급수 심화 전체 문제

If $Q(x)$ is the quadratic approximation of $f$ about $3$, then $Q(3)=f(3)$, $Q'(3)=f'(3)$, and $Q''(3)=f''(3)$. There is no guarantee that $f(x)$ and $Q(x)$ share the same third derivative, though, so we do not have enough information to know $f'''(3)$.

$$\begin{aligned} f(3)&=-3^2+6(3)=9\\ f'(3)&=\left.\frac{\mathrm{d}}{\mathrm{d}x}\left\{-x^2+6x\right\}\right|_{x=3}=\left.-2x+6\right|_{x=3}=0\\ f''(3)&=\left.\frac{\mathrm{d}^2}{\mathrm{d}{x}^2}\left\{-x^2+6x\right\}\right|_{x=3}= \left.\frac{\mathrm{d}}{\mathrm{d}x}\left\{-2x+6\right\}\right|_{x=3}=-2 \end{aligned}$$

The quadratic approximation of $f(x)$ about $x=a$ is

$$\begin{aligned} f(x) &\approx f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2 \\ \text{We subsitute $f(a)=2a+5$, $f'(a)=2$, and $f''(a)=0$:} \\ f(x) &\approx (2a+5)+2(x-a)=2x+5 \end{aligned}$$

So, our approximation is $f(x) \approx 2x+5$.

Remark: Our approximation is exact for every value of $x$. This will always happen with a quadratic approximation of a function that is quadratic, linear, or constant.

Since $T_3(x)$ is the third order Taylor polynomial for $f(x)$ about $x=1$:

• $T_3(1)=f(1)$
• $T_3'(1)=f'(1)$
• $T_3''(1)=f''(1)$
• $T_3'''(1)=f'''(1)$

In particular, $f''(1)=T_3''(1)$.

$$\begin{aligned} T_3'(x)&=3x^2-10x+9\\ T_3''(x)&=6x-10\\ T_3''(1)&=6-10=-4 \end{aligned}$$

So, $f''(1)=-4$.

Since $f'(x)=6x$, when $x=10$, $f'(10)=60$. If $\Delta y=f(11)-f(10)$, and $\Delta x = 11-10$, then the linear approximation tells us $$\Delta y \approx 60\Delta x = 60$$ So, the linear approximates estimates the error in $f(x)$ to be about 60.

Since $f''(x)=6$, the quadratic approximation (using $f'(10)=60$, $f''(10)=6$, and $\Delta x = 1$) tells us $$\Delta y \approx f'(10)\Delta x+\frac{1}{2}f''(1)\left(\Delta x\right)^2= 60\cdot 1 +\frac{1}{2}(6)(1)^2= 63$$ So, the quadratic approximates estimates the error in $f(x)$ to be about 63. (Indeed, the exact value of $f(11)-f(10)$ is 63. It is not a fluke that our estimated error, using a quadratic approximation, is exactly the same as our actual error. It is a consequence of the fact that $f(x)$ is a quadratic function.)

Whenever you approximate a polynomial with a Taylor polynomial of greater or equal degree, your Taylor polynomial is exactly the same as the function you are approximating. So, the error is zero.

The constant approximation gives

$$\begin{aligned} \sin(33)&\approx \sin (0)=0 \\ \text{while the linear approximation gives} \\ f(x)&\approx f(0)+f'(0)x\\ \sin(x)& \approx \sin (0)+\cos(0) x\\&= x\\ \sin(33)&\approx 33 \end{aligned}$$

Since $-1 \leq \sin(33)\leq 1$, the constant approximation is better. (But both are a little silly.)

The third Maclaurin polynomial for $f(x)$ is

$$\begin{aligned} f(0) + f'(0)x +\frac{f''(0)}{2}\cdot x^2 + \frac{f'''(0)}{6}\cdot x^3 &=4+3x^2+\frac{1}{2}x^3. \end{aligned}$$

The coefficient of $x$ is $f'(0)$ on one side, and $0$ on the other, so $f'(0)=0$.

The coefficient of $x^2$ is $\dfrac{1}{2}f''(0)$ on one side, and $3$ on the other, so $f''(0)=6$.

The third Maclaurin polynomial for $h(x)$ is

$$\begin{aligned} h(0) + h'(0)x +\frac{h''(0)}{2}\cdot x^2 + \frac{h^{(3)}(0)}{3!}\cdot x^3 &= 1+4x-\frac{1}{3}x^2 + \frac{2}{3}x^3 \end{aligned}$$

The coefficient of $x^3$ is $\dfrac{1}{3!}h^{(3)}(0)$ on one side, and $\dfrac{2}{3}$ on the other. Thus $\dfrac{h^{(3)}(0)}{6}=\dfrac{2}{3}$, so $h^{(3)}(0)=6\cdot\dfrac{2}{3} = 4$.

