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)   $h(x)=4x\sin(x)+e(x^{2}+2)^{2}.$
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3. (Version I) Consider the following function:

$f(x)={\begin{cases}{\sqrt {x}},&{\mbox{if }}x\geq 1,\\4x^{2}+C,&{\mbox{if }}x<1.\end{cases}}$

(a) Find a value of  $C$ which makes $f$ continuous at $x=1.$

(b) With your choice of  $C$, is $f$ differentiable at $x=1$?  Use the definition of the derivative to motivate your answer.

3. (Version II) Repeat the above for the function:

$g(x)={\begin{cases}{\sqrt {x^{2}+3}},&\quad {\mbox{if }}x\geq 1\\{\frac {1}{4}}x^{2}+C,&\quad {\mbox{if }}x<1.\end{cases}}$

(a) Find a value of  $C$ which makes $f$ continuous at 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 x=1.}

(b) With your choice of  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 C} , is 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} differentiable at 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 x=1} ?  Use the definition of the derivative to motivate your answer.

4. Use implicit differentiation to find:

an equation for the tangent line to the 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 -x^{3}-2xy+y^{3}=-1} at the point 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 (1,1)} .