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
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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
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MAT 121
Applications of Derivatives
Thursday, February 9, 2012
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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)
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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
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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
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MAT 121Thursday, February 9, 2012
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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.
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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
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MAT 121
WebAssign3.4 due Friday night
Assignments
Thursday, February 9, 2012