The accompanying graph shows the total amount of gasoline A in the gas tank of an automobile after it has been driven for t days.

c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.

a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.

b. About how fast was the object going when it hit the surface?

[Technology Exercise] 22. Make a table of values for the function at the points x=1.2, x=11/10, x=101/100, x=1001/1000, x=10001/10000, and x = 1.

a. Find the average rate of change of F(x) over the intervals [1,x] for each x≠1 in your table.

b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1.

[Technology Exercise] Let f(t) = 1/t for t≠0.

a. Find the average rate of change of f with respect to t over the intervals (i) from t=2 to t=3, and (ii) from t=2 to t=T.

b. Make a table of values of the average rate of change of f with respect to t over the interval [2,T], for some values of T approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001, and 2.000001.

c. What does your table indicate is the rate of change of f with respect to t at t=2?

The accompanying graph shows the total distance s traveled by a bicyclist after t hours.

b. Estimate the bicyclist’s instantaneous speed at the times t=1/2, t=2, and t=3.

Existence of Limits

In Exercises 5 and 6, explain why the limits do not exist.

limx→0 x/|x|

Suppose that a function f(x) is defined for all real values of x except x=c. Can anything be said about the existence of limx→c f(x)? Give reasons for your answer.

Suppose that a function f(x) is defined for all x in [-1,1]. Can anything be said about the existence of limx→0 f(x)? Give reasons for your answer.

If f(1)=5, must limx→1 f(x) exist? If it does, then must limx→1 f(x)=5? Can we conclude anything about limx→1 f(x)? Explain.