Professor Frink never anticipated Truckasaurus getting loose, but then again, he never foresaw the obvious dangers in reanimating the corpse of his long-dead father. Frinkie's not one for long-term planning. Anyway, Homer Simpson lies directly in the path of the flame-spewing juggernaut, with only the meager acceleration of the family station wagon standing between him and utter destruction.

Assume Homer's velocity (in feet per second) is given by the equation , where t is measured in seconds and . Answer the following questions based on the given information, accurate to the thousandths place.

(a) Is Homer traveling backwards at any time during the first 7 seconds of his escape attempt? If so, during what time interval(s) is the gear shift in reverse?

(b) What is Homer's average velocity from t = 2 to t = 5?

(c) At what time(s) does Homer change from accelerating to decelerating, or vice versa?

You may use a graphing calculator on this problem · Difficulty:

Solution

(a) Homer is traveling backward whenever his velocity is negative. In other words, you're looking for the intervals of v(t) when its graph falls below the x-axis. If you graph v on your calculator, you'll find 2 x-intercepts in the interval [0,7].

It's advisable to use the calculator to find them: x = .40642065 and x = 4.1819433. Since the graph is below the x-axis between those x-intercepts, Homer's in reverse on the interval (.406, 4.182).

(b) Since you're given the velocity function, and are asked to find its average value, you have to use the average value formula, which states that the average value of f on the interval [a,b] is . For this problem, a = 2 and b = 5. Since you're allowed to use a calculator, it would be a waste of time to compute the antiderivative by hand. The answer is -3.75.

(c) Homer will change from acceleration to deceleration (or vice versa) whenever the derivative of f has an x-intercept (since the derivative of the velocity function is the acceleration function). (Note that f also must cross the axis there, as well, not simply bounce off of it, or there will be no change in the sign of the graph.) The derivative of f is .

The acceleration function has only one x-intercept on the interval [0,7]: x = 2.786 seconds, at which Homer goes from decelerating to accelerating.