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Section 9.4 Taylor Series (PS4)

Activity 9.4.1.

(a)

Find the 2nd derivative of \(f(x)=x^2\text{.}\)

(b)

Find the 4th derivative of \(f(x)=x^4\text{.}\)

(c)

Find the 9th derivative of \(f(x)=x^9\text{.}\)

(d)

Which of the following represent the \(n\)th and derivative of \(f(x)=x^n\text{?}\) (Make sure it is consistent with \(n=2, 4, 9\) that you found above.)

  1. \(\displaystyle n.\)

  2. \(\displaystyle n!.\)

  3. \(\displaystyle x.\)

  4. \(\displaystyle nx.\)

(e)

What then is the \(n+1\)st derivative of \(f(x)=x^n\text{?}\) (We are differentiating with respect to \(x\text{.}\))

Activity 9.4.2.

Suppose \(p(x)=a_0+a_1x+a_2x^2+a_3x^3\) where \(p(0)=2, p'(0)=1, p''(0)=-2, p'''(0)=4\text{.}\)

(a)

What must \(a_0\) be for \(p(0)=2\text{?}\)

(b)

Using your choice of \(a_0\text{,}\) what must \(a_1\) be for \(p'(0)=1\text{?}\)

(c)

Using your choice of \(a_0, a_1\text{,}\) what must \(a_2\) be for \(p''(0)=-2\text{?}\)

(d)

Using your choice of \(a_0, a_1, a_2\text{,}\) what must \(a_3\) be for \(p'''(0)=4\text{?}\)

(e)

What is \(p(x)\text{?}\)

Activity 9.4.3.

Suppose \(f\) is a function. Then if \(f(x)\) can be represented as a power series, there is a real number \(c\) and an interval, \(I\text{,}\) centered at \(x=c\) such that \(f(x)=\displaystyle\sum_{n=0}^\infty a_n(x-c)^n=a_0+a_1(x-c)+a_2(x-c)^2+a_3(x-c)^3+\ldots\) on \(I\text{.}\)

(a)

Use term-by-term differentiation to find a power series for \(f^\prime(x)\text{.}\)

(b)

Use this power series to find \(f'(c)\text{.}\)

(c)

Use term-by-term differentiation to find a power series for \(f^{\prime\prime}(x)\text{.}\)

(d)

Use this power series to find \(f''(c)\text{.}\)

(e)

Use term-by-term differentiation to find a power series for \(f^{\prime\prime\prime}(x)\text{.}\)

(f)

Use this power series to find \(f'''(c)\text{.}\)

(g)

Deduce that \(f^{(n)}(c)=n!a_n\) and hence \(a_n=\displaystyle\frac{f^{(n)}(c)}{n!}\text{.}\)

Definition 9.4.6.

The Taylor series generated by \(f(x)\) and centered at \(x=c\) is given by \(f(x)=\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!}(x-c)^n =f(c)+f^\prime(c)(x-c)+\frac{f^{\prime\prime}(c)}{2!}(x-c)^2+\frac{f^{(3)}(c)}{3!}(x-c)^3+\ldots\text{,}\) along with its interval of convergence (or interval of validity).

When \(c=0\text{,}\) \(f(x)=\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}(x^n) =f(0)+f^\prime(0)(x)+\frac{f^{\prime\prime}(0)}{2!}(x^2)+\frac{f^{(3)}(0)}{3!}(x^3)+\ldots\) is called the Maclaurin series generated by \(f\text{.}\)

Definition 9.4.7.

\(f(x) \approx T_k(x) = \displaystyle\sum_{n=0}^k \frac{f^{(n)}(c)}{n!}(x-c)^n = f(c)+f^\prime(c)(x-c)+\displaystyle\frac{f^{\prime\prime}(c)}{2!}(x-c)^2+\displaystyle\frac{f^{(3)}(c)}{3!}(x-c)^3+\ldots+\displaystyle\frac{f^{(k)}(c)}{k!}(x-c)^k\text{,}\) where \(T_k(x)\) is called the \(k^{th}\) degree Taylor polynomial generated by \(f\) and centered at \(x=c\text{.}\)

The \(k^{th}\) degree Taylor polynomial can be seen as the “best” polynomial of degree \(k\) or less for approximating \(f(x)\) for values close to \(x=c\text{.}\) Note that polynomials of degree less than \(k\) are their own Taylor polynomials, and the \(1^{st}\) degree Taylor polynomial is also known as the linearization of \(f\text{.}\)

Activity 9.4.8.

