Question

Taylor Polynomial HW

1) Evaluate cos ( 2 π / 3 ) on your calculator and using the first 4 terms of the TP for cos x.

2) Integrate cos ( x^3 ), from 0 to π / 6, using the first 3 terms of the TP for cos x.

3) Evaluate e^x at x = .4 on your calculator and using the first 5 terms of the TP for e^ x.

4) Integrate e^x3, from 0 to .3, using the first 3 terms of the TP for ex.

5) If I integrate 1/(1-x) I will get - ln (1 - x). Integrate the given TP for 1/(1-x). What is the TP for - ln ( 1 - x )?

6) What is the value of, - ln ( 1 - x ) if x = .3? I got .3567. Use the first 4 terms of the TP you created in question 5 and see if you obtain the same result. I got .3560.
  

Here are other Taylor Polynomials for other trig functions:

tan ( x ) = x + (1/3) x3 + (2/15) x5 + (17/315) x7 + (62/2835) x9 + ....


sec ( x ) = 1 + (1/2) x2 + (5/24) x4 + (61/720) x6 + ...

7) Find the integral of tan x from 0 to π / 6. Use the first 3 terms of the TP.

8) Find a TP for sec 2 x, recall sec 2 x is the derivative of tan x.

9) On your calculator, what is the cos (π / 3)? You should get 1/2. Obviously, the sec (π / 3) is 2. Use the first 4 terms of the TP for sec x and see if the answers agree.

Answer 1

In Question 1. With the help of the Taylor series expansion of the given function f(x)= cos x. we have considered the Taylor polynomial by taking the first four terms of the expansion byand then putting x= , we will find the value cos( ) which will be same as calculated with the help of the calculator.