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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

MAT 1221Survey of Calculus

Section 3.3

Concavity and the Second Derivative Test

http://myhome.spu.edu/lauw

expectations
Expectations
  • Check your algebra.
  • Check your calculator works
  • Formally answer the question with the expected information
1 minute
1 Minute…
  • You can learn all the important concepts in 1 minute.
1 minute1
1 Minute…
  • Critical numbers – give the potential local max/mins
1 minute2
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 minute3
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 minute4
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
New - 2014
  • I will try to streamline this a bit and see if it can go better than covering all the details.
preview
Preview
  • Define
    • Second Derivatives
    • Concavities
  • Find the intervals of concave up and concave down
  • The Second Derivative Test
higher derivatives
Higher Derivatives

Given a function

which is a function.

higher derivatives1
Higher Derivatives

Given a function

concave up
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
Concavity

is concave up on

  • Potential local min.
concavity1
Concavity

is concave down on

  • Potential local max.
concavity2
Concavity

y

Concave down

Concave up

x

c

has no local max. or min.

has an inflection point at

definition
Definition
  • An inflection point is a point where the concavity changes (from up to down or from down to up)
concavity test
Concavity Test

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

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

concavity test1
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: )

slide21
Why?

implies is increasing.

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

slide22
Why?

implies is decreasing.

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

example 1
Example 1

Find the intervals of concavity and the inflection points

example 11
Example 1

1. Find ,

and the values of such that

example 12
Example 1

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

example 13
Example 1

3. Find the intervals of concavity and inflection point.

the second derivative test
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
Second Derivative Test

Suppose

If then has a local min at

y

x

c

second derivative test1
Second Derivative Test

Suppose

If then has a local max at

y

x

c

the second derivative test1
The Second Derivative Test

(c) If , then no conclusion

the second derivative test2
The Second Derivative Test

If , then no conclusion

the second derivative test3
The Second Derivative Test

If , then no conclusion

the second derivative test4
The Second Derivative Test

If , then no conclusion

the second derivative test5
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
Example 2

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

example 21
Example 2

(a) Find the critical numbers of

example 22
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
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
expectations1
Expectations
  • Follow the steps to solve the two problems