Let \(f(x)\) be a function such that:

\begin{equation*} \begin{array}{|c|c|c|c|c|c|c|} f(4) & f'(4) & f''(4) & f'''(4) & f^{(4)}(4) & f^{(5)}(4) & f^{(6)}(4) \\ \hline 0 & 1 & 2 & 3 & 4 & 5 & 6 \end{array} \end{equation*}

(a)

Find a Taylor polynomial for \(f(x)\) centered at \(x=4\) of degree \(3\text{.}\)

(b)

Using the table above, find a general closed form for \(f^{(n)}(4)\text{.}\)

(c)

Use (b) to find a Taylor series for \(f(x)\) centered at \(x=4\text{.}\)

Activity 9.4.9.

Let \(f(x)\) be a function such that:

\begin{equation*} \begin{array}{|c|c|c|c|c|c|c|} f(-2) & f'(-2) & f''(-2) & f'''(-2) & f^{(4)}(-2) & f^{(5)}(-2) & f^{(6)}(-2) \\ \hline 0 & 2 & -16 & 54 & -128 & 250 & -432 \end{array} \end{equation*}

(a)

Find a Taylor polynomial for \(f(x)\) centered at \(x=-2\) of degree \(3\text{.}\)

(b)

Using the table above, find a general closed form for \(f^{(n)}(-2)\text{.}\)

(c)

Use (b) to find a Taylor series for \(f(x)\) centered at \(x=-2\text{.}\)

Activity 9.4.10.

(a)

Find the Maclaurin series for \(f(x)=\sin(x)\) about \(x=0\text{,}\) along with the interval of validity.

(b)

Find the Taylor series generated by \(f(x)=\cos(x)\) about \(x=0\text{.}\)

(c)

Find the Maclaurin series for \(f(x)=e^x\) about \(x=0\text{,}\) along with the interval of validity.

Remark 9.4.11.

You might have seen \(\sqrt{-1}\) written as \(i\text{,}\) and know that \(z\) is a complex number if \(z=a+bi\) for some real numbers \(a\) and \(b\text{.}\) Note that \(i^2=-1\text{,}\) \(i^3=(i^2)i=-i\text{,}\) \(i^4=(i^2)^2=1\text{,}\) \(i^5=(i^4)i=i\text{,}\) and so on. This gives rise to the following notion.

Definition 9.4.12. Euler's Identity.

For any real number \(\theta\text{,}\)

\begin{align*} e^{i\theta} & = 1+\displaystyle\frac{i\theta}{1!}+\displaystyle\frac{(i\theta)^2}{2!}+\displaystyle\frac{(i\theta)^3}{3!}+\displaystyle\frac{(i\theta)^4}{4!}+\displaystyle\frac{(i\theta)^5}{5!}+\displaystyle\frac{(i\theta)^6}{6!}+\displaystyle\frac{(i\theta)^7}{7!}+\displaystyle\frac{(i\theta)^8}{8!}+\ldots\\ & = 1+i\theta-\displaystyle\frac{\theta^2}{2!}-\displaystyle\frac{i\theta^3}{3!}+\displaystyle\frac{\theta^4}{4!}+\displaystyle\frac{i\theta^5}{5!}-\displaystyle\frac{\theta^6}{6!}-\displaystyle\frac{i\theta^7}{7!}+\displaystyle\frac{\theta^8}{8!}+\ldots\\ & = \left(1-\displaystyle\frac{\theta^2}{2!}+\displaystyle\frac{\theta^4}{4!}-\displaystyle\frac{\theta^6}{6!}+\ldots\right)+i\left(\theta-\displaystyle\frac{\theta^3}{3!}+\displaystyle\frac{\theta^5}{5!}-\displaystyle\frac{\theta^7}{7!}+\ldots\right)\\ & = \cos(\theta)+i\sin(\theta). \end{align*}

Activity 9.4.13.

Use Euler's identity to evaluate \(e^{i\pi}\text{.}\)

Subsection 9.4.1 Videos

Figure 170. Video: Determine a Taylor or Maclaurin series for a function