Question

1.)Find T5(x), the degree 5 Taylor polynomial of the function f(x)=cos⁡(x) at a=0.

T5(x)=  

Find all values of x for which this approximation is within 0.003452 of the right answer. Assume for simplicity that we limit ourselves to |x|≤1.

|x|≤

2.) (1 point) Use substitution to find the Taylor series of (e^(−5x)) at the point a=0. Your answers should not include the variable x. Finally, determine the general term a_n in (e^(−5x))=∑_n=0^∞ (a_n(x^n))

e^(−5x)=  +  x +  x^2 +  x^3 + ... = ∑^∞_n=0 ___?___(x^n)

3.) Use power series operations to find the Taylor series of cos²(x) at the point a=0. Your answers should not include the variable x. Finally, determine the general term a_n in cos²(x)=∑_n=0^∞ (a_n(x^(2n))). Hint: cos²(x)=(1+cos( 2x))/2.

cos²(x)=  +  x^2 +  x^4 +  x^6 + ... = (1/2)+∑^∞_n=0 ___?___(x^(2n))

Answer 1