Q1. The coefficient of the x⁵ term in the Maclaurin polynomial for sin(2x) is

  0

  0.0083333

  0.016667

  0.26667

Q2. Given f(3)=6, f'(3)=8, and f''(3)=11, and that 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

  38.000

  79.500

  126.00
  331.50

Q3. Given that y(x) is the solution to dy/dx=y³+2, y(0)=3, the value of y(0.2) from a second order Taylor polynomial is

  4.400
  8.800

  24.46

  29.00

Q4. The series

is a Maclaurin series for the following function

  cos(x)

  cos(2x)

  sin(x)

  sin(2x)

Q5. The function

is called the error function.  It is used in the field of probability and cannot be calculated exactly for finite values of x.  However, one can expand the integrand as a Taylor polynomial and conduct integration.  The approximate value of erf(2.0) using first three terms of the Taylor series around t=0 is

  -0.75225
  0.99532

  1.5330

  2.8586

Q6. Using the remainder of Maclaurin polynomial of n^th order for f(x) defined as

  3
  5
  7

  9

Complete solution