## Chapter 10 - An Introduction to Calculus: Limits, Derivatives, and Integrals"By the beginning of the 17th century, algebra and geometry had developed to the point where physical behavior could be modeled both algebraically and graphically, each type of representation providing deeper insights into the other. New discoveries about the solar system had opened up fascinating questions about gravity and its effects on planetary motion, so that finding the mathematical key to studying motion became the scientific quest of the day. The analytic geometry of René Descartes (1596–1650) put the final pieces into place, setting the stage for Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1716) to stand “on the shoulders of giants” and see beyond the algebra- ic boundaries that had limited their predecessors. With geometry showing them the way, they created the new form of algebra that would come to be known as the calculus.
In this chapter we will look at the two central problems of motion much as Newton and Leibniz did, connecting them to geometric problems involving tangent lines and areas. We will see how the obvious geometric solutions to both problems led to algebraic dilemmas, and how the algebraic dilemmas led to the discovery of calculus. The lan- guage of limits, which we have used in this book to describe asymptotes, end behavior, and continuity, will serve us well as we make this transition." (Demana, Franklin D. "Chapter 10." Precalculus: Graphical, Numerical, Algebraic. 7th ed. Boston: Addison-Wesley, 2007. 792. Print.) |
"Windmills have long been used to pump water from wells, grind grain, and saw wood. They are more recently being used to produce electricity. The propeller radius of these windmills range from one to one hundred meters, and the power output ranges from a hundred watts to a thousand kilowatts. See page 800 in Section 10.1 for some more information and a question and answer about windmills."
(Demana, Franklin D. "Chapter 10." Precalculus: Graphical, Numerical, Algebraic. 7th ed. Boston: Addison-Wesley, 2007. 791. Print.) |

Lesson 10.1 - Limits & Motion: The Tangent ProblemThis lesson will cover average velocity, instantaneous velocity, limits revisited, connections to tangent lines, and the derivative. Lesson 10.2 - Limits & Motion: The Area ProblemThis lesson will cover distance from a constant or changing velocity, limits at infinity, connections to area, and the definite integral. |
Lesson 10.3 - More on LimitsThis lesson will cover defining a limit informally, properties of limits, limits of continuous functions, one-side & two-sided limits, and limits involving infinity. Lesson 10.4 - Numerical Derivatives and IntegralsThis lesson will cover how the previous concepts of derivatives and integrals are applied with the use of a graphing calculator. |