Honors AP Calculus / Mr. Hansen
4/15/2009

(8 pts.) Name: _________________________
Battery bonus (Mr. Hansen’s use only): ______

Test on Everything Except §12.8

 

Rules
  • You may not write calculator notation anywhere unless you cross it out. For example, fnInt(X^2,X,1,2) is not allowed; write  instead.
  • Legibility and neatness count. Do not take the time to erase large sections. Instead, mark a single “X” through anything you wish to be ignored.
  • All final answers should be circled or boxed. Simplification is not required unless the context of the problem clearly requires it. Decimal approximations in final answers must be correct to at least 3 places after the decimal point.
  • Point values are shown in parentheses to the left of each question.
  • If you need more room, write “CONT.” and go to the Continuation/Bonus Page.

 

 

1.

(36 pts., 6 pts. per part)

Function f (x) is defined on (–1.5, 1.5) by the following Maclaurin series:

 

 

(a)

Prove that f converges for all x in (–1.5, 1.5).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)

Compute the exact value of f (–1).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c)

It can be shown that  = 1, and therefore the domain of f could be extended slightly. If , we would define f (x) by its Maclaurin series, and for the left endpoint, we would simply say .

However, the Maclaurin series cannot be used to compute . Explain why not.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d)

Write the power series for , showing at least four explicit terms and the general term. Please construct your general term to have xn in it, not x to some other power.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(e)

Obtain an approximation of  by means of a degree 2 polynomial for . (Note: This is easy if you did part (d). If you could not solve part (d), you will need to use a degree 3 polynomial for f as your starting point, not a degree 2 polynomial for f.) Show your work.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(f)

Use your answers to parts (b) and (e) to write an equation of a line that approximates f (x) when x is in a neighborhood of –1. If you could not compute the answer to (b) or (e), then simply write b or e to indicate where you would insert those values.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.

(37 pts. total)

Let .

 

 

(a)
(6 pts.)

Clearly, g(x) is undefined when x = 0. However,  can be computed by L’Hôpital’s Rule. Please do so, showing your steps clearly.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(b)
(6 pts.)

Write g(x) as a power series centered about x = 0. Raise your hand if you would like to purchase an answer or if you would like to have your answer checked, since part (b) is used extensively in the questions below.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c)
(6 pts.)

Compute the limit in part (a) without using L’Hôpital’s Rule. (Use power series instead.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d)
(9 pts.)

Prove that your power series in part (b) satisfies all three requirements of the Alternating Series Test when . (Note: It is possible to show that AST is eventually satisfied for any real x, but that requires more work. Today, you need only show that AST is satisfied for x between –1 and 1, inclusive.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(e)
(10 pts.)

Use the AST error bound to determine the number of nonzero terms needed in your power series for g(x) in order to estimate g(0.8) with an absolute error of less than 0.000001. Justify your answer. Note that for full credit, you must pretend that the true value of g(0.8) is not known in advance (since obviously, you could otherwise answer the question simply by comparing g(0.8) against various polynomials until you had one that came close enough).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3-5.

“Always/Sometimes/Never” (3 pts. each)
Write A if the statement is always true, S if it is sometimes true, or N if it is never true. There is no partial credit.

 

 

___3.

A Taylor series is a Maclaurin series.

 

 

___4.

A real-valued power series that has an interval of convergence that is a proper subset of  has convergence at exactly one of the two endpoints of that interval.

 

 

___5.

If x is a real number, then the series  converges absolutely.

6.

Prove Euler’s Formula: For any complex number z, .

(12 pts.)

 

 

Note: For full credit and full rigor, you should know several facts that are listed below. If these confuse you, please ignore them. A less rigorous proof (i.e., with unjustified rearrangement of terms) will have only a 2-point penalty, and since there are 102 points possible on today’s test, that is really no penalty at all.

 

  • Rearrangement of terms within an infinite series is not permitted in general. However, it is permitted for absolutely convergent series. Thus, if you perform any rearrangement of terms, you should first justify or prove absolute convergence so that the rearrangement can be considered rigorous.
  • The complex absolute value (modulus), |z|, is a generalization of the real-number absolute value with which you are well acquainted. All you need to know for today is this: Just as a factor of 1 inside the absolute value bars does not affect real-number absolute value, a factor of 1 or i inside the complex modulus bars does not affect complex modulus.
  • A complex series converges absolutely iff its series of moduli converges.
  • Exponentials, sines, cosines, hyperbolic sines, and hyperbolic cosines converge absolutely on the complex plane. You may treat this as a given.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Continuation/Bonus Page
Use the space below if you ran out of room on an earlier page.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

BONUS

(1 point) Prove the “it can be shown” assertion in question 1(c).