Calculo

Chapter

1

Prerequisites for Calculus

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xponential functions are used to model situations in which growth or decay change dramatically. Such situations are found in nuclear power plants, which contain rods of plutonium-239; an extremely toxic radioactive isotope. Operating at full capacity for one year, a 1,000 megawatt power plant discharges about 435 lb of plutonium-239. With ahalf-life of 24,400 years, how much of the isotope will remain after 1,000 years? This question can be answered with the mathematics covered in Section 1.3.

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Section 1.1 Lines

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Chapter 1 Overview
This chapter reviews the most important things you need to know to start learning calculus. It also introduces the use of a graphing utility as a tool to investigate mathematical ideas, tosupport analytic work, and to solve problems with numerical and graphical methods. The emphasis is on functions and graphs, the main building blocks of calculus. Functions and parametric equations are the major tools for describing the real world in mathematical terms, from temperature variations to planetary motions, from brain waves to business cycles, and from heartbeat patterns to populationgrowth. Many functions have particular importance because of the behavior they describe. Trigonometric functions describe cyclic, repetitive activity; exponential, logarithmic, and logistic functions describe growth and decay; and polynomial functions can approximate these and most other functions.

1.1
What you’ll learn about • Increments • Slope of a Line • Parallel and Perpendicular Lines •Equations of Lines • Applications . . . and why Linear equations are used extensively in business and economic applications.

Lines
Increments
One reason calculus has proved to be so useful is that it is the right mathematics for relating the rate of change of a quantity to the graph of the quantity. Explaining that relationship is one goal of this book. It all begins with the slopes of lines. Whena particle in the plane moves from one point to another, the net changes or increments in its coordinates are found by subtracting the coordinates of its starting point from the coordinates of its stopping point.

DEFINITION Increments If a particle moves from the point (x1, y1) to the point (x2, y2), the increments in its coordinates are x x2 x1 and y y2 y1.

The symbols x and y are read“delta x” and “delta y.” The letter is a Greek capital d for “difference.” Neither x nor y denotes multiplication; x is not “delta times x” nor is y “delta times y.” Increments can be positive, negative, or zero, as shown in Example 1.
y P2(x 2, y2) y P1(x 1, y1) x run rise L

EXAMPLE 1 Finding Increments The coordinate increments from 4, x 2 4 3 to (2, 5) are 2, y 5 3 8.

From (5, 6) to (5, 1),the increments are x
x

Q(x 2, y1)

5

5

0,

y

1

6

5. Now try Exercise 1.

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Slope of a Line
Each nonvertical line has a slope, which we can calculate from increments in coordinates. Let L be a nonvertical line in the plane and P1(x1, y1) and P2(x2, y2) two points on L (Figure 1.1). We call y y2 y1 the rise from P1 to P2 and x x2 x1 the run from

Figure 1.1 The slope ofline L is rise y . m run x

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Chapter 1

Prerequisites for Calculus

P1 to P2. Since L is not vertical, x 0 and we define the slope of L to be the amount of rise per unit of run. It is conventional to denote the slope by the letter m.
L1 Slope m1 Slope m 2 m1 L2

DEFINITION Slope
m2 x

Let P1(x1, y1) and P2(x2, y2) be points on a nonvertical line, L. The slope of L is m rise run yx y2 x2 y1 . x1

θ1
1

θ2
1

Figure 1.2 If L1 L 2, then u1 u2 and m1 m2. Conversely, if m1 m2, then u1 u2 and L1 L 2.

A line that goes uphill as x increases has a positive slope. A line that goes downhill as x increases has a negative slope. A horizontal line has slope zero since all of its points have the same y-coordinate, making y 0. For vertical lines, x 0 and the ratio y x is...