Exam 1: Functions, Graphs, and Limits

A farmer is planning to plant a rectangular garden with an area of 400 square yards. The garden is to be fenced on all four sides. Express the number of yards of fencing required as a function of x, the long side of the fence.

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 $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & \text { if } x \leq 2 \\x + 1 & \text { if } x > 2\end{array} \right.$

A closed box with a square base is to have a volume of 40 cubic meters. The material for the top and bottom of the box costs $2 per square meter, and the material for the sides costs $4 per square meter. Express the construction cost of the box as a function of the length of its base, x.

Plot the given point in a rectangular coordinate system.(-3, 1)

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 $f ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } & \text { if } x \leq 2 \\x + 1 & \text { if } x > 2\end{array} \right.$

Find the indicated limit if it exists. $\lim _ { x \rightarrow 1 } \frac { 2 x - 1 } { x + 3 }$

Find the indicated composite function. f (4x - 5) where $f ( x ) = \frac { 1 } { x } - x$

A ball is thrown upward in such a way that t seconds later it is $H ( t ) = - 11 t ^ { 2 } + 44 t + 55$ feet above the ground. When does the ball hit the ground?

Find $\lim _ { x \rightarrow + \infty } f ( x )$ and $\lim _ { x \rightarrow - \infty } f ( x )$ . If the limiting value is infinite indicate whether it is + $\infty$ or - $\infty$ . $f ( x ) = \frac { 5 - 3 x ^ { 3 } } { 7 x ^ { 3 } - x - 2 }$

List all values of x for which f (x) is not continuous. $f ( x ) = \frac { 2 x + 1 } { x ^ { 2 } + x }$

A manufacturer's total cost consists of a fixed overhead of $250 plus production costs of $70 per unit. Express the total cost in dollars as a function of the number of units produced.

Find $\lim _ { x \rightarrow + \infty } f ( x )$ and $\lim _ { x \rightarrow - \infty } f ( x )$ . If the limiting value is infinite indicate whether it is + $\infty$ or - $\infty$ .f (x) = (5 - 9x)(x - 2)

Find the distance between the given points. (-2, 1) and (7, -5)

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 $n ^ { t h }$ trial was approximately $T ( n ) = 8 + \frac { 4 } { n } - \frac { 8 } { n ^ { 2 } }$ minutes. How long does it take the rat to traverse the maze in the second trial?

A cylindrical can is to have a volume of 45 $\pi$ 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 5 cents per square inch. Express the cost of constructing the can as a function of its radius.

A ball is thrown upward in such a way that t seconds later, it is $s = - 16 t ^ { 2 } + 96 t + 144$ feet above the ground. Sketch the graph of s(t) and determine the maximum height in feet attained by the ball.

Compute the indicated value of the given function. f (x) = | x - 3 | + x - 5; f (2)

Determine the domain of the given function. $f ( x ) = x ^ { 4 } + 2 x ^ { 3 } + 3 x + 1$

Find the indicated one-sided limit. If the limiting value is infinite, indicate whether it is + $\infty$ or - $\infty$ . $\lim _ { x \rightarrow 5^- } f ( x )$ where $f ( x ) = \left\{ \begin{array} { c l } x ^ { 2 } & \text { if } x \leq 5 \\x + 1 & \text { if } x > 5\end{array} \right.$

As a rumor spreads across a college campus, the number of people that have heard it can be modeled by the equation $N ( t ) = \frac { 7,600 t ^ { 2 } + 3,800 t } { ( t + 3 ) ^ { 2 } }$ where t is days since the rumor started spreading. What happens to the number of people that have heard the rumor in the long run (as $t \rightarrow \infty$ )?