Download Applications of the 2nd Derivative

03/16/2016
Math 131: Essential of Calculus – Unit 15
Applications of the 2nd Derivative – W 3/16/2015
1. Review: Finding intervals of increase/decreases and relative
extrema
2. The meaning of (the sign of) the 2nd Derivative
• f ′′ ( x ) > 0 implies tangent line slopes are increasing which
implies curve is concave up
• f ′′ ( x ) < 0 implies tangent line slopes are decreasing which
implies curve is concave down
3. Finding Intervals of Concavity using 2nd Derivative
4. 2nd Derivative Test for Relative Extrema
5. Inflection Points – where concavity changes
Review Problem
Find the intervals of increase and decrease for this function.
Express in interval notation.
x
y= 2
x +1
a. What do you differentiate? Why?
b. What are critical numbers? How do you find them?
c. Why do you plot the critical numbers on a number line?
d. How and why do you test values of derivative on a Sign Line?
Now find and identify the rel. maximums and rel. minimums
a. What are your candidates for relative maximums (a cap) and
relative minimums (a cup)?
b. How do you determine whether you have a cap, a cup or a
twist?
The 2nd Derivative
Again consider the function f ( x ) =
Look at the tangent lines – what do you see?
x
x +1
2
What’s happening here?
f
e
?
d
a
b
Calculate
?
2nd Derivatives & Concavity
If f ′′ ( x ) > 0 then f ′ ( x ) is increasing so the curve is
concave up.
If f ′′ ( x ) < 0 then f ′ ( x ) is decreasing so the curve is
concave down
c
d2y
What do you notice?
dx 2
Finding Intervals of Concavity
If f ′′ ( x ) is continuous (a continuous 2nd derivative!) by the IVT it
cannot change sign without first becoming 0 (or undefined)!
1. Take the 2nd Derivative
2. Find all 2nd derivative critical numbers: f ′′ ( c ) = 0 or f ′′ ( c ) DNE
3. Sketch a 2nd derivative sign line & determine intervals of
concavity
4. Express intervals of concavity in interval notation
5. Give x-y coordinates of any inflection points (if any)
Note: This is the same method used with the 1st derivative to
find intervals of increase/decrease
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03/16/2016
2nd Derivative Test for Relative Extrema
Exercise
Find intervals of increase/decrease and intervals of concavity for
x2
3
2
1. f ( x ) = x + 3 x + 1
2. f ( x ) = 2
x +3
a. Find (and )
b. Identify all 1st (and 2nd) derivative critical numbers
c. Sketch the sign line and determine all intervals of
increase/decrease (concave up concave down)
d. Express intervals of increase/decrease (concavity) in interval
notation
e. Give x-y coordinates of inflection points (if any)
Let (c, f(c)) be a critical point (i.e. f’(c) = 0 or f’(c) DNE). Then …
If f”(c) > 0 is concave up then (c, f(c)) is a relative minimum
(“cup”)
If f”(c) < 0 is concave down then (c, f(c)) is a relative maximum
(“cap”)
Otherwise the 2nd Derivative test fails (“twist”)
Counter-example: y = x3 has a critical point at x = 0
Question: To test for a relative min/max which “test” should you
use? 1st Derivative OR 2nd Derivative Test?
Inflection Points
An inflection point for a curve f(x) occurs where the concavity
changes
Continuing with the sign line created, find intervals of “concave
up” and “concave down” and identify points where concavity
changes!
Go back to a previous exercise and locate all inflection points!
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Question! x = 0 is a critical number for f ( x ) = x . Why? Is x = 0
an inflection point too?
Question! x = 0 is also a critical number for y = x4. Is it an
inflection point?
Max/Min Examples
Use the 2nd Derivative Test to identify the relative maximums
and/or minimums for the functions
1.
f ( x ) = x4 − 2 x2 + 3
2.
f ( x) = x +
1
x
For Friday – Onto Unit 16
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