For each of the following…◦ find the intervals where the graph is increasing/decreasing,◦ find all extrema

1. f(x) = -x4 + 4x 2. g(x)= x5 – 15x3 + 10

Write the equation of the tangent line to the graph off(x) = -3x2 + 4x – 1 at x = 2

Warm Up Feb. 28th

1. Y = 6x + 12. Y = 3x – 53. Y = -2x + 84. X = -15. X = 0 and x = ½6. X = 27. I: (0, ∞) D: (-∞, 0) Min @ x = 08. I: (-∞, 2) U (3, ∞) D: (2, 3) Max @ x = 2, Min @ x = 39. I: (-2, 0) U ( 2, ∞) D: (-∞, -2) U (0, 2)

Max @ x = 0, Min @ x = -2 and x = 2

Homework Questions…

Second Derivatives & Concavity

Using derivatives and knowledge about concavity to help accurately graph a polynomial

More Derivatives f '(x) represents the first derivative◦ Slope of the tangent line, instantaneous velocity or rate of change

f '' (x) represents the second derivative◦ Classify extrema or acceleration

Find the first and second derivative of the following.1) f(x) = x9 + 2x5 – 5x3 + 9

2) g(x) = 8x3 – 4x2 + 3x + 16

Concave up Concave down Concavity

The second derivative of a function can tell us whether a function is concave upward or concave downward. If a) f ''(x) > 0 for all x in an interval I, the graph is concave up on I.

b) f ''(x) < 0 for all x in an interval I, the graph is concave down on I.

Point of inflection

Point of Inflection: the point where the graph changes from concave up to concave down or vice versa

Approximate each of the following:• the point(s) of

inflection of f(x)

• Interval(s) where f(x) is concave up

• Interval(s) where f(x) is concave down

Extrema: f '(x) = 0Increasing: f '(x) > 0Decreasing: f '(x) < 0

Point of inflection: f ''(x) = 0Concave up: f ''(x) > 0Concave Down: f ''(x) < 0

Helps us get a rough idea of the

graph of the function

Allows us to graph more accurately!

Examples:1. g(x) = 1/3x3 – x2 – 3x + 2

Describe the end behavior of the graph.

What is/are the point(s) of inflection?

Where is the graph concave up?

Examples (cont.): h(x) = 0.25x4 – x3 + 1• Describe the end behavior of the graph.

• Where is the graph increasing?

• What are the extrema?

• What is/are the point(s) of inflection?

• Where is the graph concave up?

Given the graph of f(x) below, what do you know about f’ and f” at each indicated point?

A

B

C

D

E

F

G

H