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title: Calculus HW3

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# Calculus HW3
1.
    **i)** Prove $f(x)$ is strictly increasing in $[a,c)$ :
    $\text{(Prove by contradiction) Suppose NOT, then }\exists\ x,y\in[a,c)\text{ and }x<y\text{ but }f(x)\geq f(y)\\ \Rightarrow \exists\ \xi\in[x,y]\text{ such that }f'(\xi)\leq 0(\rightarrow\leftarrow)\\ \text{Similarly, }f(x)\text{ is also strictly increasing in }(c,b].$
    **ii)** Prove $f(x)$ is strictly increasing at $x=c$ :
    $\underline{\textbf{Claim}}\ f(c)<f(x),\ \forall\ x\in(c,b]\text{ and }f(c)>f(x),\ \forall\ x\in[a,c)\\
    \text{By Mean Value Theorem, }\forall\ x\in(c,b],\ \frac{f(x)-f(c)}{x-c}=f'(\xi)>0\text{ for some }\xi\in(c,x)\\
    \Rightarrow f(x)-f(c)=f'(\xi)(x-c)>0\Rightarrow f(x)>f(c),\ \forall\ x\in(c,b]\\
    \text{Similarly, }f(x)<f(c),\ \forall\ x\in[a,c).$
    Hence, $f(x)$ is strictly increasing in $[a,b].\ \blacksquare$
    
2.
    $$\lim_{x\to 0^{+}}\ln(x)\cdot\sin(x)=\lim_{x\to 0^{+}}\frac{\ln(x)}{\csc(x)}\stackrel{\left(-\frac{\infty}{\infty}\right)}{=}\lim_{x\to 0^{+}}\frac{1/x}{-\cot(x)\csc(x)}\\=\lim_{x\to 0^{+}}\frac{-\tan(x)\sin(x)}{x}=-\left(\lim_{x\to 0^{+}}\tan(x)\right)\left(\lim_{x\to 0^{+}}\frac{\sin(x)}{x}\right)=0\ \blacksquare$$
    
3.
    (1)$$\lim_{x\to\infty}\frac{x}{x+\sin(x)}=\lim_{x\to\infty}\frac{1}{1+\frac{\sin(x)}{x}}=1\ \blacksquare$$
    (2) $\text{No.  Since the limit of denominator does NOT converge, we cannot use L'Hopital rule.}\\\text{L'Hopital rule requires that both the derivative limits (of numerator/denominator) converge.}\\\text{That is, both }\lim_{x\to\infty}(x)'\text{ and }\lim_{x\to\infty}(x+\sin(x))'\text{ exist.}$
    
4.
    $$x^3+y^3+2y+3xy=19\Rightarrow 3x^2+3y^2\left(\frac{dy}{dx}\right)+2\left(\frac{dy}{dx}\right)+3y+3x\left(\frac{dy}{dx}\right)=0\\
    \Rightarrow \frac{dy}{dx}=-\frac{3x^{2}+3y}{3y^{2}+3x+2}\stackrel{(x,y)=(1,2)}{=}-\frac{9}{17}\Rightarrow\text{Suppose }L:y=\left(-\frac{9}{17}\right)x+k\\\stackrel{(x,y)=(1,2)}{\Rightarrow}2=-\frac{9}{17}+k\Rightarrow k=\frac{43}{17}\Rightarrow L:y=\left(-\frac{9}{17}\right)x+\frac{43}{17}\ \blacksquare$$