Emerging Scholars Calculus I Sample Test Problems

  1. Below is the graph of f(x). You may assume the whole graph is pictured.

    1. For which values of x is f negative?
    2. For which values of x is f increasing?
    3. For which values of x is f concave up?
    4. For which values a  of x does exist?
    5. For which values a  of x is f continuous?
    6. For which values a  of x is f differentiable?
    7. Sketch the graph of y = -2f(x+3) -4.
    8. Sketch the graph of f'(x).
    9. Sketch the graph of any antiderivative of f(x).
  2. Evaluate the limit

    numerically, graphically, and algebraically.
  3. Water is pouring into a tank at the following rates, in gal/hr, where t is measured in minutes:
    Time 0 10 25 37 46 60
    Rate 500 455 350 280 200 180

    Give upper and lower bounds for the amount of water in the tank after one hour.

  4. Approximate the integral
     
    to within an error of 0.1. Show all your preliminary work, and also show how you used your calculator output.
  5. Find the derivatives of each of the following functions by using the definition. (No shortcuts allowed!)


  6. For the function

    1. Find f', and use it to determine when f is increasing and when it is decreasing.
    2. Find f'', and use it to determine when f is concave up and when it is concave down.
    3. Evaluate the end behavior of h(x) algebraically (this should involve limits!)
    4. Use all of the above to sketch a graph of f which includes all aspects of the basic graph.
  7. Gaspar the dog is at it again! This time, he starts at home, and runs out to the highway. At some point, he sees the dog catcher, and responds appropriately. Anyways, the graph below is that of is velocity, in mi/hr, as he is running. The time t is measured in minutes, and the only curvy part of the graph is a section of a circle:

    1. At what time t is Gaspar's acceleration the greatest?
    2. How far does Gaspar run before he sees the dogcatcher?
    3. What is Gaspar's average velocity?
    4. What is Gaspar's average speed?
    5. What, approximately, is Gaspar's average acceleration?
    6. Does Gaspar make it back home?
  8. Evaluate the following antiderivatives:





  9. If F(t) is the function,

    when does F(t) achieve its global minimum on the interval [-1,3]?
  10. An oil tanker is leaking oil at the rate of R(t) = 2000e-0.2t gallons/hour. How much total oil has leaked out after 10 hours?
  11. A driver is driving at 100 mph, when he sees a police car flashing 400 ft away. He figures that if he can slow down to 70 mph in 200 ft, he won't get a ticket. If his car's brakes decelerate his speed at the rate of -10 ft/sec/sec, will he get a ticket? (You may want to use the fact that 60 mph = 88 ft/sec.)

The End!