Exam 10: Further Applications of the Derivative

Sketch the graph of the function below. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. $y = x ^ { 4 } - 4 x ^ { 3 } + 16 x$

p is in dollars and q is the number of units. Find the elasticity of the demand function $5 p + 4 q = 152$ at the price $p = \$ 30$ .

Compare dy and $\Delta y$ for $y = 3 x ^ { 2 } - 3$ at x = 2 with $\Delta x =$ dx = -0.07. Give your answers to four decimal places.

Find the limit: $\lim _ { x \rightarrow 0 ^ { - } } \left( x ^ { 1 } + \frac { 1 } { x } \right)$

A power station is on one side of a river that is 0.5 mile wide, and a factory is 6.00 miles downstream on the other side of the river. It costs $\$$ 18 per foot to run overland power lines and $\$$ 25 per foot to run underwater power lines. Estimate the value of x that minimizes the cost.

Analytically determine the location(s) of any vertical asymptote(s). $f ( x ) = \frac { 800 x + 6000 } { x ^ { 2 } - 38 x - 2867 }$

The variable cost for the production of a calculator is $\$$ 14.25 and the initial investment is $\$$ 550,000. Use differentials to approximate the change in the cost C for a one-unit increase in production when $x = 80,000$ , where x is the number of units produced.

Analyze and sketch a graph of the function $f ( x ) = \frac { x ^ { 4 } } { x ^ { 4 } + 1 }$ .

Find the dimensions of the rectangle of maximum area bounded by the x-axis and y-axis and the graph of $y = \frac { 4 - x } { 2 }$ .

If the total revenue function for a blender is $R ( x ) = 40 x - 0.25 x ^ { 2 },$ find the maximum revenue.

Sketch the graph of the function $f ( x ) = \frac { 2 + x } { 2 - x }$ using any extrema, intercepts, symmetry, and asymptotes.

A firm can produce 100 units per week. If its total cost function is $C = 600 + 1200 x$ dollars, and its total revenue function is $R = 1300 x - x ^ { 2 }$ dollars, how many units x should it produce to maximize its profit?

Analyze and sketch a graph of the function $y = x ^ { 3 } - 12 x ^ { 2 } + 3$ .

Analyze and sketch a graph of the function $y = 2 - x - x ^ { 3 }$ .

Analytically determine the location(s) of any horizontal asymptote(s). $f ( x ) = \frac { x - 80 } { x ^ { 2 } + 1100 }$

Analyze and sketch a graph of the function $y = x \sqrt { 16 - x }$ .

The cost C (in millions of dollars) for the federal government to seize p% of a type of illegal drug as it enters the country is modeled by $C = 584 / ( 100 - p )$ , for $0 \leq p \leq 100$ . Find the limit of C as $p \rightarrow 100 ^ { - }$ .

Average costs. Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by $\bar { C } ( x ) = 7 x + \frac { 1372 } { x } , \quad x > 0$ , where x is the number of machines used. How many machines give minimum average costs?

Find any horizontal asymptotes for the given function. $y = \frac { 18 x } { 9 - x ^ { 2 } }$

Find the differential dy of the function $y = 2 x ^ { 2 } - 3 x + 2$ .