Calculus
AROC
1. Given f(x) = x2 – 3x –10
a. What is the average rate of change from x = 2 to x = 7?
b. f(4) = __________
c. For what value of p is the average rate of change from x = p to x = 4 zero?
2. Today I have 1 frog. Each day I get 3 more frogs.
a. complete the table below:
days after today: 0
1
2
3
4
frogs:
b. Is the daily rate of change constant?
c. What is it?
d. What is the model for this situation?
(Define f as a function of the number of days since today)
e. Is the rate of change of the rate of change constant?
f. What is it?
3. Today I have 1 rabbit. Each day the number of rabbits I have doubles.
a. complete the table below:
days after today: 0
1
2
3
rabbits:
b. Is the daily rate of change constant?
c. What is it?
d. What is the model for this situation?
(Define r as a function of the number of days since today)
4
4. The height of a ball Barbara tosses is modeled by h(t) = 80t – 16t2.
(h is in feet, t is in seconds)
a. evaluate h(2)
What does this number represent?
b. At what time(s) was the ball on the ground?
c. At what time did the ball reach its maximum height?
What was that height?
d. Sketch:
e. Between 1 second and 3 seconds what was the ball’s average speed?
f. Between 2 seconds and _____ seconds the ball’s average speed was 0 ft/sec.
5. Find the average rate of change of g(x) = 3 x over the interval [2.5 , 4.5]
6. A baker finds that the cost of operating his bakery is given by c(x) = x2 ! 35x + 400
where C(x) is the daily cost in dollars to make x batches of bread.
a. How many batches should he make to minimize his cost?
b. What is that minimum cost?
c. What does the y-intercept of the cost function represent?
d. What is the cost per batch of bread if he produces 25 batches?
e. Suppose that the baker charges $8 per batch of bread, and therefore the amount of
revenue received is given by the function R(x) = 8x. What's the fewest number of batches
of bread he should bake to turn a profit? Will he make a profit for all numbers of batches
of bread above that number?
7. Suppose a car is driven at a constant rate of 30 mph. Sketch a graph of the distance the car
has traveled as a function of time.
8. A commuter drives to work and encounters delays at two red lights. Sketch a graph of the
distance the car has traveled as a function of time.
9. Let G(t ) = 400(15 ! t)2 be the number of gallons of water in a cistern, t minutes after an
outlet pipe is opened. Find the average rate of drainage during the first 5 minutes.
10. The graph below shows the distance traveled by a car over the first two hours of a trip.
a. Find the average velocity of the car for the entire trip.
b. Find the average velocity of the car in the second hour of the trip.
c. Is the car slowing down or speeding up during the first half hour of the trip?
d. Estimate the instantaneous velocity of the car at t=1 hour.
e. Estimate when the instantaneous velocity of the car was the same as the average velocity
for the entire trip.