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MATH 270 Week 4 iLab Latest

  • MATH 270 Week 4 iLab Latest
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MATH 270 Week 4 iLab Latest

Directions: Determine the Taylor Series for the following function:   for n = 4 where a = 1
 
1. Determine the first four derivatives of 
 
2. Evaluate  and the derivatives above at x = a = 1
 
3. The formula for the Taylor Series is  . Plug the values from question 2 into the formula. Then, simplify the fractions and write the Taylor Series approximation for  .
 
4. Enter  and the series approximation from step 3 into a graphing calculator. View the graphs and the table of values. Does the Taylor series approximation appear to converge on  for all values of x? If so, what does this mean about when I can use the Taylor series to approximate   ? If not, for what values of x can I use the series to approximate f(x)?
 
5. Under what circumstances would I be better off using a Taylor series rather than a Maclaurin series to approximate a function?
 
6. Determine a Taylor Series approximation for  where n = 4 and a = 3
 
Part II: Fourier series
 
Write the Fourier series for the following function: f(x) = 
 
1. Find   using the formula 
 
2. Find   using the formula  .
 
Note: We can use the integration formula  . To do this we will set u = kx, which makes du = kdx. To complete the substitution we need to have a k attached to both the f(x) and dx portion of the integral. Thus, we are multiplying in two k’s. To balance this we must also divide by 2 k’s outside the integration. Thus, the constant outside the integration will look like  .
 
3. Find  . using the formula  .
 
Note: We can use the integration formula  . To do this we will set u = kx, which makes du = kdx. To complete the substitution we need to have a k attached to both the f(x) and dx portion of the integral. Thus, we are multiplying in two k’s. To balance this we must also divide by 2 k’s outside the integration. Thus, the constant outside the integration will look like  .
 
4. The Fourier series definition is  . Replace   with the answers you got in questions 1, 2, and 3 to write the Fourier series. Note: If you replaced k with 2k or 2k + 1 when creating   then you must do the same with the k in the cos(kx) and/or sin(kx) in the Fourier series definition above.
 
5. Under what circumstances can you assume   = 0? What about when  ? How can you determine this from the graph of the given function? How can you determine this algebraically?

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