Exam 4: Using the Derivative

The following table shows cost and revenue for a product (in dollars). A. What is the price of the product? B. At what value of q is profit is maximized?

The concentration of alcohol in the blood (as a function of time since drinking a glass of wine) is most likely to be modeled by which of the following types of functions?

A. Find the marginal cost for q=110 when the fixed costs in dollars are 2,000, the variable costs are 100 per item, and each sells for $400. B. Find the marginal revenue under the same conditions.

Sketch the curve .

The demand equation for a product is , where q is the number of units produced, p is the price of each unit, and a and b are positive constants. Find the critical point(s) of the revenue function.

Write a formula for total cost, C, as a function of quantity r when fixed costs are $45,000 and and variable costs are $2,000 per item.

A brick is heated in an oven and taken out to cool off after a certain time. The temperature T of the brick at any time t is given by for , with T in degrees Celsius and t in minutes. What is the temperature of the brick when it is placed in the oven (to the nearest degree)?

Suppose the spread of a cold virus in an elementary school can by modeled by the equation ,where P is the number of children with the virus and t is time in days. A. How many children will eventually catch the virus? B. At the inflection point of the graph of P, how many children have caught the virus? C. If a different strain of the virus is found to fit the equation , will it eventually infect (1) more children, (2) fewer children, or (3) the same number of children?

An amusement park finds that when it charges $24 for an all-day pass, attendance is about 3700 per day. When it charges $27 , attendance is about 3400 per day. A. Estimate the elasticity for the amusement park to two decimal places. B. Is demand elastic or inelastic?

Assume that the polynomial f has exactly one local maximum, two local minima, and two inflection points. What is the largets number of zeros f could have?

A stone is thrown vertically upward so that its height, measured in feet, after t seconds is given by . What is the maximum height of the stone (in feet)?

The following figure represents the function f, with , and A. Determine the first two positive x-intercepts of f. Separate them by a comma. B. What must the value of a be so that ? C. If , calculate .

The function gives cost in dollars of producing r items. What is the marginal cost of increasing r by 1 item from the current production level of r=3?

The average cost per item to produce q items is given by for q>0. What is the marginal cost, MC, of producing q goods?

There is only one barber in a small town. Would you expect the elasticity of demand for the first barber to increase or decrease if a second barber moved into town?

Let C(q) represent the cost, R(q), the revenue and π(q) the profit, in dollars of producing q items. If and approximately, how much profit is earned by the item?

The distance, s, traveled by a runner in a 20 mile race is given in the following figure, where time, t, is in hours. At which of the following values of t is the runner's speed is the slowest?

The drug concentration curve for a drug after t hours is given by ng/ml. The minimum effective concentration is 10 ng/ml. Is the drug effective at t = 10 hours?

A youth group wishes to hold a car wash as a fundraiser. Through past experience with car washes, they have constructed the following table, which shows the price, p, charged for a car wash and the quantity, q, of cars washed at that price. A. At what price is revenue maximized? B. What is the elasticity at that price?

The demand for doughnuts at a bakery is given by , where q is the number of doughnuts sold at a price of p dollars each. A. Find the elasticity of demand to two decimal places if the price is $0.80. B. Will revenue be increased by raising or lowering the price?