2.2 The Legacy of Galileo, Newton, and Leibniz



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:

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.


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.


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See Exercises for 2.2 The Legacy of Galileo, Newton, and Leibniz (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

Interesting Application

Do radar guns measure average or instantaneous speed?

2.1 Modeling Rates of Change Table of Contents 2.3 Limits of Functions

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel