009A Sample Final 1

This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

 Problem 1 

In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.

(a)   $\lim _{x\rightarrow -3}{\frac {x^{3}-9x}{6+2x}}$

(b)   $\lim _{x\rightarrow 0^{+}}{\frac {\sin(2x)}{x^{2}}}$

(c)   $\lim _{x\rightarrow -\infty }{\frac {3x}{\sqrt {4x^{2}+x+5}}}$

 Problem 2 

Consider the following piecewise defined function:

$f(x)=\left\{{\begin{array}{lr}x+5&{\text{if }}x<3\\4{\sqrt {x+1}}&{\text{if }}x\geq 3\end{array}}\right.$

(a) Show that  $f(x)$  is continuous at  $x=3.$

(b) Using the limit definition of the derivative, and computing the limits from both sides, show that  $f(x)$  is differentiable at  $x=3$.

 Problem 3 

Find the derivatives of the following functions.

(a)   $f(x)=\ln {\bigg (}{\frac {x^{2}-1}{x^{2}+1}}{\bigg )}$

(b)   $g(x)=3\sin(4x)+4\tan({\sqrt {1+x^{3}}})$

 Problem 4 

If  $y=\cos ^{-1}(2x)$ compute  ${\frac {dy}{dx}}$  and find the equation for the tangent line at  $x_{0}={\frac {\sqrt {3}}{4}}.$

You may leave your answers in point-slope form.

 Problem 5 

A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the string has been let out?

 Problem 6 

Consider the following function:

$f(x)=3x-2\sin x+7$

(a) Use the Intermediate Value Theorem to show that  $f(x)$  has at least one zero.

(b) Use the Mean Value Theorem to show that  $f(x)$  has at most one zero.

 Problem 7 

A curve is defined implicitly by the equation

$x^{3}+y^{3}=6xy.$

(a) Using implicit differentiation, compute  ${\frac {dy}{dx}}$.

(b) Find an equation of the tangent line to the curve  $x^{3}+y^{3}=6xy$  at the point  $(3,3)$.

 Problem 8 

Let

$y=x^{3}.$

(a) Find the differential  $dy$  of  $y=x^{3}$  at  $x=2$.

(b) Use differentials to find an approximate value for  $1.9^{3}$.

 Problem 9 

Given the function  $f(x)=x^{3}-6x^{2}+5$,

(a) Find the intervals in which the function increases or decreases.

(b) Find the local maximum and local minimum values.

(c) Find the intervals in which the function concaves upward or concaves downward.

(d) Find the inflection point(s).

(e) Use the above information (a) to (d) to sketch the graph of  $y=f(x)$.

 Problem 10 

If a resistor of  $R$  ohms is connected across a battery of  $E$  volts with internal resistance  $r$  ohms, then the power (in watts) in the external resistor is

$P={\frac {E^{2}R}{(R+r)^{2}}}.$

If  $E$  and  $r$  are fixed but  $R$  varies, what is the maximum value of the power?