009A Sample Final A

This is a sample final, and is meant to represent the material usually covered in Math 9A. Moreover, it contains enough questions to represent a three hour test. An actual test may or may not be similar.

1. Find the following limits:

(a)   $\lim _{x\rightarrow 0}{\frac {\tan(3x)}{x^{3}}}.$

(b)   $\lim _{x\rightarrow -\infty }{\frac {\sqrt {x^{6}+6x^{2}+2}}{x^{3}+x-1}}.$

(c)   $\lim _{x\rightarrow 3}{\frac {x-3}{{\sqrt {x+1}}-2}}.$

(d)   $\lim _{x\rightarrow 3}{\frac {x-1}{{\sqrt {x+1}}-1}}.$

(e)   $\lim _{x\rightarrow \infty }{\frac {5x^{2}-2x+3}{1-3x^{2}}}.$

2. Find the derivatives of the following functions:

(a)   $f(x)={\frac {3x^{2}-5}{x^{3}-9}}.$

(b)   $g(x)=\pi +2\cos({\sqrt {x-2}}).$

(c)   $h(x)=4x\sin(x)+e(x^{2}+2)^{2}.$

3. Consider the following function:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\begin{cases} \sqrt{x}, & \quad\mbox{if \ensuremath{x\geq1}}\\ 4x^{2}+C, & \quad\mbox{if }x<1. \end{cases}}