극한·미분 심화 전체 문제

Newton’s method seeks to approximate a solution $f(x)=0$ that starts with an initial approximation ${x}_{0}$ and successively defines a sequence ${x}_{n+1}={x}_{n}-\frac{f({x}_{n})}{{f}^{'}({x}_{n})}.$ For the given choice of $f$ and ${x}_{0},$ write out the formula for ${x}_{n+1}.$ If the sequence appears to converge, give an exact formula for the solution $x,$ then identify the limit $x$ accurate to four decimal places and the smallest $n$ such that ${x}_{n}$ agrees with $x$ up to four decimal places.

[T] $f(x)={(x-1)}^{2}-2,$ ${x}_{0}=2$

In the following exercises, write the appropriate $\varepsilon$-$\delta$ definition for each of the given statements.

$\lim_{x\to a}f(x)=N$

In the following exercises, write the appropriate $\varepsilon$-$\delta$ definition for each of the given statements.

$\lim_{t\to b}g(t)=M$

In the following exercises, write the appropriate $\varepsilon$-$\delta$ definition for each of the given statements.

$\lim_{x\to c}h(x)=L$

In the following exercises, write the appropriate $\varepsilon$-$\delta$ definition for each of the given statements.

$\lim_{x\to a}\varphi (x)=A$

[T] In the following exercises, use a graphing calculator to find a number $\delta$ such that the statements hold true.

$|\sin (2x)-\frac{1}{2}|<0.1,$ whenever $|x-\frac{\pi }{12}|<\delta$

[T] In the following exercises, use a graphing calculator to find a number $\delta$ such that the statements hold true.

$|\sqrt{x-4}-2|<0.1,\,\text{whenever}\,|x-8|<\delta$

Using precise definitions of limits, determine $\lim_{x\to 0}f(x)$ for $f(x)=\left\{\begin{array}{l}x\,\text{if}\,x\,\text{is rational} \\ 0\,\text{if}\,x\,\text{is irrational}\end{array}\right..$ (Hint: Break into two cases, x rational and x irrational.)

$\lim_{x\to a}[cf(x)]=cL$ for any real constant c (Hint: Consider two cases: $c=0$ and $c\ne 0.)$

What is the linear approximation for any generic linear function $y=mx+b?$

When is the linear approximation exact?

For the following exercises, find the linear approximation $L(x)$ to $y=f(x)$ near $x=a$ for the function.

$f(x)=x+{x}^{4},a=0$

For the following exercises, find the linear approximation $L(x)$ to $y=f(x)$ near $x=a$ for the function.

$f(x)=\frac{1}{x},a=2$

For the following exercises, find the linear approximation $L(x)$ to $y=f(x)$ near $x=a$ for the function.

$f(x)=\tan x,a=\frac{\pi }{4}$

For the following exercises, find the linear approximation $L(x)$ to $y=f(x)$ near $x=a$ for the function.

$f(x)=\sin x,a=\frac{\pi }{2}$

For the following exercises, find the linear approximation $L(x)$ to $y=f(x)$ near $x=a$ for the function.

$f(x)=x\sin x,a=2\pi$

For the following exercises, find the linear approximation $L(x)$ to $y=f(x)$ near $x=a$ for the function.

$f(x)={\sin }^{2}x,a=0$

For the following exercises, compute the values given within 0.01 by deciding on the appropriate $f(x)$ and $a,$ and evaluating $L(x)=f(a)+{f}^{'}(a)(x-a).$ Check your answer using a calculator.

[T] ${(2.001)}^{6}$

For the following exercises, compute the values given within 0.01 by deciding on the appropriate $f(x)$ and $a,$ and evaluating $L(x)=f(a)+{f}^{'}(a)(x-a).$ Check your answer using a calculator.

[T] $\sin (0.02)$

For the following exercises, compute the values given within 0.01 by deciding on the appropriate $f(x)$ and $a,$ and evaluating $L(x)=f(a)+{f}^{'}(a)(x-a).$ Check your answer using a calculator.

[T] $\cos (0.03)$