In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point.
2. f(x)=1+x+x2 at a=−1
4. f(x)=sin(2x) at a=π2
6. f(x)=lnx at a=1
8. f(x)=ex at a=1
In the following exercises, verify that the given choice of n in the remainder estimate |Rn|≤M(n+1)!(x−a)n+1, where M is the maximum value of |f(n+1)(z)| on the interval between a and the indicated point, yields |Rn|≤11000. Find the value of the Taylor polynomial pn of f at the indicated point.
10. [T] (28)13;a=27,n=1
12. [T] e2; a=0,n=9
13. [T] cos(π5);a=0,n=4
14. [T] ln(2);a=1,n=1000
16. Integrate the approximation ex≈1+x+x22+⋯+x6720 evaluated at −x2 to approximate ∫10e-x2dx.
In the following exercises, find the smallest value of n such that the remainder estimate |Rn|≤M(n+1)!(x−a)n+1, where M is the maximum value of |f(n+1)(z)| on the interval between a and the indicated point, yields |Rn|≤11000 on the indicated interval.
18. f(x)=cosx on [−π2,π2],a=0
20. f(x)=e-x on [−3,3],a=0
In the following exercises, the maximum of the right-hand side of the remainder estimate |R1|≤max|f''(z)|2R2 on [a−R,a+R] occurs at a or a±R. Estimate the maximum value of R such that max|f''(z)|2R2≤0.1 on [a−R,a+R] by plotting this maximum as a function of R.
22. [T] sinx approximated by x, a=0
24. [T] cosx approximated by 1,a=0
In the following exercises, find the Taylor series of the given function centered at the indicated point.
26. 1+x+x2+x3 at a=−1
28. cosx at a=2π
30. cosx at x=π2
32. ex at a=1
34. 1(x−1)3 at a=0
In the following exercises, compute the Taylor series of each function around x=1.
36. f(x)=2−x
38. f(x)=(x−2)2
40. f(x)=1x
42. f(x)=x4x−2x2−1
44. f(x)=e2x
[T] In the following exercises, identify the value of x such that the given series ∞∑n=0an is the value of the Maclaurin series of f(x) at x. Approximate the value of f(x) using S10=10∑n=0an.
46. ∞∑n=02nn!
48. ∞∑n=0(−1)n(2π)2n+1(2n+1)!
The following exercises make use of the functions S5(x)=x−x36+x5120 and C4(x)=1−x22+x424 on [-π,π].
50. [T] Plot cos2x−(C4(x))2 on [-π,π]. Compare the maximum difference with the square of the Taylor remainder estimate for cosx.
52. [T] Compare S5(x)C4(x) on [−1,1] to tanx. Compare this with the Taylor remainder estimate for the approximation of tanx by x+x33+2x515.
54. (Taylor approximations and root finding.) Recall that Newton’s method xn+1=xn−f(xn)f′(xn) approximates solutions of f(x)=0 near the input x0.
- If f and g are inverse functions, explain why a solution of g(x)=a is the value f(a)off.
- Let pN(x) be the Nth degree Maclaurin polynomial of ex. Use Newton’s method to approximate solutions of pN(x)−2=0 for N=4,5,6.
- Explain why the approximate roots of pN(x)−2=0 are approximate values of ln(2).
In the following exercises, use the fact that if q(x)=∞∑n=1an(x−c)n converges in an interval containing c, then limx→cq(x)=a0 to evaluate each limit using Taylor series.
56. limx→0ln(1−x2)x2
58. limx→0+cos(√x)−12x
Candela Citations
- Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-2/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction