Concavity and the second derivative
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Concavity and the Second Derivative. mysite.oswego308.org/.../3.4-. Relative max. f (0) = 1. Relative min. f (4) = -31. The First Derivative. Find the intervals of increase and decrease of. + 0 - 0 +. 0 4. inc: (-∞,0)(4,∞) dec: (0,4).

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Concavity and the second derivative

Concavity and the Second Derivative

mysite.oswego308.org/.../3.4- ...


The first derivative

Relative max. f (0) = 1

Relative min. f (4) = -31

The First Derivative

Find the intervals of increase and decrease of

+ 0 - 0 +

0 4

inc: (-∞,0)(4,∞)

dec: (0,4)

http://math.colstate.edu/calmada/courses/math1125/ch5%20sect%201-2-3-4.ppt#338,1,Chapter 5 Applications of the Derivative Sections 5.1, 5.2, 5.3, and 5.4


Concavity of a Function

As you look at the graph of a function … 

… if the function CURVES UP, like a cup,

we say the function is _______________.

  …if the function CURVES DOWN,

like a frown,

we say the function is _______________.

CONCAVE UP

CONCAVE DOWN



is positive

is positive

is negative

is negative

is zero

Second derivative:

Curve is concave up.

Curve is concave down.

Possible inflection point

(where concavity changes).

+

+



Example:

inc: (-∞,0)(2,∞)

dec: (0,2)

positive

negative

positive


Example:

up: (1,∞)

down: (-∞,1)


Example: Determine the points of inflection and discuss the concavity of the graph of

(x^4-4x^3)/15

Points of inflection


Homework
Homework: concavity of the graph of

  • Page 195

  • #5-10, 21, 22


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