## Problem Set: Taylor and Maclaurin Series

In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point.

1. $f\left(x\right)=1+x+{x}^{2}$ at $a=1$

2. $f\left(x\right)=1+x+{x}^{2}$ at $a=-1$

3. $f\left(x\right)=\cos\left(2x\right)$ at $a=\pi$

4. $f\left(x\right)=\sin\left(2x\right)$ at $a=\frac{\pi }{2}$

5. $f\left(x\right)=\sqrt{x}$ at $a=4$

6. $f\left(x\right)=\text{ln}x$ at $a=1$

7. $f\left(x\right)=\frac{1}{x}$ at $a=1$

8. $f\left(x\right)={e}^{x}$ at $a=1$

In the following exercises, verify that the given choice of n in the remainder estimate $|{R}_{n}|\le \frac{M}{\left(n+1\right)\text{!}}{\left(x-a\right)}^{n+1}$, where M is the maximum value of $|{f}^{\left(n+1\right)}\left(z\right)|$ on the interval between a and the indicated point, yields $|{R}_{n}|\le \frac{1}{1000}$. Find the value of the Taylor polynomial pn of $f$ at the indicated point.

9. [T] $\sqrt{10};a=9,n=3$

10. [T] ${\left(28\right)}^{\frac{1}{3}};a=27,n=1$

11. [T] $\sin\left(6\right);a=2\pi ,n=5$

12. [T] e2; $a=0,n=9$

13. [T] $\cos\left(\frac{\pi }{5}\right);a=0,n=4$

14. [T] $\text{ln}\left(2\right);a=1,n=1000$

15. Integrate the approximation $\sin{t}\approx t-\frac{{t}^{3}}{6}+\frac{{t}^{5}}{120}-\frac{{t}^{7}}{5040}$ evaluated at πt to approximate ${\displaystyle\int }_{0}^{1}\frac{\sin\pi t}{\pi t}dt$.

16. Integrate the approximation ${e}^{x}\approx 1+x+\frac{{x}^{2}}{2}+\cdots+\frac{{x}^{6}}{720}$ evaluated at −x2 to approximate ${\displaystyle\int }_{0}^{1}{e}^{\text{-}{x}^{2}}dx$.

In the following exercises, find the smallest value of n such that the remainder estimate $|{R}_{n}|\le \frac{M}{\left(n+1\right)\text{!}}{\left(x-a\right)}^{n+1}$, where M is the maximum value of $|{f}^{\left(n+1\right)}\left(z\right)|$ on the interval between a and the indicated point, yields $|{R}_{n}|\le \frac{1}{1000}$ on the indicated interval.

17. $f\left(x\right)=\sin{x}$ on $\left[\text{-}\pi ,\pi \right],a=0$

18. $f\left(x\right)=\cos{x}$ on $\left[-\frac{\pi }{2},\frac{\pi }{2}\right],a=0$

19. $f\left(x\right)={e}^{-2x}$ on $\left[-1,1\right],a=0$

20. $f\left(x\right)={e}^{\text{-}x}$ on $\left[-3,3\right],a=0$

In the following exercises, the maximum of the right-hand side of the remainder estimate $|{R}_{1}|\le \frac{\text{max}|f\text{''}\left(z\right)|}{2}{R}^{2}$ on $\left[a-R,a+R\right]$ occurs at a or $a\pm R$. Estimate the maximum value of R such that $\frac{\text{max}|f\text{''}\left(z\right)|}{2}{R}^{2}\le 0.1$ on $\left[a-R,a+R\right]$ by plotting this maximum as a function of R.

21. [T] ex approximated by $1+x,a=0$

22. [T] $\sin{x}$ approximated by x, $a=0$

23. [T] $\text{ln}x$ approximated by $x - 1,a=1$

24. [T] $\cos{x}$ approximated by $1,a=0$

In the following exercises, find the Taylor series of the given function centered at the indicated point.

25. ${x}^{4}$ at $a=-1$

26. $1+x+{x}^{2}+{x}^{3}$ at $a=-1$

27. $\sin{x}$ at $a=\pi$

28. $\cos{x}$ at $a=2\pi$

29. $\sin{x}$ at $x=\frac{\pi }{2}$

30. $\cos{x}$ at $x=\frac{\pi }{2}$

31. ${e}^{x}$ at $a=-1$

32. ${e}^{x}$ at $a=1$

33. $\frac{1}{{\left(x - 1\right)}^{2}}$ at $a=0$ (Hint: Differentiate $\frac{1}{1-x}.$)

34. $\frac{1}{{\left(x - 1\right)}^{3}}$ at $a=0$

35. $F\left(x\right)={\displaystyle\int }_{0}^{x}\cos\left(\sqrt{t}\right)dt;f\left(t\right)=\displaystyle\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{n}}{\left(2n\right)\text{!}}$ at $a=0$ (Note: $f$ is the Taylor series of $\cos\left(\sqrt{t}\right).$)

In the following exercises, compute the Taylor series of each function around $x=1$.

