1 Minute…
• You can learn all the important concepts in 1 minute.
• We are going to develop the theory carefully so that it works for all the functions that we are interested in.
• There are a few definitions…
New - 2014
• I will try to streamline this a bit and see if it can go better than covering all the details.
Preview
• Define
• Second Derivatives
• Concavities
• Find the intervals of concave up and concave down
• The Second Derivative Test
Higher Derivatives

Given a function

which is a function.

Higher Derivatives

Given a function

Concave Up

(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 calledconcave 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

y

Concave down

Concave up

x

c

has no local max. or min.

has an inflection point at

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 f is concave downward on .

Concavity Test

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

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

Why? (Hint: )

Why?

implies is increasing.

i.e. the slope of tangent lines is increasing.

Why?

implies is decreasing.

i.e. the slope of tangent lines is decreasing.

Example 1

Find the intervals of concavity and the inflection points

Example 1

1. Find ,

and the values of such that

Example 1

2. Sketch a diagram of the subintervals formed by the values found in step 1. Make sure you label the subintervals.

Example 1

3. Find the intervals of concavity and inflection point.

The Second Derivative Test

Suppose is continuous near .

(a) If and , then has a local minimum at c.

(b) If and , then f has a local maximum at .

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

Second Derivative Test

Suppose

If then has a local min at

y

x

c

Second Derivative Test

Suppose

If then has a local max at

y

x

c

The Second Derivative Test

(c) If , then no conclusion

The Second Derivative Test

If , then no conclusion

The Second Derivative Test

If , then no conclusion

The Second Derivative Test

If , then no conclusion

The Second Derivative Test

Suppose is continuous near .

(a) If and , then has a local minimum at c.

(b) If and , then f has a local maximum at .

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

Example 2

Use the second derivative test to find the local max. and local min.

Example 2

(a) Find the critical numbers of

Example 2

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

• The local max. value of f is
• The local min. value of f is
Review
• Example 1 & 2 illustrate two different but related problems.
• 1. Find the intervals of concavity and inflection points.
• 2. Find the local max. /min. values
Expectations
• Follow the steps to solve the two problems