극한·미분 심화 전체 문제

converges; Direct Comparison Test with $xe^{-x}$.

converges; Direct Comparison Test with $e^{-x}$.

converges; Direct Comparison Test with $xe^{-x}$.

converges; Direct Comparison Test with $1/(x^2-1)$.

diverges; Direct Comparison Test with $x/(x^2+1)$.

converges; Direct Comparison Test with $1/e^x$.

converges; Limit Comparison Test with $1/e^x$.

(a) We can plug any number into the cosine function, and it will return a number in $[-1,1]$. The domain of $\arcsin x$ is $[-1,1]$, so any number we plug into cosine will give us a valid number to plug into arcsine. So, the domain of $f(x)$ is all real numbers.

(b) We can plug any number into the cosine function, and it will return a number in $[-1,1]$. The domain of $\mathop{\mathrm{arccsc}} x$ is $(-\infty,-1] \cup [1,\infty)$, so in order to have a valid number to plug into arccosecant, we need $\cos x = \pm 1$. That is, the domain of $g(x)$ is all values $x=n\pi$ for some integer $n$.

(c) The domain of arccosine is $[-1,1]$. The domain of sine is all real numbers, so no matter what number arccosine spits out, we can safely plug it into sine. So, the domain of $h(x)$ is $[-1,1]$.

The tangent line is horizontal when $0=y'=a-\sin x$. That is, when $a=\sin x$.

• If $|a|>1$, then there is no value of $x$ for which $a=\sin x$, so the curve has no horizontal tangent lines.
• If $|a| = 1$, then there are infinitely many solutions to $a=\sin x$, but only one solution in the interval $[-\pi,\pi]$: $x=\arcsin(a)=\arcsin(\pm1)=\pm\frac{\pi}{2}$. Then the values of $x$ for which $a=\sin x$ are $x=2\pi n +a \frac{\pi}{2}$ for any integer $n$.
• If $|a|<1$, then there are infinitely many solutions to $a=\sin x$. The solution in the interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ is given by $x=\arcsin(a)$. The other solution in the interval $\left(-\pi,\pi\right)$ is given by $x=\pi-\arcsin(a)$, as shown in the unit circles below.

So, the values of $x$ for which $x=\sin a$ are $x=2\pi n+\arcsin(a)$ and $x=2\pi n + \pi - \arcsin (a)$ for any integer $n$.

Remark: when $a=1$, then $$2\pi n+\arcsin(a) = 2\pi n + \dfrac{\pi}{2}=2\pi n +\pi -\left(\dfrac {\pi}{2}\right)=2\pi n+\pi-\arcsin(a).$$ Similarly, when $a=-1$, $$2\pi n+\arcsin(a) = 2\pi n - \dfrac{\pi}{2}=2\pi (n-1) +\pi -\left(-\dfrac {\pi}{2}\right) =2\pi(n-1)+\pi-\arcsin(a).$$ So, if we try to use the descriptions in the third bullet point to describe points where the tangent line is horizontal when $|a|=1$, we get the correct points but each point is listed twice. This is why we separated the case $|a|=1$ from the case $|a|<1$.

Using the chain rule,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\left\{\arcsin\left(\frac{x}{3}\right)\right\}&=\frac{1}{\sqrt{1-\left(\frac{x}{3}\right)^2}}\cdot \frac{1}{3}\\ &=\frac{1}{3\sqrt{1-\frac{x^2}{9}}}\\ &=\frac{1}{\sqrt{9-x^2}} \end{aligned}$$

Since the domain of arcsine is $[-1,1]$, and we are plugging in $\dfrac{x}{3}$ to arcsine, the values of $x$ that we can plug in are those that satisfy $-1 \le \dfrac{x}{3} \leq 1$, or $-3\leq x \leq 3$. So the domain of $f$ is $[-3,3]$.

Using the quotient rule,

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\left\{\dfrac{\arccos t}{t^2-1}\right\}&= \frac{(t^2-1)\left(\frac{-1}{\sqrt{1-t^2}}\right)-(\arccos t)(2t)}{(t^2-1)^2} \end{aligned}$$

The domain of arccosine is $[-1,1]$, and since $t^2-1$ is in the denominator, the domain of $f$ requires $t^2-1 \neq 0$, that is, $t \neq \pm 1$. So the domain of $f(t)$ is $(-1,1)$.

The domain of $\mathop{\mathrm{arcsec}} x$ is $|x| \geq 1$: that is, we can plug into arcsecant only values with absolute value greater than or equal to one. Since $-x^2-2 \leq -2$, every real value of $x$ gives us an acceptable value to plug into arcsecant. So, the domain of $f(x)$ is all real numbers.

To differentiate, we use the chain rule. Remember $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left\{\mathop{\mathrm{arcsec}} x\right\} = \dfrac{1}{|x|\sqrt{x^2-1}}$.