The third order Taylor polynomial for $h(x)$ about $x=2$ is $$h(2) + h'(2)(x-2) +\frac{h''(2)}{2}\cdot (x-2)^2 + \frac{h'''(2)}{6}\cdot (x-2)^3$$ The coefficient of $(x-2)$ is $h'(2)$ in the definition, and $\dfrac{1}{2}$ in the given function, so $h'(2)=\dfrac{1}{2}$.

The coefficient of $(x-2)^2$ is $~\dfrac{1}{2}h''(2)~$ in the definition, and $0$ in the given function, so $h'(2)=0$.

(a) If Olaf is old, and I am even older, then I am old as well.

(b) If Olaf is old, and I am not as old, then perhaps I am old as well (just slightly less so), or perhaps I am young. There is not enough information to tell.

(c) If Yuan is young, and I am older, then perhaps I am much older and I am old, or perhaps I am only a little older, and I am young. There is not enough information to tell.

(d) If Yuan is young, and I am even younger, then I must also be young.

Another way to think about this is with a timeline of birthdates. People born before the threshold are old, and people born after it are young.

If I'm born before (older than) Olaf, I'm born before the threshold, so I'm old.

If I'm born after (younger than) Yuan I'm born after the threshold, so I'm young.

If I'm born after Olaf or before Yuan, I don't know which side of the threshold I'm on. I could be old or I could be young.

The integral test does not apply because $f(x)$ is not decreasing.

One possible answer: $\displaystyle\sum_{n=1}^\infty \dfrac{1}{n^2}$. This series converges (it's a $p$–series with $p=2>1$), but if we take the ratio of consecutive terms: $$\lim_{n \to \infty}\frac{a_{n+1}}{a_n} = \lim_{n \to \infty}\frac{n^2}{(n+1)^2}=1$$ The limit of the ratio is 1, so the ratio test is inconclusive.

By the divergence test, for a series $\sum a_n$ to converge, we need $\lim\limits_{n \to \infty} a_n=0$. That is, the magnitude (absolute value) of the terms needs to be getting smaller. If $\displaystyle\lim_{n \to \infty}\left|\frac{a_n}{a_{n+1}}\right|<1$ or (equivalently) $\displaystyle \lim_{n \to \infty}\left|\frac{a_{n+1}}{a_{n}}\right|>1$, then $|a_{n+1}|>|a_n|$ for sufficiently large $n$, so the terms are actually growing in magnitude. That means the series diverges, by the divergence test.

The terms of the series only see a small portion of the domain of the integral. We can try to think of a function $f(x)$ that behaves “nicely" when $x$ is a whole number (that is, it produces a sequence whose sum converges), but is more unruly when $x$ is not a whole number.

For example, suppose $f(x)=\sin(\pi x)$. Then $f(x)=0$ for every integer $x$, but this is not representative of the function as a whole. Indeed, our corresponding series has terms $\{a_n\}=\{0,0,0,\ldots\}$.

• $\displaystyle\int_{1}^\infty \sin(\pi x)\,\mathrm{d}x = \lim_{R \to \infty} \left[-\frac{1}{\pi}\cos (\pi x)\right]_1^R = \lim_{R \to \infty}\Big[-\cos (\pi R)\Big]-\frac{1}{\pi}$

Since the limit does not exist, the integral diverges.

• $\displaystyle\sum_{n=1}^{\infty}\sin(\pi n) = \sum_{n=1}^\infty 0 = 0$. The series converges.

If $x=c$, then

$$\begin{aligned} f(x)&=A_a(c-c)^a+A_{a+1}(c-c)^{a+1}+A_{a+2}(c-c)^{a+2}+\cdots\\ &=A_a \cdot 0+A_{a+1}\cdot 0+A_{a+2}\cdot 0+\cdots\\ &=0 \end{aligned}$$

So, $f(x)$ converges (to the constant 0) when $x=c$. (Had we allowed $a=0$, it would be possible for $f(x)$ to converge to a nonzero number $A_0$, because we use the convention $0^0=1$.)

Depending on the sequence $\{A_n\}$, it's possible that $f(x)$ diverges for all $x \neq c$. For example, suppose $A_n=n!$, so $f(x)=\displaystyle\sum_{n=0}^\infty n!(x-c)^n$. If $x \neq c$, then the limit $\displaystyle\lim_{n\to\infty} \left|\frac{(n+1)!(x-c)^{n+1}}{n!(x-c)^n}\right|=\lim_{n\to\infty}(n+1)|x-c|$ is infinity, since $x-c\neq 0$. So, the series diverges.

We've now shown that the series definitely converges at $x=c$, but at any other point, it may fail to converge.

Following how a Taylor series is constructed, the Taylor series and the function agree at the point chosen as the centre. So, $T(5)=\arctan^3\left(e^5+7\right)$.

If we were evaluating a Taylor series at a point other than its centre, we would generally need to check that (a) the series converges, and (b) it converges to the same value as the function we used to create it.