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**Problem Set # 1 Multiple Choice Test **

**COMPLETE SOLUTION SET**

**1. The coefficient of the x ^{5} 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

Hence, the coefficient of the x^{5} term is 0.26667.

**2. Given f (3) = 6 , f â€²(3) = 8 , f â€²â€²(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

Since all the derivatives higher than second are zero,

**3. Given that y(x ) is the solution to 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

**4. The series**** 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).

**5. The function** **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

The first three terms of the Taylor series for around t = 0 are

The first three terms of the Taylor series are

Hence

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

**6. Using the remainder of Maclaurin polynomial of n ^{th} order for f (x ) defined as**

the order of the Maclaurin polynomial at least required to get an absolute true error of at most 10^{ âˆ’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)| â‰¤ 1 and | cos( x) |â‰¤ 1 ).

(A) 3

(B) 5

(C) 7

(D) 9

**Solution : **The correct answer is (B).

So when is

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