Honors AP Calculus / Mr. Hansen

Name: ____________________

4/14/2010

 

 

Test through §12-6 (No Calculator Permitted)

 

The time limit for the first question, which is an AP-style free-response question, is 15 minutes. You may not work on any other problems during this time. After 15 minutes, you will be given a set of 7 AP-style multiple-choice questions, for which the time limit is also 15 minutes. In the unlikely event that you finish those questions early, you may return to the free-response question if you wish.

After 30 minutes, all AP-style questions will be collected and replaced with non-AP questions. You should spend the remaining 20 minutes of the period working on the non-AP questions.

Scoring: 30 points (curved) for Part I, 30 points (curved) for Part II, 40 points (not curved) for Part III. There is a penalty for wrong guesses only in the middle section (AP multiple-choice).

 

 

 

Part I: Free Response in AP Style (30% of test score).

 

 

1.

(a) Write the first four terms and the general term for the Maclaurin series expansion of .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) For the series in part (a), find the common ratio.

 

 

 

 

 

 

 

 

 

 

(c) For the series in part (a), find the radius of convergence.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d) The series in part (a) can be differentiated and modified to find a Maclaurin series for function g(x),
         where . Write the first four nonzero terms of the Maclaurin series for g(x).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(e) Use the series in part (a) to write a Maclaurin series for  Give at least four terms.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(f) Show that the interval of convergence for h(x) in part (e) is .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Part II: Multiple Choice in AP Style (30% of test score).

 

 

2.

[See AP Calculus Course Description at

http://apcentral.collegeboard.com/apc/public/repository/ap-calculus-course-description.pdf

for this question.]

 

 

3.

[See AP Calculus Course Description at

http://apcentral.collegeboard.com/apc/public/repository/ap-calculus-course-description.pdf

for this question.]

 

 

4.

If the series converges absolutely, then . . .

(A)    the series may or may not converge
(B)    the series surely converges, but the error after n terms may or may not be bounded by |tn + 1|
(C)    the series surely converges, and the error after n terms is surely bounded by |tn + 1|
(D)    the series  may diverge
(E)    the series  surely diverges

 

5.

Let n be a fixed positive integer. “Expression 1” refers to , and “Expression 2” refers to . Which expression is greater?

(A)    Expression 1, since n can be arbitrarily large.
(B)    Expression 2, since Expression 1 is finite but Expression 2 is infinite.
(C)    Neither, since Expression 1 is finite and Expression 2 converges to a finite value.
(D)    Answering the question requires knowing the value of n.
(E)    The answer to the question cannot be determined, even if the value of n is known.

 

 

 

 

6.

The series  converges to

(A)    0
(B)    a negative value between −1 and 0
(C)    a positive value between 0 and 1
(D)    a positive value greater than or equal to 1
(E)    DNE (not convergent)

 

 

 

 

7.

Where does the power series  converge?

(A)    (−1, 1)
(B)    [−1, 1)
(C)   
(D)   
(E)   

 

 

 

 

8.

The sine of 1 radian, if estimated with a third-degree Taylor polynomial centered on x = 0, will be accurate to within what error tolerance? Choose the smallest value that serves as an upper bound for the absolute value of the error. (For example, if the absolute value of the error is bounded by 0.008, you would choose C, since 0.008 < 0.01, and you would avoid D and E, since they are too small. You would also avoid A and B, since although they are upper bounds, they are too large. C is the smallest value that serves as an upper bound for the size of the error.)

(A)    0.5
(B)    0.1
(C)    0.01
(D)    0.001
(E)    0.0001

 

 

Part III: Free Response, Non-AP (40% of test score).
Point values are shown in parentheses in the left margin.

 

 

9.

(23 pts.)

Earlier in the year, we defined , .

 

 

(a)

Use one quick line of algebra to show that these definitions imply the identity exp(x) = sinh x + cosh x.

 

 

 

 

 

 

 

 

 

 

 

 

(b)

“Formal substitution” means _____________________________________________________

 

 

 

____________________________________________________________________________ .

 

 

(c)

Using the sinh, cosh, and exp power series we learned, use formal substitution to show that the identity in part (a) holds for power series as well. Warning: Infinite series cannot be combined safely in this way unless the series are absolutely convergent. Luckily, that is the case here!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d)

How do you know that the identity you proved (or tried to prove) in part (c) is valid for all real x?

 

 

 

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(e)

It turns out that the identity exp(x) = sinh x + cosh x is valid not only for all real x, but for all complex values of x as well. Why is this to be expected? (Please use the phrase “radius of convergence” in your answer.)

 

 

 

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10.

(9)

An extremely common problem in statistics is to calculate a so-called “normal probability,” which is the area under the curve  between two given values of x.

One obvious way to calculate such an area is to use the TI-83/84 fnInt feature (MATH 9), which employs adaptive quadrature. However, even though adaptive quadrature is faster and more accurate than simpleminded trapezoid rule or midpoint rule techniques, there is a much better way for statisticians to find the area under the f (x) curve.

 

 

(a)

Without using any mathematical symbols, equations, or numbers, other than perhaps the letter f, briefly explain what it is that the TI calculators do in order to compute normal probabilities. Remember, whatever it is, it runs much faster than adaptive quadrature (MATH 9).

_____________________________________________________________________________

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(b)

The calculator function that computes the area under the f (x) curve is called normalcdf and is found under the 2nd VARS menu. Unfortunately, it was not possible for the designers of the TI-83/84 to program the normalcdf command so that it would use an exact antiderivative of f and simply apply FTC1. If they had programmed a closed-form function G that was an exact antiderivative of f, then the normal probability from x = a to x = b would exactly equal G(b) − G(a). That is what FTC1 tells us, and what could be easier than that? Yet this was not done. Why not?

_____________________________________________________________________________

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11.

(8)

(a)

Use your memorized Taylor series for ln x to write ln 8 as a series of constants. Simplify your answer, and write out at least six terms or, if you prefer, the sigma notation. Hint: ln 8 = ln 23. Do not plug 8 into the Taylor series.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Show that the series requested in part (a) is not absolutely convergent.