Calculus 3 5

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```							       3.5 Concavity and Inflection
Points
Tues Nov 29
Do Now
Find the 2nd derivative of each function
1) f (x)  3x  2x  4
2

2) f (x)  x 4  3x 3  5x  2



HW Review: worksheet p.287
#19-24, 27-28, 31-38
Applications of the 2nd
derivative
• So far, we’ve only talked about one
application of the 2nd derivative, which
is the acceleration function
• The second derivative can also be used
to describe the behavior of functions as
well.
Concavity and Inflections
• The 1st derivative is used to describe
slope. But since it is also a function, it
also has its own “slope” or derivative.
• The 2nd derivative can be used to
model the behavior of the slope, as it is
ALSO changing with the function
– Some slopes can be steep, while others
rather flat
Concavity
• The 2nd derivative can be used to
describe concavity
• Concavity is the rate at which the slope
increases or decreases
• There are two types of concavity
– Concave up (looks like a smile)
– Concave down (looks like a frown)
Concavity and f’’(x)
• Thm- Suppose f(x) is differentiable on
an interval I and f ’’(x) exists,
– If f ’’(x) > 0, then the graph is concave up
– If f ’’(x) < 0, then the graph is concave
down
Note: Second derivative only
Inflection Points
• An inflection point is a point on the
graph where a graph alternates
between concave up and concave down
• Think of the inflection point as the 2nd
derivative’s critical point
• We will find all of our inflection points
when f ’’(x) = 0
Example 5.1
• Determine where the graph is concave
up and concave down
f (x)  2x 3  9x 2  24x 10


Example 5.2
• Determine where the graph is concave
up and down, and find any inflection
points     f (x)  x 4  6x 2 1


Ex 5.3
• Determine the concavity and inflection
points of     f (x)  x 4


Closure
• Journal Entry: What can the 2nd
derivative tell us about the behavior of
the function?

• HW: p.284 #9-14 (Don’t sketch the
graphs)
3-5 2nd Derivative Test
Wed Nov 30
• Do Now
• Find the intervals of concavity and
inflection points of the following function

f (x)  x  4 x 1
4       3


HW Review p.284 #9-14
• book
2nd Derivative Test
• The 2nd Derivative can also be used to
determine if a critical point is a local
max or min.
• Thm- Suppose that f(x) is continuous on
an interval (a,b) and f’( c) = 0, then
– If f’’( c) < 0, then c is a local max
• Concave down means a local max!
– If f’’( c) > 0, then c is a local min
• Concave up means a local min!
Practice
• Bookwork p.285 #15, 16, 19-28
Closure:
• Hand in: Find intervals of increase,
decrease, and concavity, local extrema,
and inflection points of
f (x)  x  4 x 1
4       2

• HW: p.285 #15, 16, 19-28


Sketching Curves
Thurs Dec 01
• Do Now

```
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