Exam 12: Applications of the Derivative

The estimated monthly sales of Mona Lisa paint-by-number sets is given by the formula $q = 120 e ^ { p - \frac { 5 p ^ { 2 } } { 2 } }$ Where q is the demand in monthly sales and p is the retail price in yen. At what price will revenue be a maximum Round the answer to the nearest hundredth.

Find the coordinates of all relative and absolute extrema. ​ $g ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } - 27 }$ ​ ​

Find the exact location of all the relative and absolute extrema of the function. $f ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } - 48 }$ Select all correct answers.

A general hyperbolic demand function has the form to follow. $q = \frac { m } { p ^ { z } }$ (s, m - nonzero constants) Obtain a formula for the elasticity of demand at a unit price of p.

A point is moving along the circle $x ^ { 2 } + ( y - 3 ) ^ { 2 } = 5$ In such a way that its x-coordinate is decreasing at a rate of 2 units per second. At what rate is the y-coordinate decreasing at the instant when the point has reached $( - 2,4 )$

Hercules films is deciding on the price of the video release of its film "Son of Frankenstein". Its marketing people estimate that at a price of p dollars, it can sell a total of $q = 10,000 - 200 p$ copies. What price will bring in the greatest revenue Round your answer to the nearest cent

Find the coordinates of all relative and absolute extrema. $g ( x ) = \frac { x ^ { 3 } } { x ^ { 2 } - 48 }$

Find the exact location of all absolute extrema of the function $f ( x ) = x ^ { 3 } - 12 x$ with domain $[ - 7,7 ]$ . Select all correct answers.

The fruit yield per tree in an orchard that contains 60 trees is 240 pounds per tree each year. Due to crowding, the yield decreases by 2 pounds per season per every additional tree planted. How many additional trees should be planted for a maximum total annual yield ?

You can now sell 120 cups of lemonade at 40 cents per cup, but demand is dropping at a rate of 6 cups per week each week. Assuming that raising the price doesn't affect the demand, how fast do you have to raise your price if you want to keep the revenue constant ?

Find the exact location of all the relative and absolute extrema of the function with domain $[ - 1,1 ]$ . Round your answers to two decimal places. $f ( x ) = e ^ { 3 x } - 3 x$ Select all correct answers.

You manage a small antique store that owns a collection of Louis XVI jewelry boxes. Their value v is increasing according to the formula $v = \frac { 12,000 } { 1 + 500 e ^ { - 0.5 t } }$ Where t is the number of years from now. You anticipate an inflation rate of 1% per year, so that the present value of an item that will be worth $v in t years' time is given by $p = v ( 1.01 ) ^ { - t }$ In how many years from now will the greatest rate of increase of the present value of your antiques be attained Round your answer to two decimal places.

Find the exact location of all the relative and absolute extrema of the function with domain $( - \infty , \infty )$ . $f ( x ) = e ^ { - 5 x ^ { 2 } }$

Demand for your tie-dyed T-shirts is given by the formula ? $p = 5 + \frac { 100 } { \sqrt { q } }$ Where p is the price in dollars you can charge to sell q T-shirts per month. If you currently sell T-shirts for $10 each and you raise price by $1 each month, how fast will the demand drop

Find the exact location of all the absolute extrema of the function with domain $( - 7 , \infty )$ . ? $f ( x ) = x ^ { 4 } - 8 x ^ { 3 }$

Calculate $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ . $y = - 14 x ^ { 2 } + 15 x$

Combined SAT scores in the United States could be approximated by $T ( t ) = - 0.0101 t ^ { 3 } + 0.5856 t ^ { 2 } - 9.6 t + 1038$ $( 0 \leq t \leq 23 )$ In the years 1967 - 1990. Here t is the number of years since 1967, and T is the combined SAT score average for the United States. Find all points of inflection of the graph of T, and interpret the result. Round coordinates to two decimal places if necessary.

Find the exact location of all the relative and absolute extrema of the function. $f ( x ) = \frac { x ^ { 2 } - 32 } { x + 6 }$ Select all correct answers.

The population P is currently 20,000 and growing at a rate of 1,000 per year. What is the mathematical notation for the rate ?

I want to fence in a rectangular vegetable patch. The fencing for the east and west sides costs $4 per foot, while the fencing for the north and south sides costs only $3 per foot. I have a budget of $120 for the project. What is the largest area I can enclose ?