The distance in miles of a car from its starting point after t hours is given by the formulaf(t)=40t.

(a) What is the average rate of change off(t)betweent=4andt=6

(b) What is the average rate of change off(t)betweent=4andt=5

(c) What is the average rate of change off(t)betweent=4andt=4.1

(d) On graph paper sketch y=f(t) for0t4. What is the slope of your graph?

(e) What are the units of the slope?

A.Miles B.Square miles C.Hours D.Miles per hour E.Hours per mile F.Miles per hour squared

3.A tanker ship is off-loading its oil tank at a terminal. It takes 40 minutes to fill a 700 gallon tank. What is the average flow rate of oil into the truck in gallons per hour?

4.A car drives at 50 miles per hour for 1 hour then at 70 miles per hour for the next 30 minutes and then is stationary for 30 minutes. On a piece of graph paper, sketch a graph showing distance traveled against time taken. What was the average speed over the entire two hours?

6.Suppose that an object is moving along the ground and that the distance (in centimeters) it has travelled after t seconds is given by the formulaf(t)=t2cm. Calculate the average speed during the 1 millionth of a second after t=2.

8.

The temperature at t hours after noon is given by the function f(t) in degrees Fahrenheit.

(a) What are the units off(t) A.Degrees Fahrenheit squared B.Degrees Fahrenheit per hour C.Degrees Fahrenheit D.Hours per Degree Fahrenheit E.Degrees Fahrenheit per second F.Hours G.Seconds per degree Fahrenheit

(b) What is the meaning of the statement thatf(7)=3 A.The temperature is decreasing at a rate of 7 degrees Fahrenheit every 3 hours. B.At 7pm the temperature is decreasing at a rate of 3 degrees Fahrenheit per hour. C.The temperature at 8pm was three degrees lower than the temperature at 7pm. D.At 3pm the temperature is decreasing at a rate of 7 degrees Fahrenheit per hour. E.The temperature is decreasing at a rate of 3 degrees Fahrenheit every 7 hours.

(c) What is the significance off(t)being positive A.The temperature is increasing t hours after noon. B.The temperature is above boiling t hours after noon. C.Time t is after noon. D.The temperature is above freezing t hours after noon.

9.

The total profit that a company has made measured in millions of dollars isp(t)where t is the time measured in years with year zero corresponding to 1990.

(a) What is the meaning ofp(5)=0 A.The rate at which the company was making money was not changing at the start of 1995. B.The company had a total profit of 5 million dollars at the start of 1990. C.The company's total profit was 0 at the start of 1995. D.The company neither made not lost money during 1995. E.The company's total profit was decreasing by 5 million dollars a year at the start of 1990. F.The company was neither making nor losing money at the start of 1995.

(b) What are the units ofp(t) A.Years per million dollars B.Years C.Millions of dollars per year D.Millions of dollars per year per year E.Millions of dollars

(c) What does it mean ifp(t)is positive? A.The rate at which the company is making money is increasing at time t. B.The company is not in debt at time t. C.The company is making money at time t. D.Time t is after 1990.

10.

The population (in millions) of a certain country t years after 1900 is given by the functionp(t). There were 480 million in 1900. Ifp(t)=7throughout the time span 1900 to 1910, what was the population of the country in 1907? (Hint: What was the practical signifigance ofp(t)=7)

12.

The volume (inm3) of water in my (large) bathtub when I pull out the plug is given byf(t)=4t2(t is in minutes). This formula is only valid for the two minutes it takes my bath to drain.

(a) Find the average rate the water leaves my tub betweent=1andt=2 (b) Find the average rate the water leaves my tub betweent=1andt=1.1 (c) What would you guess is the exact rate water leaves my tub at t=1 (d) In this bit h is a very small number. Find the average rate the water leaves my tub betweent=1andt=1+h(simplify as much as possible) (e) What do you get if you put inh=0in the answer to (d)?

14.

The volume (inm3) of water in my (large) bathtub when I pull out the plug is given byf(t)=9t2(t is in minutes). This formula is only valid for the 3 minutes it takes my bath to drain.