36. $f\left(x\right)=2-x$

37. $f\left(x\right)={x}^{3}$

38. $f\left(x\right)={\left(x - 2\right)}^{2}$

39. $f\left(x\right)=\text{ln}x$

40. $f\left(x\right)=\frac{1}{x}$

41. $f\left(x\right)=\frac{1}{2x-{x}^{2}}$

42. $f\left(x\right)=\frac{x}{4x - 2{x}^{2}-1}$

43. $f\left(x\right)={e}^{\text{-}x}$

44. $f\left(x\right)={e}^{2x}$

[T] In the following exercises, identify the value of x such that the given series $\displaystyle\sum _{n=0}^{\infty }{a}_{n}$ is the value of the Maclaurin series of $f\left(x\right)$ at $x$. Approximate the value of $f\left(x\right)$ using ${S}_{10}=\displaystyle\sum _{n=0}^{10}{a}_{n}$.

45. $\displaystyle\sum _{n=0}^{\infty }\frac{1}{n\text{!}}$

46. $\displaystyle\sum _{n=0}^{\infty }\frac{{2}^{n}}{n\text{!}}$

47. $\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}{\left(2\pi \right)}^{2n}}{\left(2n\right)\text{!}}$

48. $\displaystyle\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^{n}{\left(2\pi \right)}^{2n+1}}{\left(2n+1\right)\text{!}}$

The following exercises make use of the functions ${S}_{5}\left(x\right)=x-\frac{{x}^{3}}{6}+\frac{{x}^{5}}{120}$ and ${C}_{4}\left(x\right)=1-\frac{{x}^{2}}{2}+\frac{{x}^{4}}{24}$ on $\left[\text{-}\pi ,\pi \right]$.

49. [T] Plot ${\sin}^{2}x-{\left({S}_{5}\left(x\right)\right)}^{2}$ on $\left[\text{-}\pi ,\pi \right]$. Compare the maximum difference with the square of the Taylor remainder estimate for $\sin{x}$.

50. [T] Plot ${\cos}^{2}x-{\left({C}_{4}\left(x\right)\right)}^{2}$ on $\left[\text{-}\pi ,\pi \right]$. Compare the maximum difference with the square of the Taylor remainder estimate for $\cos{x}$.

51. [T] Plot $|2{S}_{5}\left(x\right){C}_{4}\left(x\right)-\sin\left(2x\right)|$ on $\left[\text{-}\pi ,\pi \right]$.

52. [T] Compare $\frac{{S}_{5}\left(x\right)}{{C}_{4}\left(x\right)}$ on $\left[-1,1\right]$ to $\tan{x}$. Compare this with the Taylor remainder estimate for the approximation of $\tan{x}$ by $x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}$.

53. [T] Plot ${e}^{x}-{e}_{4}\left(x\right)$ where ${e}_{4}\left(x\right)=1+x+\frac{{x}^{2}}{2}+\frac{{x}^{3}}{6}+\frac{{x}^{4}}{24}$ on $\left[0,2\right]$. Compare the maximum error with the Taylor remainder estimate.

54. (Taylor approximations and root finding.) Recall that Newton’s method ${x}_{n+1}={x}_{n}-\frac{f\left({x}_{n}\right)}{f\prime \left({x}_{n}\right)}$ approximates solutions of $f\left(x\right)=0$ near the input ${x}_{0}$.

1. If $f$ and $g$ are inverse functions, explain why a solution of $g\left(x\right)=a$ is the value $f\left(a\right)\text{of}f$.
2. Let ${p}_{N}\left(x\right)$ be the $N\text{th}$ degree Maclaurin polynomial of ${e}^{x}$. Use Newton’s method to approximate solutions of ${p}_{N}\left(x\right)-2=0$ for $N=4,5,6$.
3. Explain why the approximate roots of ${p}_{N}\left(x\right)-2=0$ are approximate values of $\text{ln}\left(2\right)$.

In the following exercises, use the fact that if $q\left(x\right)=\displaystyle\sum _{n=1}^{\infty }{a}_{n}{\left(x-c\right)}^{n}$ converges in an interval containing $c$, then $\underset{x\to c}{\text{lim}}q\left(x\right)={a}_{0}^{}$ to evaluate each limit using Taylor series.

55. $\underset{x\to 0}{\text{lim}}\frac{\cos{x} - 1}{{x}^{2}}$

56. $\underset{x\to 0}{\text{lim}}\frac{\text{ln}\left(1-{x}^{2}\right)}{{x}^{2}}$

57. $\underset{x\to 0}{\text{lim}}\frac{{e}^{{x}^{2}}-{x}^{2}-1}{{x}^{4}}$

58. $\underset{x\to {0}^{+}}{\text{lim}}\frac{\cos\left(\sqrt{x}\right)-1}{2x}$