Question 1 [36 marks]

a) [3 marks]

Evaluate $\lim_{n\rightarrow\infty}\sqrt[n]{2^n+4^n+8^n}$.

b) [3 marks]

Evaluate $\lim_{x\rightarrow\infty}\left(1-\frac3x\right)^{2x}$.

c) [3 marks]

Evaluate $\lim_{x\rightarrow0}\frac{\tan x-\sin x}{2x^3}$.

d) [3 marks]

$f(x)=\ln\tan x$. Find $f'(x)$.

e) [3 marks]

$f(x)=\mathrm{e}^{\sqrt x}$. Find $f'(4)$.

f) [3 marks]

Evaluate $\int\frac{\mathrm{d}x}{\sqrt{3x+5}}$.

g) [3 marks]

Evaluate $\int x\mathrm{e}^{-x}\mathrm{d}x$.

h) [3 marks]

Evaluate $\int_1^\mathrm{e}\ln x\,\mathrm{d}x$.

i) [3 marks]

Evaluate $\int_{\frac\pi4}^{\frac\pi2}\frac{\mathrm{d}x}{\tan^2x}$.

j) [3 marks]

Evaluate $\int_{-\frac\pi2}^{\frac\pi2}(x^3+\sin x)\cos x\,\mathrm{d}x$.

k) [3 marks]

Evaluate $\int_1^{+\infty}\frac{\mathrm{d}x}{x^3}$.

l) [3 marks]

Find the general solution of the equation $y''+3y'=0$.

Question 2 [12 marks]

a) [6 marks]

Suppose that $y=f(x)$ is defined by $\mathrm{e}^y+xy=\mathrm{e}$. Find $\left.\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\right|_{x=0}$.

b) [6 marks]

Suppose that $\begin{cases}x=t\mathrm{e}^t\\\mathrm{e}^{ty}=y+t^2-2\end{cases}$. Find $\left.\frac{\mathrm{d}y}{\mathrm{d}x}\right|_{x=0}$.

Question 3 [14 marks]

a) [6 marks]

Evaluate $\int\frac1{x(1+\ln x)}\,\mathrm{d}x$.

b) [8 marks]

Suppose that $f(x)=\begin{cases}x\mathrm{e}^{-x^2},&x\ge0\\\frac1{1+\cos x},&-1<x<0\end{cases}$. Determine the type of discontinuity (first kind or second kind) at $x=0$ and find $\int_{-1}^2f(x)\,\mathrm{d}x$.

Question 4 [14 marks]

a) [6 marks]

Evaluate $\int_0^{+\infty}\frac{\mathrm{e}^x}{(1+\mathrm{e}^x)^2}\,\mathrm{d}x$.

b) [8 marks]

Suppose that a curve passes through the $(0,0)$ point. The slope of the tngent line at any point $(x,y)$ of the curve is $3x+y$. Find tha area $A$ of the region enclose by the curve, the x-axis, and $x=1$.

Question 5 [12 marks]

a) [6 marks]

Suppose that $y(x)$ is defined by $\int_0^xf(t)\,\mathrm{d}t=xf(yx)$ where $f(x)=\mathrm{e}^x$. Find $\lim_{x\rightarrow0}y(x)$.

b) [6 marks]

Suppose that $\lim_{x\rightarrow+\infty}\left(\frac{x-a}{x+a}\right)^x=-\int_a^{+\infty}x\mathrm{e}^{-2x}\mathrm{d}x$. Find $a$.

Question 6 [6 marks]

Suppose that $f(x)=\int_x^{x+\frac\pi2}\left|\sin t\right|\,\mathrm{d}t$.

a) Prove that $f(x)$ is a periodic function. [2 marks]

b) Find the global maximum and minimum of $f(x)$. [4 marks]

Question 7 [6 marks]

Suppose that $f(x)$ is twice diferentiable on $[-2,2]$ satisfying $\left|f(x)\right|\le1$, $f(-2)=f(0)=f(2)$, and $[f(0)]^2+[f'(0)]^2=3$. Prove that there exists a $\xi\in(-2,2)$ such that $f(\xi)+f''(\xi)=0$.