Taylor and Maclaurin Series

1. Perform operations on a known Maclaurin series (i.e., use the provided table) to find the power series representation for each function \(f\). Include the interval of convergence.

(a) \(\displaystyle{f(x)= \frac{3}{2-x}}\)  solution

(b) \(\displaystyle{f(x)= \frac{x}{1-2x}}\)   solution

(c) \(\displaystyle{ f(x)=\frac{5x}{x^2+4}}\)  solution

(d) \(\displaystyle f(x)= x^2 e^{-2x}\)  solution

(e) \(\displaystyle f(x)= 3\sin(2x)\)  solution

(f) \(\displaystyle f(x)= 4\cos(x^2)\)  solution

(g) (i) \(\displaystyle f(x)= xe^{-3x}\)   (ii) \(\displaystyle f(x)= 4x \sin(2x)\)   (iii) \(\displaystyle f(x)= 2x \cos(x^2) \)  solution

2. Use the Taylor series formula (the definition of a Maclaurin series) to find the Maclaurin series of the following functions.

(a) \( f(x)=e^{3x}\)  solution

(b) \( f(x)=e^{-5x}\)  solution

(c) \(f(x)=xe^x \)   solution

3. Find the Taylor series for \(f(x)=\cos x \) centered at \( a= \pi\).  solution

Taylor Polynomials

1. Find the degree 3 Taylor polynomial centered at \(a=1\) for  \(f(x)=x^3-4x-7\). Then multiply out and simplify your answer. (Did the result surprise you?)   solution

2. Find  the degree 2 Taylor polynomial for \( f(x) = x^4-x+1\) centered at \(a=-1\).  solution

3. Consider the function \(f(x)=\sqrt[3]{x}\). (a) Find the degree 2 Taylor polynomial, \(T_2(x)\), for \(f\) centered at \(a=8\). (b) Use the polynomial you found in part (a) to approximate the value of \(\sqrt[3] 7\).  solution

4. Find the degree 0, degree 1, degree 2, and degree 3 Taylor polynomials centered at \(a=4\) for the function given by \(f(x)=\sqrt{x}\). Use each of these to get successively better approximations for \(\sqrt{5}\).  solution

5. Find the degree 3 Taylor polynomial, \(T_3(x)\), for \(f(x)=e^{-x}\, \sin(x)\) centered at \(a=0\). solution 1  solution 2

6. Find the degree 3 Taylor polynomial, \(T_3(x)\), for \(f(x)=xe^{-4x}\) centered at \(a=0\).   solution