Chapter 8

Show that the Taylor expansion of the following functions at the given points converges for all points \(x\) : a. \(f(x)=\sin x\) at the point \(x_{0}=0\). b. \(f(x)=\cos x\) at the point \(x_{0}=\pi\).

Prove that $$ 1+\frac{x}{2}-\frac{x^{2}}{8}<\sqrt{1+x}<1+\frac{x}{2} \quad \text { if } x>0 $$ In particular, show that \(1.375<\sqrt{2}<1.5\).

For each of the following pairs of functions, determine its highest order of contact at the indicated point: a. \(f(x)=x^{2}\) and \(g(x)=\sin x\) for all \(x ; x_{0}=0\). b. \(f(x)=e^{x^{2}}\) and \(g(x)=1+2 x^{2}\) for all \(x ; x_{0}=0\). c. \(f(x)=\ln x\) and \(g(x)=(x-1)^{3}+\ln x\) for all \(x>0 ; x_{0}=1\). d. \(f(x)=\ln x\) and \(g(x)=(x-1)^{200}+\ln x\) for all \(x>0 ; x_{0}=1\).

$$ \text { Show that for } \beta=-1 \text { , the Binomial Expansion reduces to the Geometric Series. } $$

Prove that $$ 1+\frac{x}{3}-\frac{x^{2}}{9}<(1+x)^{1 / 3}<1+\frac{x}{3} $$ if \(x>0\)

Compute the third Taylor polynomial for each of the following functions at the indicated point: a. \(f(x)=\int_{0}^{x} 1 /\left(1+t^{2}\right) d t\) for all \(x ; x_{0}=0\). b. \(f(x)=\sin x\) for all \(x ; x_{0}=0\). c. \(f(x)=\sin x+x^{200}\) for all \(x ; x_{0}=0\) d. \(f(x)=\sqrt{2-x}\) for all \(x<2 ; x_{0}=1\).

Define \(f(x)=x^{6} e^{x}\) for all \(x .\) Find the sixth Taylor polynomial for the function \(f\) at \(x_{0}=0\)

Prove that if the functions \(g:[a, b] \rightarrow \mathbb{R}\) and \(h:[a, b] \rightarrow \mathbb{R}\) are continuous, with \(h(x) \geq 0\) for all \(x\) in \([a, b],\) then there is a point \(c\) in \((a, b)\) such that $$ \int_{a}^{b} h(x) g(x) d x=g(c) \int_{a}^{b} h(x) d x $$

Suppose that the function \(F: \mathbb{R} \rightarrow \mathbb{R}\) has derivatives of all orders and that $$ \left\\{\begin{array}{l} F^{\prime \prime}(x)-F^{\prime}(x)-F(x)=0 \quad \text { for all } x \\ F(0)=1 \quad \text { and } \quad F^{\prime}(0)=1 \end{array}\right. $$ Find a recursive formula for the coefficients of the \(n\) th Taylor polynomial for \(F\) at \(x=0 .\) Show that the Taylor expansion converges at every point

Suppose that the function \(g: \mathbb{R} \rightarrow \mathbb{R}\) has derivatives of all orders and that for each natural number \(n\) there are positive numbers \(c_{n}\) and \(\delta_{n}\) such that $$|g(x)| \leq c_{n}|x|^{n} \quad \text { if }|x|<\delta_{n}$$ Prove that for each natural number \(n, g^{(n)}(0)=0\).

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has a second derivative and that $$ \left\\{\begin{array}{lc} f^{\prime \prime}(x)+f(x)=e^{-x} & \text { for all } x \\ f(0)=0 \quad \text { and } & f^{\prime}(0)=2 \end{array}\right.$$ Find the fourth Taylor polynomial for \(f: \mathbb{R} \rightarrow \mathbb{R}\) at \(x=0\).

Apply the Cauchy Integral Remainder Theorem in the analysis of the expansion $$ \ln (1+x)=\sum_{k=1}^{\infty}(-1)^{k+1} \frac{x^{k}}{k} \quad \text { if }-1

Let \(I\) be a neighborhood of the point \(x_{0}\) and let \(n\) be a natural number. Suppose that the function \(f: I \rightarrow \mathbb{R}\) has \(n+1\) derivatives. Show that the Lagrange Remainder Theorem is equivalent to the following: For each number \(h\) such that \(x_{0}+h\) is in \(I\) there is a number \(\theta\), strictly between 0 and 1 , such that $$ f\left(x_{0}+h\right)=\sum_{k=0}^{n} \frac{f^{(k)}\left(x_{0}\right)}{k !} h^{k}+\frac{1}{(n+1) !} f^{(n+1)}\left(x_{0}+\theta h\right) h^{n+1} $$

By replacing \(x\) by \(x_{0}+\left(x-x_{0}\right)\) and using the Binomial Formula, show that any polynomial \(p\) can be expressed in powers of \(x-x_{0}\) in the form $$ p(x)=c_{0}+c_{1}\left(x-x_{0}\right)+\cdots+c_{n}\left(x-x_{0}\right)^{n} $$

A number \(x_{0}\) is said to be a root of order \(k\) of the polynomial \(p\) provided that \(k\) is a natural number such that \(p(x)=\left(x-x_{0}\right)^{k} r(x),\) where \(r\) is a polynomial and \(r\left(x_{0}\right) \neq 0 .\) Prove that \(x_{0}\) is a root of order \(k\) of the polynomial \(p\) if and only if $$ p\left(x_{0}\right)=p^{\prime}\left(x_{0}\right)=\cdots=p^{(k-1)}\left(x_{0}\right)=0 $$ and $$ p^{(k)}\left(x_{0}\right) \neq 0 $$

$$ \text { For what values of } r \text { does the sequence }\left\\{n^{3} r^{n}\right\\} \text { converge? } $$

a. Show that for a natural number \(n\). $$ (1+x)^{n}=1+\left(\begin{array}{l} n \\ 1 \end{array}\right) x+\left(\begin{array}{l} n \\ 2 \end{array}\right) x^{2}+\cdots+\left(\begin{array}{c} n \\ n-1 \end{array}\right) x^{n-1}+x^{n} $$ b. Use part (a) to provide another proof of the Binomial Formula.

vSuppose that each of the functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) has \(n+1\) continuous derivatives. Prove that \(f\) and \(g\) have contact of order \(n\) at 0 if and only if $$ \lim _{x \rightarrow 0} \frac{f(x)-g(x)}{x^{n}}=0 $$

Use the Lagrange Remainder Theorem to verify the following criterion for identifying local extreme points: Let \(I\) be a neighborhood of the point \(x_{0}\) and let \(n\) be a natural number. Suppose that the function \(f: I \rightarrow \mathbb{R}\) has \(n+1\) derivatives and that \(f^{(n+1)}: I \rightarrow \mathbb{R}\) is continuous. Assume that \(f^{(k)}\left(x_{0}\right)=0\) if \(1 \leq k \leq n\) and that \(f^{(n+1)}\left(x_{0}\right) \neq 0\) a. If \(n+1\) is even and \(f^{(n+1)}\left(x_{0}\right)>0,\) then \(x_{0}\) is a local minimizer. b. If \(n+1\) is even and \(f^{(n+1)}\left(x_{0}\right)<0,\) then \(x_{0}\) is a local maximizer. c. If \(n+1\) is odd, then \(x_{0}\) is neither a local maximizer nor a local minimizer.