Exam 8: Appendix: Algebra Review

Find the points of intersection (if any) of the given pair of curves.y = x + 7 and y = 2x + 4

Let $f ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + 13$ . Find all critical points of f and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Find the composite function $f ( g ( x ) )$ . $f ( u ) = \frac { 1 } { u } , g ( x ) = x + 3$

The equation of the tangent line to $f ( x ) = 2 \ln x ^ { 3 }$ at x = e is

Find the indicated limit if it exists. $\lim _ { x \rightarrow 4 } \frac { \sqrt { x } - 2 } { x - 4 }$

Find the slope (if possible) of the line that passes through the given pair of points. (1, 0) and (18, 12)

$\lim _ { x \rightarrow 3 ^ { - } } f ( x ) = 3$ where $f ( x ) = \left\{ \begin{aligned}x & \text { if } x < 3 \\x + 1 & \text { if } x \geq 3\end{aligned} \right.$

Find all critical points of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x$ , and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

The owner of an appliance store expects to sell 600 toasters this year. Each toaster costs her $8 dollars to purchase, and each time she orders a shipment of toasters, it costs $28. In addition, it costs $1 a year to store each toaster. Assuming the toasters sell out at a uniform rate and that the owner never allows herself to run out of toasters, how many toasters should be ordered in each shipment to minimize the annual cost? (Round any fractional amounts.)

The clothes dryers at a laundromat run for 45 minutes. You arrive at the laundromat and find that all of the dryers are being used. Use an appropriate uniform density function to find the probability that a dryer chosen at random will finish its cycle within 5 minutes.

Evaluate the improper integral: $\int _ { 0 } ^ { \infty } x ^ { 5 } e ^ { - x ^ { 6 } / 4 } d x$ Round to two decimal places, if necessary.

Let f (x, y) = x ln(1 + 2x - 5y). Find $f _ { x x } ( x , y )$ .

To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time required for the rat to traverse the maze on the nth trial was approximately $T ( n ) = 5 + \frac { 2 } { n } - \frac { 4 } { n ^ { 2 } }$ minutes. How many minutes does it take the rat to traverse the maze on the 2nd trial?

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1.f (x) =3x³ + 2x² - 8

If $x ^ { 2 } y + x y ^ { 2 } = 7$ , then $\frac { d y } { d x } = 2 x y + y ^ { 2 }$

Evaluate $\int \frac { \ln 3 x } { x ^ { 2 } } d x$ .

Find the indicated limit if it exists. $\lim _ { x \rightarrow 6 } \frac { x - 6 } { x ^ { 2 } - 36 }$

$f ( x ) = \left\{ \begin{array} { l l } \frac { 1 } { 2 } e ^ { - x / 2 } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$ is a probability density function for a particular random variableX $P ( X \geq 4 )$ . Use integration to find

Evaluate the following integral: $\int 7 e ^ { 4 x } d x$