###### tags: `課業`
# Santa calculus problems

[Answer](https://hackmd.io/@HNO2/BJBBGVMTw)

## Speedrun
* $\frac{d}{dx}(\cos x)$
* $\int \sin (2x) dx$
* $\frac{d}{dx}(\tan x)$
* $\frac{d}{dx}(\tan^{-1} x)$
* $\int \cot x \csc x dx$
* $\int \sec x dx$
---
* $\frac{d}{dx}(\cot (x^2))$
* $\int \tan x dx$
* $\frac{d}{dx}(\ln x)$
* $\int_{-2}^{2} \frac{\sin (x^3)}{x^8+1} dx$
* $\int_1^2 x\lfloor 2x \rfloor dx$
* $\frac{d}{dx}(\frac{1}{x})$
---
* $\int \frac{1}{x} dx$
* $\int \sin x \cos x dx$
* $\int_0^2 \sqrt{4-x^2} dx$
* $\int \ln x dx$
* $\frac{d}{dx}({5}^{x^2})$
* If $f(x)=x^7+x^5+x^3+1$, then $(f^{-1})'(4)=$ ?
---
* $\lim\limits_{x \rightarrow 0} x \sin (\frac{1}{x})$
* $\lim\limits_{x \rightarrow \infty} x \sin (\frac{1}{x})$
* $\lim\limits_{x \rightarrow 0} \frac{x}{\tan x}$
* $\lim\limits_{x \rightarrow \infty} \frac{\lfloor x \rfloor}{x}$
* $\lim\limits_{x \rightarrow \infty} x^\frac{1}{\ln x}$
* $\lim\limits_{x \rightarrow \infty} (1-\frac{2}{x})^{x}$

## 單選
1. If the integral $\int_0^\infty(\frac {6x} {x^2+4} - \frac 1 {ax+1}) dx$ is **convergent**, find $a$.
A. $\frac 1 6$
B. $\frac 1 3$
C. $3$
D. $6$

2. Let $f(x) = \frac{(2x+1)(2x+2)\cdots(2x+n)}{(x+1)(x+2)\cdots(x+n)}$, find $f'(0)$.
A. $\sum\limits_{k=1}^n \frac {1}{k}$
B. $\sum\limits_{k=1}^n k$
C. $1$
D. $2$

3. For the $y = \frac{x^2}2$, find **arc length** for $x \in [0, 3]$:
A. $\frac 3{14}$
B. $\frac{3\sqrt{10}}2 + \frac12 \ln(3+\sqrt{10})$
C. $\frac37$
D. $3\sqrt{10}$

4. Which of the following statement is **true**?
A. If $\lim \limits_{x \to 0}f(x)=0$ and $\lim \limits_{x \to 0}g(x)=2$, then $\lim \limits_{x \to 0} g(f(x))=2$.
B. If $|f(x)|$ is a differentiable function on $\mathbb{R}$, then $f(x)$ is a differentiable function on $\mathbb{R}$.
C. $f(x)$ is a differentiable function on $(0,1)$ and a continuous function on $[0,1]$. If $f(0)=0$, then there is a real number $a$ in $(0,1)$ such that $f'(a)=f(1)$.
D. If $f(x)$ and $f'(x)$ are both differentiable functions on $\mathbb{R}$, and $f'(x)>0$ for all $x$, then $\lim\limits_{x \to \infty} f(x) = \infty$.

5. Find the largest $\delta$ such that if $0 < \lvert x-1 \rvert < \delta$ then $\lvert f(x)-1 \rvert < 0.2$, where \begin{equation} f(x) = \begin{cases} \sqrt[3]{x},\quad x \ge 1 \\ 2 - \sqrt{x}, \quad x < 1\end{cases} \end{equation}
A. $\frac{19}{100}$
B. $\frac{21}{100}$
C. $\frac{9}{25}$
D. $\frac{91}{125}$

6. Let $R$ be the region enclosed by $y = \ln (x+7), \space y = \ln (x+4), \space y = 0, \space x = t$. If $V(t)$ is the volume of the solid obtained by rotating $R$ about the **y-axis**, then the limit $\lim\limits_{t \to \infty}(\frac d{dt}V(t))$ = ?
A. $2\pi$
B. $\pi$
C. $3\pi$
D. $6\pi$

7. How many horizontal, vertical and slant **asymptotes** does the function $f(x) = \frac{5(x-2) \sqrt{x^2+1}+x^3-6x^2+9x-3}{x^2-6x+8}$ has?
A. 2
B. 3
C. 4
D. 5

8. Evaluate
\begin{equation}
\lim_{x \rightarrow \frac{\pi}{2}} \frac{\tan(\frac{x}{2}) - \csc x}{\sin x-\cos x}
\end{equation}
A. $\frac 1 4$
B. $\frac {-1} 4$
C. $1$
D. $0$

