Warm up Problems1.If , find f (x).

2.If , find f (x).

3.Given the graph of f (x), find all intervals where f (x) is increasing.

12 3f x x

23 2f x x

2

-2

5 10

f '(x)

Interpreting the Derivative

New Notation: If y = f (x), then

derivative of

with respect

to

dyf x y

dxx

x a

dyf a

dx

3 23d

x xdx

Ex. The cost C, in dollars, of building a new Avengers facility that has area A square feet is given by C = f (A). What are the units of f (A)?

Ex. The cost, in dollars, for the seven dwarves to extract T tons of ore from their mine is given by M = f (T). What does f (2000) = 100 mean?

Ex. Suppose P = f (t) is the population of Springfield, in millions, t years since 1990. Explain f (15) = -2.

Derivative of the DerivativeWe can find the derivative of f (x):

f (x) = the second derivative of f

If s(t) = position, then

velocitys t v t

accelerations t a t

Note: If f > 0, then f is increasing.If f < 0, then f is decreasing.

Thm. If f > 0, then f is concave up.

If f < 0, then f is concave down.

Concave up means that the graph lies above its tangent line and below its secant line

Ex. Given the graph of f, determine if each is positive, negative, or zero.

f f f

A

B

C

D

10

5

-5

-10

-2 2

f

A

B

C

D

Ex. Given the graph of f , answer the following:

a) Where is f decreasing?

b) Where is f concave up?

20

-20

-40

2 4 6 8

f'

Warm up Problems1. If y = x3 + 3x, find .

2. If , find g(8).

3. Draw a graph of f so that f > 0 and f < 0 for all x.

4. If the position of a particle is given by x(t) = t2, is the speed increasing or decreasing at t = -3? What about t = 2?

dy

dx

43g x x

Next class we will review, and Monday will be our next chapter test.