thursday, february 9, 2012mat 121. thursday, february 9, 2012mat 121
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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