2.) Let \( f \) be the function given by: \( f(x)=\sin \left(x^{2}\right)+\cos (x) \). The graph of \( f^{(5)}(x) \) is shown above. a) Write the first four nonzero terms of the Taylor series for \( \sin (x) \) about \( x=0 \) and write the first four nonzero terms of the Taylor series for \( \sin \left(x^{2}\right) \) about \( x=0 \). c) Find the value of \( f^{(5)}(0) \). d) Let \( P(x) \) be the fourth-degree Taylor polynomial for \( f \) about \( x=0 \). Using information from the given graph, show that \( \left\lvert\, P_{4}\left(\frac{1}{4}\right)-f\left(\frac{1}{4}\right)<\frac{1}{3000}\right. \). b) Write the first four nonzero terms of the Taylor series for \( \cos (x) \) about \( x=0 \). Use this series and the series for \( \sin \left(x^{2}\right) \) you found in part a to write the first four nonzero terms of the Taylor series about \( x= \) 0 for \( f \).
Added by Connie A.
Close
Step 1
a) The Taylor series for \( \sin (x) \) about \( x=0 \) is given by: \[ \sin (x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots \] Show more…
Show all steps
Your feedback will help us improve your experience
Mengchun Cai and 92 other Calculus 2 / BC educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Using Maclaurin's series, find the first 4 (non zero) terms for the function $f(x)=\sin x$ $$ \begin{array}{cc} f(x)=\sin x & f(0)=\sin 0=0 \\ f^{\prime \prime}(x)=-\sin x \quad f^{\prime \prime}(0)=-\sin 0=0 \\ f^{\prime \prime \prime}(x)=-\cos x \quad f^{\prime \prime \prime}(0)=-\cos 0=-1 \\ f^{\mathrm{iv}}(x)=\sin x \quad & f^{\mathrm{i}}(0)=\sin 0=0 \\ f^{\mathrm{v}}(x)=\cos x & f^{\mathrm{v}}(0)=\cos 0=1 \\ f^{\mathrm{vi}}(x)=-\sin x & f^{\mathrm{vi}}(0)=-\sin 0=0 \\ f^{\mathrm{vii}}(x)=-\cos x & f^{\mathrm{vii}}(0)=-\cos 0=-1 \end{array} $$ Substituting the above values into Maclaurin's series of equation (5) gives: $$ \begin{aligned} \sin x=0+x(1)+\frac{x^{2}}{2 !}(0)+\frac{x^{3}}{3 !}(-1)+\frac{x^{4}}{4 !}(0) \\ +\frac{x^{5}}{5 !}(1)+\frac{x^{6}}{6 !}(0)+\frac{x^{7}}{7 !}(-1)+\cdots \\ \text { i.e. } \sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\cdots \end{aligned} $$
Given $f(x)=\left\{\begin{aligned} x, & 0<x<1 \\-2, & 1<x<2 \end{aligned}\right.$ (a) Sketch at least three periods of the graph of the function represented by the sine series for $f(x)$. Without finding any series, answer the following questions: (b) To what value does the sine series in (a) converge at $x=1 ?$ At $x=2 ?$ At $x=0$ ? At $x=-1$ (c) If the given function is continued with period 2 and then is represented by a complex exponential series $\sum_{\pi}^{\alpha}-\infty \bar{i}_{n} e^{i n \pi x}$, what is the value of $\sum_{n^{-}-i x}^{x}\left|c_{n}\right|^{2}$ ?
FOURIER SERIES
Miscellaneous problems
(a) Find the Taylor polynomials up to degree 5 for $ f (x) = sin x $ centered at $ a = 0. $ Graph $ f $ and these polynomials on a common screen. (b) Evaluate $ f $ and these polynomials at $ x = \pi/4, \pi/2, $ and $ \pi $. (c) Comment on how the Taylor polynomials converge to $ f(x). $
Infinite Sequences and Series
Applications of Taylor Polynomials
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Watch the video solution with this free unlock.
EMAIL
PASSWORD