Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

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Problem Set # 1 Multiple Choice Test 

COMPLETE SOLUTION SET

1. The coefficient of the x5 term in the Maclaurin polynomial for sin (2 x ) is

(A) 0
(B) 0.0083333
(C) 0.016667
(D) 0.26667

Solution The correct answer is (D).
The Maclaurin series for sin(2 x ) is

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Hence, the coefficient of the x5 term is 0.26667.

2. Given f (3) = 6 , f ′(3) = 8 , f ′′(3) = 11 , and all other higher order derivatives of f (x ) are zero at x = 3 , and assuming the function and all its derivatives exist and are continuous between x = 3 and x = 7 , the value of f (7 ) is

(A) 38.000
(B) 79.500
(C) 126.00
(D) 331.50

Solution: The correct answer is (C).
The Taylor series is given by

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Since all the derivatives higher than second are zero,

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

3. Given that y(x ) is the solution toQuestions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev y (0) = 3 the value of y(0.2) from a second order Taylor polynomial around x=0 is

(A) 4.400
(B) 8.800
(C) 24.46
(D) 29.00

Solution: The correct answer is (C).
The second order Taylor polynomial is

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

4. The seriesQuestions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev is a Maclaurin series for the following function

(A) cos(x)
(B) cos(2 x)
(C) sin (x)
(D) sin (2 x)

Solution: The correct answer is (B).

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

5.   The functionQuestions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev dt is called the error function.  It is used in the field of probability and cannot be calculated exactly.  However, one can expand the integrand as a Taylor polynomial and conduct integration.  The approximate value of erf (2.0) using the first three terms of the Taylor series around t = 0 is

(A) -0.75225
(B) 0.99532
(C) 1.5330
(D) 2.8586

Solution: The correct answer is (A).
Rewrite the integral as

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The first three terms of the Taylor series for  Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRevaround t = 0 are

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

The first three terms of the Taylor series are

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Hence

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Note:  Compare with the exact value of erf (2)


6.  Using the remainder of Maclaurin polynomial of nth order for f (x ) defined as

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

the order of the Maclaurin polynomial at least required to get an absolute true error of at most 10 −6 in the calculation of sin (0.1) is (do not use the exact value of sin (0.1) or cos(0.1) to find the answer, but the knowledge that |sin( x)| ≤ 1 and | cos( x) |≤ 1 ).
(A) 3
(B) 5
(C) 7
(D) 9

Solution : The correct answer is (B).

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

So when is 

Questions:Taylor Series - Mathematical Methods of Physics, UGC - NET Physics Physics Notes | EduRev

But since the Maclaurin series for sin (x ) only includes odd terms, n ≥ 5 .

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