Honors AP Calculus / Mr. Hansen
4/13/2011

Name: ___________KEY___________
Bonus (for Mr. Hansen’s use only): ________

Test on Chapter 12 (Calculator for First 15 Minutes)

 

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.
  • Adequate justification is required for free-response questions.
  • All final answers in free-response portions should be circled or boxed.
  • Decimal approximations must be correct to at least 3 places after the decimal point.

 

 

1.

In the subject of statistics (STAtistics), the probability density function for the standard normal curve is given by the function  Our goal in parts (b) through (d) will be to estimate f (0.5).

 

 

(a)

Write the Maclaurin series (first 3 nonzero terms and general term) for f (x).

 

 

 

Let u =  and let n consistently start from 0. Then

 

 

 

Scoring rubric:
<1>      first term completely correct
<1>      second and third terms both correct (but allow point if  was consistently omitted)
<1>      general term completely correct, or if n is consistently replaced with 2n, n + 1, etc.

 

 

(b)

Find an absolute-value error bound for f (0.5) after n nonzero terms. Your answer should be in terms of n, the number of nonzero terms added up. Describe what method you are using (Lagrange, AST, or other), and provide adequate justification. If you need more room, write “OVER.”

 

 

 

AST method (recommended):

 

The series in (a) is strictly alternating , decreasing in abs. value  (since numerator is decreasing while denom. is increasing), and has limit 0  (since numerator limit is 0 and denominator increases). Therefore, |error| is bounded by abs. value of first omitted term. After n nonzero terms (numbered 0 through n − 1, and having degree 0 through 2n − 2), we have  

 

 

 

Scoring rubric:
<1>      at least one AST check completely correct
<1>      other two AST checks completely correct

<2>      expression (permit ECF from part (a), as long as definition of n is respected; otherwise, 1 off per error, e.g., failing to plug in 0.5 for x)

 

(c)

Using the error bound in part (b), determine the number of nonzero terms of part (a) needed to ensure accuracy of f (0.5) within 0.001.

 

 

 

 

 



This inequality is first true for n = 4. Answer: .

 

 

 

Scoring rubric:
<1>      inequality
<1>      answer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(d)

Use your calculator’s value for f (0.5), namely 0.3520653267643, to determine how many terms of part (a) are actually needed in order to obtain accuracy of f (0.5) within 0.001. Everyone can do this, even those who could not answer parts (b) and (c). Show a little bit of work (a table, perhaps, or a few partial sums if you prefer).

 

 

 


n

nonzero term #


exponent

term
value

partial
sum


|error|

|error|
< 0.001?

 

0

1

0

0.398942

0.39894228

0.046876954

no

 

1

2

2

−0.04987

0.349074495

0.002990831

no

 

2

3

4

0.003117

0.352191232

0.000125905

no

 

3

4

6

−0.00013

0.352061368

< 4·10−6

yes

 

4

5

8

0.00000406

0.352065426

< 1·10−7

yes

 

 

 

Answer: .

 

 

 

Scoring rubric:
<1>      any work showing an effort to compare partial sum with 0.3520653 . . .
<1>      answer

 

 

 

 

 

 

 

 

 

 

2.

Solution and scoring rubric are #6 at this link.

 

 

3.

Solution and scoring rubric are #3 at this link.