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AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

Thursday, December 15, 2005

Antiderivative

In today’s class, we went to @_@group work.
Will’s pet Bob~^.^~
Will the wallaby has been looking for the perfect pet for a very long time and he has finally found it – Bob the boa constrictor. Now Willy has to make a close, rectangular cage for Bob but it has to be 4000 cubic feet in volume and the length has to be 5times that of the height of the cage. The material to make the cage costs $0.25 per square foot.
1) What are the dimensions of the least expensive cage?
2) How much does it cost?
Given V=4000
L=5h
V =L * W * h
4000 = 5h w h
4000 = 5h^2 w
W= 800/(h^2)

Closed rectangular cage
S. A = 2 L h + 2 L h + 2 h w
S.A = 2 (5h) w + 2(5h) h + 2 h w
S.A = 10 h w + 10 h^2 + 2 h w
S.A = 12 h w +10 h^2

Cost =1/4 surface area
Cost = 1/4 (12 h w + 10 h^2)
C (h) = 3h (800/h^2) + 5/2 h^2
C (h) = 24000 h^-1 + 5/2 h^2
C’ (h) = -24000 h^-2 + 5h
= -5 h^ -2 (480-h^3)
- - +
--------------1--------------------1---------------------
2 roots: 0 3root480

By the first derivative test, cost is Mimi zed when h = 3 root 480 feet

L= 5 h= 39.15 feet
W = 800/h^2 = 13.04 feet
Cost = 3h (800/h^2) + 5/2 h^2 = $ 331.05


Antiderivatives
Given the function f, determine a function F whose derivative is f. That is, F’ = f. If F’= f, then the function F is called an antiderivative of f.

A function f has an antiderivative it will have a whole family of them and each member of the family can be obtained from one of them by the addition of an appropriate constant.

If f’(X) = 2x, find different possibilities for f, the parent function.
1) f (x) = x^2
2) f (x) = x^2 + e
3) f (x) = x^2 + 10^100
4) f(x) = x^2 + 1 …

Next scrible Prince. +_+



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1 Comments:

  • At 10:07 PM CST, Blogger Mr. Kuropatwa said…

    Beautifully done Xun! Lots of detail but also briefly and clearly explained. You've set a new standard for our class. Bravo!!

     

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