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\left\{\mathop{\mathrm{arcsec}}(-x^2-2)\right\}&= \dfrac{1}{|-x^2-2|\sqrt{(-x^2-2)^2-1}}\cdot(-2x)\\ &=\dfrac{-2x}{(x^2+2)\sqrt{x^4+4x^2+3}}. \end{aligned}$$

We use the chain rule, remembering that $a$ is a constant.

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\left\{\frac{1}{a}\arctan\left(\dfrac{x}{a}\right)\right\}&= \frac{1}{a}\cdot\frac{1}{1+\left(\frac{x}{a}\right)^2}\cdot \frac{1}{a}\\ &=\frac{1}{a^2+x^2} \end{aligned}$$

The domain of arctangent is all real numbers, so the domain of $f(x)$ is also all real numbers.

We differentiate using the product and chain rules.

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\left\{ x\arcsin x + \sqrt{1-x^2}\right\}&= \arcsin x + \frac{x}{\sqrt{1-x^2}}+ \frac{-2x}{2\sqrt{1-x^2}}\\ &=\arcsin x \end{aligned}$$

The domain of $\arcsin x$ is $[-1,1]$, and the domain of $\sqrt{1-x^2}$ is all values of $x$ so that $1-x^2 \geq 0$, so $x$ in $[-1,1]$. Therefore, the domain of $f(x)$ is $[-1,1]$.

We differentiate using the chain rule:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}x}\{\arctan(x^2)\}&=\frac{2x}{1+x^4} \end{aligned}$$

This is zero exactly when $x=0$.

Using formulas you should memorize from this section, $$\frac{\mathrm{d}}{\mathrm{d}x}\{\arcsin x + \arccos x\}=\frac{1}{\sqrt{1-x^2}}+\frac{-1}{\sqrt{1-x^2}}=0$$

Remark: the only functions with derivative equal to zero everywhere are constant functions, so $\arcsin x + \arccos x$ should be a constant. Since $\sin \theta = \cos \left(\frac{\pi}{2}-\theta\right)$, we can set

$$\begin{aligned} \sin\theta&=x & \cos\left(\frac{\pi}{2}-\theta\right)&=x\\ \text{where $x$ and $\theta$ are the same in both expressions, and $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$. Then} \\ \arcsin x &=\theta & \arccos x &= \frac{\pi}{2}-\theta \\ \text{We note here that arcsine is the inverse of the sine function \textit{ restricted to} $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. So, since we restricted $\theta$ to this domain, $\sin \theta=x$ really does imply $\arcsin x = \theta$. (For an example of why this matters, note $\sin(2\pi)=0$, but $\arcsin (0)=0 \neq 2\pi$.) Similarly, arccosine is the inverse of the cosine function restricted to $[0,\pi]$. Since $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$, then $0 \leq (\frac{\pi}{2}-\theta) \leq \pi$, so $\cos\left(\frac{\pi}{2}- \theta\right) =x$ really does imply $\arccos x=\frac{\pi}{2}-\theta$.} \\ \end{aligned}$$

So, $$\arcsin x+\arccos x =\theta+\frac{\pi}{2}-\theta =\frac{\pi}{2}$$ which means the derivative we were calculating was actually just $\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left\{\dfrac{\pi}{2}\right\}=0$.

Using the chain rule, $$y'=\frac{-{\frac{1}{ x^2}}}{\sqrt{1-\left({\frac{1}{ x}}\right)^2}}=\frac{-1}{x^2\sqrt{1-\frac{1}{x^2}}}.$$

Using the chain rule, $$y'=\frac{-{\frac{1}{ x^2}}}{1+\left({\frac{1}{ x}}\right)^2}=\frac{-1}{x^2+1}.$$

Using the product rule:

$$\begin{aligned} \displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\left\{(1+x^2)\arctan x\right\} &=2x\arctan x+(1+x^2)\frac{1}{1+x^2}\\ &=2x\arctan x+1\end{aligned}$$

The line $y=2x+9$ has slope $2$, so we must find all values of $x$ between $-1$ and $1$ ($\arcsin x$ is only defined for these values of $x$) for which $\frac{\mathrm{d}}{\mathrm{d}x}\{\arcsin x\}=2$. Evaluating the derivative:

$$\begin{aligned} y&=\arcsin x\\ 2=y'&=\frac{1}{\sqrt{1-x^2}}\\ 4&=\frac{1}{1-x^2}\\ \frac{1}{4}&=1-x^2\\ x^2&=\frac{3}{4}\\ x&=\pm\frac{\sqrt{3}}{2}\\ (x,y)&=\pm\big(\frac{\sqrt{3}}{2},\frac{\pi}{3}\big) \end{aligned}$$