2.2 The Legacy of Galileo, Newton, and Leibniz

Summary

The relationship between average velocity and instantaneous velocity is further developed. The notion of a limit is introduced informally. Some of the contributions of Galileo and Newton are mentioned, and especially Newton's role as one of the co-founders of calculus.

By the end of your studying, you should know:

How to compute the average velocity of a function over an interval.
How to compute as a limit of average velocities the instantaneous velocity at a point.

On-screen applet instructions: This applet shows the average velocity over the interval a to a+h, where a can be chosen from the pull down list. A single click in the graph gives an enlarged picture around the point x = a. Another click restores the original size. The value of h can be set on the slider. Click here for further instructions.

Examples

Two taxicab drivers decide to race their cabs. The first driver has a 30-second head start, and accelerates at 1 meter per second per second. The second driver accelerates at 2 meters per second per second. How many seconds will it take for the second driver to catch the first?

The distance an object falls in t seconds is given by the formula

where g is the force of gravity. If a penny is dropped from the top of the Empire State Building (350 meters tall), what is its average velocity? What is its average velocity on the [340 meters, 350 meters], that is, during its final 10 meters before it hits the ground?

The gravitational acceleration on Mars is about 3.7 meters per second per second. If a Martian juggler were to throw a ball straight up at a rate of 25 meters per second, how high would the ball go? How long would it take until it came down again? Use the formulas v(t) = v₀ – at for the velocity and d(t) = v₀t – (1/2)at² for the height of the ball.