A Preview of Calculus

Chapter 2.1

What is Calculus?

Calculus is the mathematics of change• An object traveling at a constant velocity can be analyzed with precalculus

mathematics. To analyze the velocity of an accelerating object, you need calculus

• The sloe of a line can be analyzed with precalculus mathematics. To analyze the slope of a curve, you need calculus

• A tangent line to a circle can be analyzed with precalculus mathematics. To analyze a tangent line to a general graph, you need calculus

• The area of a rectangle can be analyzed with precalculus mathematics. To analyze the area under a general curve, you need calculus.

The Tangent Line Problem

• The transition from precalculus mathematics to calculus requires that we know and understand the limit process

• The tangent line problem originated in ancient times

• Greek mathematicians knew how to find the tangent to a circle at any point on the circle

• More generally, they wanted to find the tangent line to any curve

• The Greek scientist and mathematician Archimedes actually succeeded in finding tangent lines to many curves

• But each curve required a different method; what was wanted and needed was a general method for finding such tangent lines for any curve

The Tangent Line Problem

The Tangent Line Problem

The Tangent Line Problem

The Tangent Line Problem

The Tangent Line Problem

The Tangent Line Problem

• The problem of finding the tangent line at a point on a curve is equivalent to finding the slope of the tangent line at that point

• But this means that the problem comes down to finding the slope of a line knowing only one point on the line!

• Can we start out with a “best guess”?

The Tangent Line Problem

The Tangent Line Problem

The Tangent Line Problem

• In the previous animation, the secant line approximates the desired tangent line

• The approximations get better as the value of approaches zero

• However, cannot equal zero otherwise

• Later in this chapter you will see how we can use the limit process to avoid this indeterminate form

The Area Problem

• The ancient Greeks were able to find the area of any rectilinear (straight-edged) figure

• Aside from a circle, finding the areas of curvilinear figures was difficult

• Archimedes managed to find areas for many curved figures, but as before a general method for finding the area of curvilinear figures eluded him

• He used a method that came to be called the Method of Exhaustion, which was reminiscent of the limit process

The Area Problem

The Area Problem

The Area Problem

The Area Problem

The Limit Process

• To find the tangent line, we allow a second point on a curve to approach the desired point from both the left and the right so that the secant lines through the points approach the desired tangent line

• To find the area under a curve, we use rectangle to approximate the area and allow the number of rectangles approach infinity (or what is equivalent, we allow the area of the base to approach zero)

• In both cases, we will need to learn how to handle infinity

• Historically, this was the missing concept when the calculus was discovered independently in the 17th by Isaac Newton and Gottried Liebniz

• It took nearly 200 years before the limit concept was formulated as a useable mathematical concept; in underlies the whole of calculus

Exercise 2.1

• Page 67, #1-11