Rates of Change and Limits Homework

Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet.

(1) Estimate the limits, if they exist, by using a table of values to two decimal places:

    (a)         (b)     (c)

(2)  Sketch the graph of and Then identify the values of for which and exist given

    (a)     (b)

(3) Find so that the function satisfies

(4) Estimate the limits by using tables of values for

    (a)           (b)

Then using long division (or synthetic division if you know it) explain why one of the limits exists and the other does not.

(5) Consider the function Estimate by evaluating at -values near 0. Sketch the graph of

(6) Explain why does not exist.

(7) Evaluate the function for and Guess the value of Evaluate the function for and Guess again.

(8) The tabular approach is a convenient device for discussing limits informally, but if it is not used carefully, it can be misleading. For example, for let



    (a) Construct a table showing the value of and for and Based on this table what would you say about
    
    (b) Construct a table showing the value of and for Based on this table what would you say about
    
    (c) Based on your results in (a) and (b) what do you conclude about
    

(9) If an arrow is shot upward on the moon with a velocity of its height in meters after seconds is given by

    (a) Find the velocity of the arrow after
    
    (b) Find the velocity of the arrow when
    
    (c) When will the arrow hit the moon?
    
    (d) With what velocity will the arrow hit the moon?

(10) If a cylindrical tank holds gallons of water, which takes 1 hour to drain from the bottom of the tank, then Torricelli's Law gives the volume of water remaining in the tank after minutes as



Find the rate at which the water is flowing out of the tank after 20 minutes.

(11) To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. How high is the building if the splash is seen seconds after the stone is dropped?  

(12) Verify that the average over the time interval is the same as the instantaneous velocity at for the position function

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