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# Week 8 Problem Set C.
## Reading.
Thomas Ch. 3.9 on linearization and linear approximations.
## Problems.
### Finding linearizations.
For the following, find the linearization $L(x)$ of the function $f(x)$ at the point $x=a$.
1. $f(x)=x^{3}-2x+3$, $a=2$.
2. $\displaystyle f(x)=x + \frac{1}{x}$, $a=1$.
3. $f(x) = \tan(x)$, $a=\pi$.
4. **Common linear approximations at $x=0$.** Find the linearizations of the following functions at $x=0$:
1. $\sin(x)$
2. $\cos(x)$
3. $\tan(x)$
### Linearization for approximations.
In the following, first find a linearization at a suitably chosen **integer** near $x_{0}$ at which the given function and its derivative are easy to calculate, then use your linearization to estimate $f(x_{0})$.
1. $f(x)=x^{2}+2x$, $x_{0}=0.1$
2. $f(x) = 2x^{2}+4x-3$, $x_{0}=-0.9$
3. $f(x)=\sqrt[3]{x}$, $x_{0}=8.5$
4. Show that the linearization of $f(x) = (1+x)^{k}$ at $x=0$ is $L(x)=1+kx$.
5. **Faster than a calculator.** Use the approximation $(1+x)^{k}\approx 1+kx$ when $x$ is near 0 to estimate the following:
1. $(1.0002)^{50}$
2. $\sqrt[3]{1.009}$
6. Find the linearization $L(x)$ of $\displaystyle f(x)= \frac{x}{x+1}$ at $x=1$. Then use this linearization to approximate $f(1.3)$.
### Higher order approximations.
In class we mentioned that linearizations are first order approximations for a function. As it turns out, there are higher order approximations as well, and quite often this is implemented computationally to calculate values in a computer!
For instance, the second order approximation of $f(x)$ at $x=a$ is $$
T_{2}(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}
$$
and the third order approximation of $f(x)$ at $x=a$ is $$
T_{3}(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^{2} +\frac{f'''(a)}{3!}(x-a)^{3}
$$
And so on.
1. Find the **third order approximation** $T_{3}(x)$ of $f(x)=\sin(x)$ at $x=0$.
2. Find the **third order approximation** $T_{3}(x)$ of $f(x)=\cos(x)$ at $x=0$.
3. Use the **third order approximation** of $\sin(x)$ at $x=0$ to estimate $\sin(0.1)$. Also, use the **linearization** of $\sin(x)$ at $x=0$ to estimate $\sin(0.1)$. Which one is closer to the calculator value of $\sin(0.1)$? Notice: You have to be in **radian mode** when using the calculator!
You will learn more about this in Math 8! How exciting!
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