4.4: The Binomial Theorem and Applications of Taylor Series
- Page ID
- 163299
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Binomial Series
For any real number \( r\), the Maclaurin series for \( f(x)=(1+x)^r\) is called the binomial series. It converges to \( f\) for \( |x| \lt 1\), and we write\[(1+x)^r=\sum_{n=0}^ \infty \binom{r}{n}x^n=1+rx+\dfrac{r(r−1)}{2!}x^2+ \cdots +r\dfrac{(r−1) \cdots (r−n+1)}{n!}x^n+ \cdots \nonumber \]for \( |x| \lt 1\).
Expand\[ \left( \dfrac{1}{2}x^{1/3} - \dfrac{3}{5} y \right)^4 \nonumber \]
Find the first three nonzero terms of the Maclaurin series for\[ f(x) = \dfrac{x^4}{\sqrt{49 + x^2}}. \nonumber \]
- Hints
-
\( \frac{x^4}{7} + \displaystyle \sum_{n = 1}^{\infty} \frac{(-1)^n 1 \cdot 3 \cdot 5 \cdots (2n - 1)}{n! 7^{2n + 1} 2^n} x^{2n + 4} \)
Approximate the value of the definite integral to within \( 5 \times 10^{-6} \).\[ \int_0^{0.4} \sqrt{1 + x^4} \, dx \nonumber \]
A Summary of Common Functions Expressed as Taylor Series
| Function | Maclaurin Series | Interval of Convergence |
|---|---|---|
| \( f(x)=\frac{1}{1−x}\) | \(\displaystyle \sum_{n=0}^ \infty x^n\) | \( −1<x<1\) |
| \( f(x)=e^x\) | \(\displaystyle \sum_{n=0}^ \infty \frac{x^n}{n!}\) | \( − \infty <x< \infty \) |
| \( f(x)=\sin x\) | \(\displaystyle \sum_{n=0}^ \infty (−1)^n\frac{x^{2n+1}}{(2n+1)!}\) | \( − \infty <x< \infty \) |
| \( f(x)=\cos x\) | \(\displaystyle \sum_{n=0}^ \infty (−1)^n\frac{x^{2n}}{(2n)!}\) | \( − \infty <x< \infty \) |
| \( f(x)=\ln(1+x)\) | \(\displaystyle \sum_{n=0}^ \infty (−1)^{n+1}\frac{x^n}{n}\) | \( −1<x \leq 1\) |
| \( f(x)=\tan^{−1}x\) | \(\displaystyle \sum_{n=0}^ \infty (−1)^n\frac{x^{2n+1}}{2n+1}\) | \( −1 \leq x \leq 1\) |
| \( f(x)=(1+x)^r\) | \(\displaystyle \sum_{n=0}^ \infty \binom{r}{n}x^n\) | \( −1<x<1\) |
Applications Involving Taylor Series
Recentering Polynomials
Recenter the polynomial at \( a = 1 \) and use it to evaluate \( P(1.1) \), \( P(1.01) \), and \( P(1.001) \).\[ P(x) = x^4 - 4x^2 + 5. \nonumber \]
Building Data Models
The computer in a car captures the following information:
At a given instant, the car is moving with speed \(60 \, m/s\), acceleration \(6 \, m/s^2\), jerk \(2 \, m/s^3\), and the rate that the jerk is changing is between \(-3 \, m/s^4\) and \(-1 \, m/s^4\).
- Use a third-degree Taylor polynomial to estimate how far the car moves in the next second.
- How far out can we predict if we want the error to be less than 0.1 m?