Exam 2: Rate of Change: the Derivative

The cost of mining a ton of coal is rising faster every year. Suppose is the cost of mining a ton of coal at time t. Which of the following must be concave up? Select all that apply.

The function is graphed below. Which is larger, or ?

Let represent the number of students enrolled in school in the year t. If enrollment is decreasing steadily, then _____0 and _____0. (Enter "<",">", or "=")

Cost and revenue functions for a certain chemical manufacturer are given in the following figure. Should the company increase production beyond 25 tons?

To produce 250 items the total cost is $4700 and the marginal cost is $15. Which estimate is more likely to be accurate, one for producing 251 items, or one for producing 500 items?

Given the following data about the function f, the equation of the tangent line at x=3.2 is approximately y = _____x+_____. Use the nearest right-hand value to make your estimate.

Assume that f and g are differentiable functions defined on all of the real line. f and g can satisfy: for all x and for all x.

A certain bacterial colony was observed for several hours and the following conditions were reported. Let be the number of bacteria present after t hours. There were 1000 bacteria after 5 hours. The growth rate was never negative and never exceeded 100 per hour. The growth rate was decreasing for the first 5 hours. At 7 hours, the growth rate was zero. Is it possible that ?

To study traffic flow along a major road, the city installs a device at the edge of the road at 4:00 am. The device counts the cars driving past, and records the total periodically. The resulting data is plotted on a graph, with time (in hours) on the horizontal axis and the number of cars on the vertical axis. The graph is shown below. It is a graph of the function = Total number of cars that have passed by after t hours. When is the traffic flow the greatest?

The cost of mining a ton of coal is rising faster every year. Suppose is the cost of mining a ton of coal at time t. Which of the following must be increasing? Select all that apply.

The following table gives the cost and revenue, in dollars, for different production levels, q. What are the fixed costs?

Consider the two functions shown below. A. B.

A certain bacterial colony was observed for several hours and the following conditions were reported. Let be the number of bacteria present after t hours. There were 1000 bacteria after 5 hours. The growth rate was never negative and never exceeded 100 per hour. The growth rate was decreasing for the first 5 hours. At 7 hours, the growth rate was zero. Is it possible that ?

Recently Esther swam a lap in an Olympic swimming pool (the length of the pool is 50 meters, and the length of a lap is 100 meters); her times for various positions s (in meters from her starting point) during the lap are given in the following table. Her approximate velocity at time t=3.2 seconds was _____ m/sec. Round to 3 decimal places.

The graph of is shown in the following figure. Give an estimate for

Sketch a graph with the following conditions: and .

Consider the function f sketched in the following figure. Do you expect to be positive, negative, or zero?

Given the following data about the function f, give the average rate of change of f between x=3.2 and x=3.8. Round to 2 decimal places.

The following table shows the number of oranges sold in one month, , against the price per bag, p (in cents). Find an approximation for . Use the nearest right-hand value to make your estimate.

Cost and revenue functions for a certain chemical manufacturer are given in the following figure. Marginal revenue at 20 tons is about how much?