9. Assume that $f$ is continuous and satisfies the following equation
\begin{equation}
\lim_{n \to \infty} \frac{f(\frac{\sin x}n) + f(\frac{2\sin x}n) + \cdots + f(\frac{n\sin x}n)}{n} = \csc x \cdot e^{\sin x}
\end{equation}
for any real number $x \ne (n + \frac 1 2)\pi, n \in \mathbb{Z}$. Find $f(\frac 1 {\sqrt{2}})$.
A. $\frac 1 {\sqrt{2}}e^{\frac 1 {\sqrt{2}}}$
B. $\sqrt{2}e^{\frac 1 {\sqrt{2}}}$
C. $\frac 1 {\sqrt{2}}e$
D. $e^{\frac 1 {\sqrt{2}}}$

10. The area bounded by $x = 0, \space x = n^2, \space y = 0, \space y = \lfloor \sqrt{x} \rfloor$ is
\begin{equation}
\int^{n^2}_0 \lfloor \sqrt{x} \rfloor dx \quad n \in \mathbb{N}
\end{equation}
It is equivalent to
A. $\frac{n(n-1)(n+1)}{2}$
B. $\frac{n(n-1)(4n+1)}{6}$
C. $\int^{n}_0 \lfloor u \rfloor du$
D. $\int^{n}_0 u\lfloor u \rfloor du$

## 多選

11. Let $F(x),f(x)$ be two diffentiable functions on $\mathbb{R}$ and $F'(x)=f(x)$. Which of the following statements are **true**?
A. If $\lim\limits_{x \rightarrow \infty} F(x)=\infty$, then $\lim\limits_{x \rightarrow \infty} f(x) > 0$.
B. If $\lim\limits_{x \rightarrow \infty} f(x) > 0$, then $\lim\limits_{x \rightarrow \infty} F(x) = \infty$.
C. If $F(x)\geq \frac{x^2}{2} \space \forall x \in \mathbb{R}$, then $f(x) \geq x\space \forall x \in \mathbb{R}$.
D. If $f(x)\geq x \space \forall x \in \mathbb{R}$, then $F(x) \geq \frac{x^2}{2}\space \forall x \in \mathbb{R}$.

12. Consider the function below:
$f(x)=\begin{cases}
x \sin(\frac{1}{x^2}) & \text{if} \space x \neq0 \\
0 & \text{if} \space x=0 \\
\end{cases}$
Which of the following statements are **true**?
A. $f$ is differentiable at $x=0$.
B. $f$ has infinitely many local maxima.
C. $f$ has horizontal asymptotes.
D. $f$ has slant asymptotes.

13. Which of the following integrals **converge**?
A. $\int_0^{1} \frac{\sin \sqrt [3] x}{x} dx$
B. $\int_0^{\infty} \frac{1}{x \sqrt{x+1}} dx$
C. $\int_1^{\infty} \frac{\ln x}{x^{0.8}} dx$
D. $\int_{1}^{\infty} e^{-x^{2+ \sin x}} dx$

14. For the parametric curve $(x(\theta), y(\theta)) = (1-\sin(\theta), \theta + \cos(\theta))$, which of the followings are **true**? <br>
A. $\frac{dy}{dx} = \frac{1-\sin\theta}{\cos\theta}$ .<br>
B. There are one horizontal tangent point and one vertical tangent point in $\theta \in [0, 2\pi]$. <br>
C. Arc length for $\theta \in [0, \frac{\pi}{2}]$ is $2 - \sqrt 2$. <br>
D. Surface area obtained by rotating it about y-Axis for $\theta \in [0, \frac{\pi}{2}]$ is $\frac{32-20\sqrt 2}{3} \pi$.

15. Which of the following statements are **true**?
A. If $(a,f(a))$ is a local maximum of $f(x)$ and $f'(a)$ exists, then $f'(a)=0$.
B. If $f''(a)=0$, then $(a,f(a))$ is an inflection point of $f(x)$.
C. If $(a,f(a))$ is an inflection point of $f(x)$, then $(a,f'(a))$ is a local extreme value of $f'(x)$.
D. If $(a,f'(a))$ is a local extreme value of $f'(x)$, then $(a,f(a))$ is an inflection point of $f(x)$.

## 計算證明
1. For the definite integral
\begin{equation}
\int^{\frac \pi 2}_0 \sin^n x dx \quad \text{for } n \ge 1
\end{equation}
you need to:

(A)
Find a recursive expression of this indefinite integral.
\begin{equation}
I_n = \int \sin^n x dx \quad
\end{equation}
Hint. $I_n = (...)I_{n-2} + (...), \space (...)$ is a function about $x, \space n$.

(B)
Evaluate
\begin{equation}
\int^{\frac \pi 2}_0 \sin^n x dx \quad \text{for } n \ge 1
\end{equation}

2. For $e^y = \frac 1 {(x+1)^2}$, find:
(A) The **arc length** for $x \in [\sqrt{5}-1, \sqrt{12}-1]$
(B) The **surface area** by rotating it about $x = -1$ for $x \in [1, 3]$