Exam 2: Functions, Limits and the Derivative

Complete the table by computing at the given values of x.Use these results to estimate (if it exists).

Let f be the function defined by Find and = __________ = __________ = __________ = __________

Determine whether the statement is true or false. If f is a function, then f (a + b) = f (a) + f (b).

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. f(x) = -10 = __________

Use the graph of the function f to find and if the limit exists.

Find functions f and g such that .

Use the graph of the function f to find and if the limit exists.If it does not, answer none. __________ __________ __________

Productivity and yield of cultivated crops are often reduced by insect pests.The following graph shows the relationship between the yield of a certain crop, , as a function of the density of aphids x.(Aphids are small insects that suck plant juices.) Here, is measured in kilograms/ 4,000 square meters, and x is measured in hundreds of aphids/bean stem.By computing the slopes of the respective tangent lines, estimate the rate of change of the crop yield with respect to the density of aphids when that density is 200 aphids/bean stem and when it is 800 aphids/bean stem.

Find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line.

Suppliers of transistor radios will market 10,000 units when the unit price is $55 and 62,500 units when the unit price is $100.Determine the supply function if it is known to have the form where x is the quantity supplied and p is the unit price in dollars.

During the construction of a high-rise building, a worker accidentally dropped his portable electric screwdriver from a height of 400 ft.After t sec, the screwdriver had fallen a distance of ft. How long did it take the screwdriver to reach the ground? __________ sec What was the average velocity of the screwdriver between the time it was dropped and the time it hit the ground? __________ ft/sec What was the velocity of the screwdriver at the time it hit the ground? __________ ft/sec

Determine whether the equation defines y as a linear function of x.

The total worldwide box-office receipts for a long-running blockbuster movie are approximated by the function where is measured in millions of dollars and is the number of months since the movie's release. What are the total box-office receipts after the first month? The second month? The third month? Round the answer to the nearest tenth, if necessary. What will the movie gross in the long run (when is very large)? $ __________ million

Determine all values of x at which the function is discontinuous. __________

For the pair of supply and demand equations where x represents the quantity demanded in units of a thousand and p the unit price in dollars, find the equilibrium quantity and the equilibrium price. Equilibrium quantity: __________ thousand units Equilibrium price: $ __________

The owner of a luxury motor yacht that sails among the 4,000 Greek islands charges $800/person/day if exactly 20 people sign up for the cruise.However, if more than 20 people sign up (up to the maximum capacity of 90) for the cruise, then each fare is reduced by $4 for each additional passenger.Assume at least 20 people sign up for the cruise and let x denote the number of passengers above 20.Find a function R giving the revenue/day realized from the charter.

Find function f such that and

Determine all values of x at which the function is discontinuous.

A study of driving costs of 1992 model subcompact (four-cylinder) cars found that the average cost (car payments, gas, insurance, upkeep, and depreciation), measured in cents/mile, is approximated by the function where x denotes the number of miles (in thousands) the car is driven in a year.What happens to the average cost as the number of miles driven increases without bound? Express your answer to the nearest tenth, if necessary. It approaches __________ .

Let . Find the zeros of f in by solving the equation .