Exam 1: Functions, Graphs, and Limits

At a certain factory, the total cost of manufacturing q units during the daily production run is $C ( q ) = q ^ { 2 } + 2 q + 297$ dollars. On a typical workday, q(t) = 17t units are manufactured during the first t hours of a production run. How many dollars are spent during the first 3 hours of production?

Between August 29 and September 4, 2005, the price of gasoline increased steadily due to the impact of hurricane Katrina. The average price per gallon in Cincinnati on August 29 was $2.60, and on September 4 if was $3.10. Write a linear function C(x) that describes the cost per gallon of gas in Cincinnati in terms of days after August 29. Round the slope to three decimal places, if necessary. Use your function to find how much gas would have cost on October 1 if the price had continued to increase at that rate.

Find the indicated composite function.f (x + 6) where $f ( x ) = \frac { 1 } { x }$

Compute the indicated value of the given function. $f ( t ) = \left\{ \begin{aligned}- 2 t + 8 & \text { if } t < - 1 \\t ^ { 2 } + 9 & \text { if } t \geq - 1\end{aligned} \right.$ ; f (-3)

Plot the given point in a rectangular coordinate system.(1, -5)

Complete the table by evaluating f (x) at the specified values of x. Then use the table to estimate the indicated limit, or show that it does not exist. $f ( x ) = 3 x ^ { 2 } - 6 x , \lim _ { x \rightarrow 4 } f ( x )$ x 3.9 3.99 3.999 4 4.001 4.01 4.1 f(x) \times

Find the indicated one-sided limit. If the limiting value is infinite, indicate whether it is + $\infty$ or - $\infty$ . $\lim _ { x \rightarrow 2 ^ { - } } f ( x )$ where $f ( x ) = \left\{ \begin{aligned}x + 5 & \text { if } x < 2 \\x ^ { 2 } & \text { if } x \geq 2\end{aligned} \right.$

Complete the table by evaluating f (x) at the specified values of x. Then use the table to estimate the indicated limit, or show that it does not exist. $f ( x ) = \frac { 3 x ^ { 2 } - 48 } { x + 4 } ; \lim _ { x \rightarrow - 4 } f ( x )$ x -4.1 -4.01 -4.001 -4 -3.999 -3.99 -3.9 f(x) \times

Find the points of intersection (if any) of the given pair of curves. $y = x ^ { 2 }$ and y = 3x - 2.

The supply and demand functions, S(x) and D(x), respectively, for a certain commodity are given in terms of the level of production x. S(x) = 2x + 30, D(x) = 360 - x. Find the value of x_e for which equilibrium occurs.

A ball is thrown upward in such a way that t seconds later it is $H ( t ) = - 16 t ^ { 2 } + 64 t + 80$ feet above the ground. When does the ball hit the ground?

Complete the table by evaluating f (x) at the specified values of x. Then use the table to estimate the indicated limit, or show that it does not exist. $f ( x ) = 2 x ^ { 2 } - 9 x , \quad \lim _ { x \rightarrow 5 } f ( x )$ x 4.9 4.99 4.999 5 5.001 5.01 5.1 f(x) \times

List all values of x for which f (x) is not continuous. $f ( x ) = \frac { 5 x - 10 } { x - 2 }$

A cylindrical can is to have a volume of 36 $\pi$ cubic inches. The cost of the material used for the top and bottom of the can is 4 cents per square inch, and the cost of the material used for the curved side is 3 cents per square inch. Express the cost in cents of constructing the can as a function of its radius.

Find the difference quotient, $\frac { f ( x + h ) - f ( x ) } { h }$ . $f ( x ) = \frac { 5 } { x }$

Find the slope and y-intercept of the line 3y - 7x = 1.

Compute the indicated value of the given function. $f ( t ) = \left\{ \begin{aligned}- 4 t + 5 & \text { if } t < - 1 \\t ^ { 2 } + 6 & \text { if } t \geq - 1\end{aligned} \right.$ ; f (-3)

Find the indicated limit if it exists. $\lim _ { x \rightarrow 1 } \frac { 2 x - 1 } { x + 4 }$

Find the points of intersection (if any) of the given pair of curves.y = 3x - 1 and y = 3 - 5x

Two car rental agencies are competing. One agency rents cars for 35 dollars per day and 35 cents a mile; the other agency rents cars for 30 dollars per day and 50 cents a mile. For a 9 day trip, how many miles must you travel to have the total cost be the same with each agency? Round to the nearest whole mile , if necessary.