Applied Calculus (MAT 121)Dr. Day Friday Feb 10, 2012

Applying Derivatives (Rate of Change) to Business and Economics (3.4)

Derivatives of Derivatives (3.5)

Assignments

Friday, February 10, 2012

MAT 121

MAT 121

Review: Limits and Derivatives

1. What do we mean when we say we are “determining the limit of a function at a point?”

a) Show a graphical and a numerical example to illustrate this.

b) Write a sentence or two to tie it all together.

2. What is our purpose in determining the derivative of a function? Why bother?

3. State the formal definition for the derivative of a function f(x).

Friday, February 10, 2012

MAT 121

Applications of Derivatives

Friday, February 10, 2012

MAT 121

Derivatives of Composite

Functions (2.2,3.3)

Friday, February 10, 2012

The derivative of a composite function is :

The derivative of the outside functionevaluated at the inside function

multiplied bythe derivative of the inside function.

If h(x) = f (g(x)),

thenh (x) = f (g(x)) g (x)

MAT 121

Derivatives of Composite

Functions (2.2,3.3)

Friday, February 10, 2012

The derivativeof the outside function

evaluated atthe inside function

multiplied bythe derivative of

the inside function.

For

h(x) = 7x 2 + 3x 3

inside function : u(x) 7x 2 + 3x its derivative : u (x) 14x + 3

outside function : f (u) u3

its derivative : f (u) 3u2

So with h(x) = 7x 2 + 3x 3 h(u) = u 3

h (u) = 3 u 2 u and

h (x) = 3 7x 2 + 3x 214 x + 3

MAT 121

Derivatives of Composite

Functions (3.3)

Friday, February 10, 2012

(1) y 5x 3 3x 4 4

(2) f (x) 4 x 1

(3) P(t) 3t 1

2 t

2

MAT 121Friday, February 10, 2012

MAT 121Friday, February 10, 2012

THE CHAIN RULEWORDS BY: JOHN A. CARTER TUNE: "CLEMENTINE"Here's a function in a function And your job here is to findThe derivative of the whole thingWith respect to x inside.

Call the outside f of uAnd call the inside u of x.Differentiate to find df/duAnd multiply by du/dx.

Use the chain rule.Use the chain rule.Use the chain rule whene'er you find The derivative of a function compositionally defined.

MAT 121

Using Rates of Change in Business and

Economics 3.4) For the Acrosonic model F loudspeaker system, the

relationship between its unit price p, in dollars, and the quantity demanded x, is modeled by the function p = −0.02x +400, for 0 ≤ x ≤ 20,000. Create the revenue function R for the model F system. Calculate the marginal revenue function R’. Compute R’(2000) and interpret that result within the context

of the problem.

Suppose we also know that the cost function, to produce x units of the model F system, is C(x) = 100x + 200,000. Create the profit function P(x). Determine the marginal profit function P’. Calculate P’(2000) and interpret that result. Look at a graph of the profit function P and describe what is

shows within the context of this problem. Friday, February 10, 2012

MAT 121

Derivatives of Derivatives (3.5)

Friday, February 10, 2012

(1) y 5x 3 3x 4 4

(2) f (x) 4 x 1

(3) P(t) 3t 1

2 t

2

MAT 121

Position, Velocity, Acceleration

The distance s (in feet) covered by a car t seconds after starting is given by the following function:

s = -t3 + 13t2 + 18t     (0 ≤ t ≤ 6) Determine a general expression for the car's acceleration at any time t (0 ≤ t ≤ 6).

s ''(t) = At what time t does the car begin to decelerate? (Round your answer to one decimal place.)

Friday, February 10, 2012

MAT 121

WebAssign3.4 due Monday night3.5 due Tuesday night

WA Quiz #3: due Sunday night!

Test #2: Wednesday, Feb 15

Assignments

Friday, February 10, 2012