Honors AP Calculus / Mr. Hansen |
Name: ____________________ |
4/15/2010 |
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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. |
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Part I: Free Response
in AP Style (30% of test score). |
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1. |
(a) Use the fact that |
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(b) State the common ratio, radius of convergence, and interval of convergence for the series in part (a). |
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(c) Integrate your series in (a) term-by-term to develop a series expansion for ln x that is almost entirely in powers of (1 − x). |
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(d) For the series in part (c), find the interval of convergence, paying special attention to the behavior at the endpoints of the interval. |
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Part II: Multiple Choice in AP Style (30% of test score). |
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2. |
[See AP Calculus Course Description at |
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3. |
[See AP Calculus Course Description at |
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4. |
If the series |
5. |
Let n be a fixed
positive integer. “Expression 1” refers to |
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6. |
The series |
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7. |
Where does the power series |
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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.) |
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Part III: Free Response,
Non-AP (40% of test score). |
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9.(a) (6) |
State the radii of convergence (not intervals of
convergence) for the standard Maclaurin series for
exp(x), sin x, and cos x. |
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(b) |
Explain why it is no surprise that these series are
convergent not only on |
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____________________________________________________________________________ |
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____________________________________________________________________________ |
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(c) |
“Formal substitution” means _____________________________________________________ |
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____________________________________________________________________________ . |
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(d) |
Use formal substitution to write cos(2i) as a series of constants. Simplify your answer for full credit. The symbol i, as you know, is the complex unit, the principal root of the quadratic equation x2 + 1 = 0. |
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(e) |
Prove, formally, that for any value |
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(f) |
Regardless of whether you succeeded or failed in part (e),
assume that Euler’s formula is valid. Now use it to prove Euler’s identity, a famous equation
that unifies the five most fundamental constants in mathematics: |
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10. (8) |
Suppose that you are interviewing for a job as a
programmer with TI or one of the other large calculator manufacturers. The company
is planning to develop a scientific calculator that will return the values of
the hyperbolic functions (sinh, cosh,
etc.) with unheard-of accuracy for a handheld calculator: error not to exceed
10−20 in absolute value. This is about a million times more
accurate than our TI-83/84 units, which are, at best, accurate on hyperbolic
functions only to about 10−14. |
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