Honors AP Calculus / Mr. Hansen

Name: _________________________

Part I. Free Response (26 points) with calculator

1.

Perhaps missing from our discussion of Taylor series and error bounds have been practical applications. One of the most important functions of interest to statisticians and engineers is the function that your TI-83 calls "normalcdf," which gives the area under the normal probability curve from A to B (where A and B are measured in standard deviations) and which is defined as follows:

normalcdf(A,B) = 1/Ö(2p)*fnInt(e^(-X²/2),X,A,B)

(a)

Compute the area under the normal curve from 2 to 3 standard deviations using both methods on the TI-83. Area = ___________________ . Which one runs faster? ____________ (No work required.)

(b)

Use a Taylor series to write the integral. Then use a suitable improper integral to find an upper bound for the magnitude of the error after 12 terms. [Use any method. The improper integral method is invalid here because the series alternates.]

2.(a)

Write the Taylor series for sin(x/2), expanded about 0.

(b)

Use 3 terms of your series in part (a) to estimate sin(3/2). Show your work.

(c)

Use the Lagrange error term for your series to find the number of terms needed to compute sin(3/2) correct to an error of no more than 0.000001. Show your work.

3.

EXTRA CREDIT (2 pts.): Explain briefly (< 1 sentence) why one method in #1(a) is faster than the other.

Part II (24 points): Multiple Choice WITHOUT CALCULATOR
Scoring: 6 points for a correct answer, –2 points for an incorrect answer, 0 points for an omission.

4.

See #20 on p.377 of the Barron's review book.

5-7.

See #24-26 on p.378 of Barron's.