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Honors AP Calculus / Mr. Hansen 1/12/2003 |
Name: _________________________ |
Solution Key to Problem 13 on Practice Midterm
[Expressions in brackets are optional, not required for credit.]
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(a) |
F(x) = 1.48 2/(x^2 + 1) |
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(b) |
[The expression W(x) = ∫1x F(t) dt is an antiderivative of F(x) whenever x Î [1, 7.3]. Reason: FTC2.] [Since W(1) = 388.214 (given), the constant of integration for ∫F(t) dt must be 388.214. Reason: The definite integral from 1 to 1 is 0, so all the value of W(1) must come from the constant.] Final answer: Whenever 1 £ x £ 7.3, W(x) = 388.214 + ∫1x F(t) dt. [Proof that this function W(x) is correct: In the equation W(x) = 388.214 + ∫1x F(t) dt, the LHS and RHS have the same derivative, which by FTC2 is simply F(x). Also, the LHS and RHS have the same value when x = 1, namely 388.214. Therefore, by the Uniqueness Theorem for Derivatives (p.265), the LHS and RHS are equal for all x in the domain of interest.] [Editorial comment: This technique of forming an arbitrary antiderivative is very powerful and gives engineers, mathematicians, and AP Calculus students a general method of constructing a function that describes some phenomenon, given only an initial condition and a function for the derivative. However, what would you do if you weren't given a function for the derivative, but only the derivative at a number of discrete points? (Think for a minute before reading on.) . . . . That's right, you'd use Euler's Method or a more sophisticated o.d.e. solver to estimate the function values of interest.] |
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(c) |
W ´(x) = F(x) by FTC2. |
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(d) |
If you believe FTC2, the proof is immediate. Here’s another possible explanation: W is an accumulator function, and work accumulates at a varying rate. What determines the rate of accumulation? Certainly not the change in x, since after all, x is just changing in a boring incremental fashion (little bits of x from left to right, being multiplied by force and added up in a Riemann-sum manner). Yes, the accumulation of work depends on how the force changes. If force were constant (think Form II science), then work through any intermediate point is simply the product of force times distance, i.e., work would accumulate linearly. But since force is not constant, work will not accumulate in a linear fashion. Work accumulates at a rate (W ´) that is exactly specified by force function F. Yet another possible explanation would be to state the bracketed proof of correctness for W given in part (b). |
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(e) |
We say L = limx® c f(x) iff "e >0, $d >0 ' 0 < |x – c| < d Þ |L – f(x)| < e. |