Applied Calculus (MAT 121)Dr. Day Thursday Feb 9, 2012

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

Derivatives of Derivatives (3.5)

Assignments

Thursday, February 9, 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).

Thursday, February 9, 2012

MAT 121

Applications of Derivatives

Thursday, February 9, 2012

MAT 121

Derivatives of Composite

Functions (2.2,3.3)

Thursday, February 9, 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)

Thursday, February 9, 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)

Thursday, February 9, 2012

(1) y 5x 3 3x 4 4

(2) f (x) 4 x 1

(3) P(t) 3t 1

2 t

2

MAT 121Thursday, February 9, 2012

MAT 121Thursday, February 9, 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. Thursday, February 9, 2012

MAT 121

WebAssign3.4 due Friday night

Assignments

Thursday, February 9, 2012