Exam 12: Applications of the Derivative

A general linear demand function has the form to follow. $q = k p + b$ (k and b constants, with $k \neq 0$ ) Obtain a formula for the elasticity of the demand at a unit price p.

Minimize $F = x ^ { 2 } + y ^ { 2 }$ with $x + 2 y = 20$ .

The weekly sales in Honolulu Red Oranges is given by the following equation. $q = 1,070 - 24 p$ Calculate the elasticity of demand for a price of $27 per orange. Round your answer to the two decimal places.

Assume that the demand function for tuna in a small coastal town is given by $p = \frac { 50,000 } { q ^ { 1.5 } }$ Where q is the number of pounds of tuna that can be sold in 1 month at the price of p dollars per pound. The town's fishery finds that the demand for tuna is currently 900 pounds per month and is increasing at a rate of 200 pounds per month. How fast is the price changing Round your answer to the three decimal places.

There are presently $N = 300$ cases of Bangkok flu, and the number is growing by 40 new cases every month. Rewrite the rate in mathematical notation.

The consumer demand curve for tissues is given by $q = ( 72 - p ) ^ { 2 }$ Where p is the price per case of tissues and q is the demand in weekly sales. At what price should tissues be sold to maximize the revenue Round your answer to the nearest cent.

The radius of a circular puddle is growing at a rate of $8 \frac { \mathrm { cm } } { \mathrm { s } }$ . How fast is the area growing at the instant when it equals 20 cm² Round your answer to the nearest whole number.

The graph of the second derivative, $f ^ { \prime \prime } ( x )$ , is given. Determine the x-coordinates of all points of inflection of f(x), if any. (Assume that f(x) is defined and continuous everywhere in $[ - 3,3 ]$ .)

As a new owner of the supermarket, you have inherited a large inventory of unsold imported Limburger cheese, and you would like to price it so that your revenue from selling it is as great as possible. Previous sale figures of the cheese are shown in the following table. Price per pound, \ 3.00 \ 4.00 \ 5.00 Monthly sales (pounds), q 407 301 223 Use the sale figures for the $3 and $5 per pound prices to construct a demand function of the form $q = A e ^ { - b p }$ where A and b are constants you must determine. (The values of A, b were rounded.)

Find the exact location of all the relative and absolute extrema of the function. $f ( x ) = \frac { x ^ { 2 } + 81 } { x ^ { 2 } - 81 } ; - 86 \leq x \leq 86 , x \neq \pm 9$

Combined SAT scores in the United States could be approximated by ​ $T ( t ) = - 0.01774 t ^ { 3 } + 0.5928 t ^ { 2 } - 10.91 t + 1025$ ​ in the years 1964 - 1986. Here t is the number of years since 1964, 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. ​ Enter your answers rounded to the nearest hundredth.

You have been hired as a marketing consultant to Johannesburg Burger Supply, Inc., and you wish to come up with a unit price for its hamburgers in order to maximize weekly revenue. To make life as simple as possible, you assume that the demand equation for Johannesburg hamburgers has the linear form $q = m p + b$ Where p is the price per hamburger, q is the demand in weekly sales, and m and b constants are certain constants you must determine. Your market studies reveal the following sales figures: when the price is set at $4 per hamburger, the sales amount to 5,920 per week, but when the price is set at $8 per hamburger, the sales drop to zero. Now estimate the unit price in order to maximize annual revenue and predict what the weekly revenue will be at that price. Round your answers to the nearest cent.

The FeatureRich Software Company sells its graphing program, Dogwood, with a volume discount. If a customer buys x copies, then that customer pays $600 \sqrt { x }$ . It costs the company $30,000 to develop the program and $5 to manufacture each copy. If just one customer were to buy all the copies of Dogwood, how many copies would the customer have to buy for FeatureRich Software's average profit per copy to be maximized

Find the exact location of all relative and absolute extrema of the function $f ( x ) = 8 x ^ { 2 } + 48 x + 9$ with domain $[ - 9,9 ]$ .

The graph of a function $f ( x ) = x ^ { 2 } \left( x ^ { 2 } - 24 \right)$ is given. ​ ​ Find the coordinates of all points of inflection of this function (if any). ​ Please enter your answer as ordered pairs in the form (x, y) separated by commas.

You have been hired as a marketing consultant to Johannesburg Burger Supply, Inc., and you wish to come up with a unit price for its hamburgers in order to maximize weekly revenue. To make life as simple as possible, you assume that the demand equation for Johannesburg hamburgers has the linear form $q = m p + b$ Where p is the price per hamburger, q is the demand in weekly sales, and m and b constants are certain constants you must determine. Your market studies reveal the following sales figures: when the price is set at $4 per hamburger, the sales amount to 1,420 per week, but when the price is set at $5 per hamburger, the sales drop to zero. Use these data to calculate the demand equation.

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

Find the exact location of all the relative and absolute extrema of the function with domain $( - \infty , \infty )$ . $g ( x ) = \frac { 1 } { 4 } x ^ { 4 } - \frac { 2 } { 3 } x ^ { 3 } + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 }$

Locate all maxima in the graph. Select all correct answers.

A general quadratic demand function has the form to follow. $q = c p ^ { 2 } + l p + r$ (c, l, r are constants, $l \neq 0$ ) Obtain a formula for the elasticity of demand at a unit price p.