(a) Find the average rate the water LEAVES my tub between t=1 and t=2. m3/min

(b) Find the average rate the water leaves my tub between t=1 and t=1.06. m3/min

(c) What would you guess is the exact rate water leaves my tub at t=1. m3/min

(d) In this bit h is a very small number. Find the average rate the water leaves my tub between t=1 and t=1+h. (simplify as much as possible) m3/min

(e) What do you get if you put in h=0 in the answer to (d)? m

15.

An ice cube is melting. The mass of the ice cube after t minutes is m(t) grams. You are told that the rate of change of m(t) is -3 grams/min

(a) How much mass does the ice cube lose in 5 minutes? grams

(b) If the ice cube starts out with a mass of 70 grams how long until it has all melted? minutes

16.

cceleration is the rate of change of velocity. The velocity (positive means down) of a pumpkin thrown off the top of Cheadle hall t seconds after launch is 32t ft/sec (until it hits the ground).

(a) What is the average rate of change of velocity between t=1 and t=2? feet per second per second

(b) What is the average rate of change of velocity between t=1 and t=1.1? feet per second per second

(c) If the pumpkin lands after 2 seconds, what is the speed of the pumpkin when it hits? feet per second

17.

The functionf(t)=2t2.

(a) What is the average rate of change of f(t) between t=1 and t=1.001?

(b) What is the average rate of change of f(t) between t=2 and t=2.001?

(c) What is the average rate of change of f(t) between t=3 and t=3.001?

(d) What is the average rate of change of f(t) between t=4 and t=4.001?

19.

For this question refer to the graph for question 7 on page 128 of your textbook.

(a) On what intervals is the derivative positive (use I for infinity and -I for negative infinity, and enter the leftmost interval first)?

(b) Where is the derivative zero (enter the smaller number first)? x= x=

22.

Let A(r) be the area in square meters enclosed by a circle of radius r meters.

(a) What are the units ofA(r) A.meters per minute B.cubic meters C.meters D.square meters

(b) What is the meaning of the statement thatA(3)=6 A.The area increases by 6square meters whenever the radius is increased by three meters. B.When the radius is 3 meters the area is increasing at a rate of 6square meters per meter of radius. C.When the radius is 6meters the area is increasing at a rate of 3 square meters per meter of radius. D.The area increases by 3 square meters whenever the radius is increased by 6meters. E.When the radius is 3 meters the area is increasing at a rate of 6square meters per three meters of radius. F.The area increases by 3 square meters whenever the radius is increased by one meter. G.The area increases by 6square meters whenever the radius is increased by one meter.

23.

The perimeter of a rectangle is28cm. If the area of the rectangle is40cm2find the length and width of the rectangle.

Let the length be the smaller of the two numbers, and the width the larger.

length =cm width =cm

24.

A tourist travels 1500 miles using two planes. The second plane averages 50 miles per hour faster than the first plane. The tourist uses the slower plane for the first 500 and the faster plane for the next 1000 miles. The total flying time is 6.5 hours. What is the speed of the first plane?

Speed of first plane =mph

25.

It takes George 1 hour longer to mow the lawn than it takes Henry. Working together, using two mowers, they can mow the lawn in 1 hour and 12 minutes. How long would it take Henry to mow the lawn by himself? [hint: Letxbe the time taken by George. How much of the lawn does George mow in 1 hour? How much does Henry do in 1 hour? How much do they mow together in 1 hour?]

Time it takes Henry =hours

26.

The sum of the reciprocals of two consecutive odd integers is 32/255. Find the two integers.

Smaller number = Larger number =

27.

Let A,B,C and D be four points on thex-axiswith coordinates 3,x, -3, and 20 respectively. If the distance between A and C is the same as the distance between B and D, find the two possible values for the coordinatexof B.

Smaller value of x = Larger value of x =

28.

Let A,B,C and D be four points on thex-axiswith coordinates 3,x, -3, and 20 respectively. If the distance between A and C is the same as the distance between B and D, find the two possible values for the coordinatexof B.

Smaller value of x = Larger value of x =

29.

