Exam 10: Further Applications of the Derivative

This problem contains a function and its graph, where $A = 20$ Use the graph to determine, as well as you can, the horizontal asymptote. Check your conclusion by using the function to determine the horizontal asymptote analytically. $f ( x ) = \frac { 4 ( x - 3 ) } { x - 2 }$

Use analytic methods to find the limit as $x \rightarrow + \infty$ for the given function. $f ( x ) = \frac { 3000 x } { 3900 - 15 x }$

Use the graph $f ^ { \prime \prime }$ to sketch the graph of $f$ .

A firm has total revenue given by $R ( x ) = 300 x - 45.5 x ^ { 2 } - x ^ { 3 } \text { dollars }$ for x units of a product. Find the maximum revenue from sales of that product.

Find the limit. $\lim _ { x \rightarrow \infty } \frac { 7 x + 4 } { 8 x ^ { 2 } + 8 }$

Complete the table for the function $y = \sqrt { x }$ . Let x = 4. dx=\Deltaxdy\Deltay\Deltay-dydy/\Deltay 1.00000 0.50000 0.10000

The revenue R for a company selling x units is $R = 900 x - 0.3 x ^ { 2 }$ . Use differentials to approximate the change in revenue if sales increase from $x = 1000$ to $x = 1100$ units.

A business has a cost (in dollars) of $C = 0.8 x + 100$ for producing x units. What is the limit of $\bar { C }$ as x approaches infinity?

Analyze and sketch a graph of the function $f ( x ) = \frac { x ^ { 2 } - 4 x + 12 } { x + 2 }$ .

Complete the table for the function $y = \frac { 1 } { x }$ . Let x = 4. dx=\Deltax dy \Deltay \Deltay-dy dy/\Deltay 5.00000 2.50000 0.50000

The measurement of the edge of a cube is found to be 8 inches, with a possible error of 0.08 inch. Use differentials to estimate the propagated error in computing (a) the volume of the cube and (b) the surface area of the cube. Give your answers to two decimal places.

Find the speed v, in miles per hour, that will minimize costs on a 125-mile delivery trip. The cost per hour for fuel is $C = \frac { v ^ { 2 } } { 800 }$ dollars, and the driver is paid $W = \$ 8$ dollars per hour. (Assume there are no costs other than wages and fuel.)

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 400 square meters.

Match the function $f ( x ) = \frac { 2 x ^ { 2 } } { x ^ { 2 } + 2 }$ with one of the following graphs.

Find any vertical asymptotes for the given function. $y = \frac { 5 x - 5 } { x + 2 }$

Sketch the graph of the relation $x ^ { 2 } y = 1$ using any extrema, intercepts, symmetry, and asymptotes.

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window (see figure). Find the dimensions of a Norman window of maximum area if the total perimeter is 24 feet. Round yours answers to two decimal places.

A rancher has 320 feet of fencing to enclose two adjacent rectangular corrals (see figure). What dimensions should be used so that the enclosed area will be a maximum?

Compare dy and $\Delta y$ for $y = 3 x ^ { 4 } + 1$ at x = 1 with dx = 0.03. Give your answers to four decimal places.

A function and its graph are given. Use the graph to find the horizontal asymptotes, if they exist. Confirm your results analytically. $f ( x ) = \frac { 24 } { x + 2 }$