A swimming pool can be filled by either or both of two pipes of different diameters. It takes the smaller pipe twice the time that the larger pipe takes to fill the pool. If water flows through both pipes it take213hours to fill the pool. How long will it take the larger pipe to fill the pool? [hint: let x be the amount of the pool filled by the small pipe in one hour]

Time =hours

30.

Car A and car B both travel 120 miles. Car A drove at an average speed of 10 miles per hour faster than car B and made the trip in one hour less time. Find the average speeds of both cars.

Car A =mph Car B =mph

31.

Four tasks, A, B, C, and D, must be performed. They take 6 hours, 4 hours, 3 hours, and 2 hours respectively. Tasks can be performed simultaneously but there are certain constraints. Task B cannot be started until half of task A has been completed. Task C cannot be started until 7 hours in total have been spent on task A and B combined. An hour before task C ends, task D is started. When is the soonest task D can be completed? Draw a suitable diagram showing the time period each task takes.

D ends after_______ hours.

32.

It is required to make 8 grams of certain chemical compound called Z. This is made from compounds W, X, and Y in the ratio of 2:1:3. The compound Y is itself made from W and X. To make 6 grams of Y requires 4 grams of W and 2 grams of X. How much W and X is required to make the required amount of Z.

W = ______grams X = ________ grams

33.

There are 3 beakers each of which contains saline solution. Beaker A initially contains 3 liters of 15\% salt solution. Beaker B initially contains 2 liters of 30\% salt solution. Beaker C initially contains 4 liters of 0\% salt solution. Two liters are transferred from A to B and the result is thoroughly mixed. Then one liter is transferred from B to C and the result mixed. Finally two liters are transferred from C back to A. What is the percentage concetration of salt in A after all this?

Percentage concentration of salt in A =_________ \%

34.

The total number of people at a football game was 5600. Field-side tickets were 40 dollars and end-zone tickets were 20 dollars. If the total amount of money received for the tickets was 140000 how many of each kind of ticket were sold.

Field-side = End-zone =

35.

10 adults and 5 children can watch a movie for 150 dollars. But 8 adults and 10 children can watch the movie for 156 dollars. How much would it cost for 6 adults and 10 children to watch the movie?

Cost = ______dollars

36.

What amounts of 45% pure silver and 50% pure silver should be mixed to obtain 14 grams of 49% pure silver?

45% silver = _______grams 50% silver = _______grams

37.

A tank initially contains 800 liters of pure water. Then water containing 6 mg of detergent per liter starts to enter the tank at the rate of 30 liters per hour.

(a) How long until the average concentration of detergent in the tank is 2 mg per liter?hours (b) How long until the average concentration of detergent in the tank is x mg per liter?hours (use 'x' in your equation) (c) Sketch a graph showing the function you obtained in (b). Put x on the horizontal axis and t on the vertical axis. (d) What does your answer to part (b) give when x = 7.hours Do you notice anything strange? Can you explain this?

38.

If 1000 liters of water with an unknown chlorine content are combined with 500 liters of water with 45 ppm (parts per million) of chlorine the result is 65 ppm chlorine. What is the concentration of chlorine in the unknown water?

Concentration = _______ppm

39.

Car A starts in Sacramento at 11am. It travels along 400 mile route to Los Angeles at 30 mph. Car B starts from Los Angeles at noon and travels to Sacramento along the same route at 75 mph. The route goes past Fresno which is 150 miles along the route from Los Angeles. How far from Fresno are the cars when they meet?

Distance = _______ miles

40.

A railroad and a highway intersect at right angles. A train is 10 miles from the intersection at the same moment that a car is 6 miles from the intersection. Both are traveling at 30 mph. How long until they are 4 miles apart?

There are two answers: First time = ________minutes Second time = ________minutes

41.

Express the length of the perimeter of a circle in terms of the area of the circle.

Use A for the area in your answer. Perimeter =

42.

A box with rectangular sides has width twice the length of the base. The volume is 24 cubic inches and the total surface area of all six sides is 52 square inches. Write down a (cubic) equation whose solution is the length of the box.

Use L for the length of the box. Equation: 0 =