Chapter 1

Prerequisites for Calculus

Section 1.1 LinesCalculus – The Study of change

Relating the rate of change of a quantity to the graph of a quantity all begins with the slopes of lines.

When a particle in a plane moves from one point to another; the net change is called an increment.

Increments If a particle moves from the point (x1, y1) to the

point (x2, y2), the increments in its coordinates are Δx = x2 – x1

Δy = y2 – y1

Example 1: Find the increments of x and y from (4, -3) to (2, 5)

Slope of a line A slope can be calculated from increments in coordinates.

We call Δy the rise and Δx the run.

Let P1(x1, y1) and P2(x2, y2) be points on a non-vertical line L. The slope of L is M =

Slope ExampleFind the slope of the the two points (1,2) and (4,3)

Parallel and Perpendicular Lines Parallel lines have the same slope

Perpendicular lines

If line one has slope m1 and line two has slope m2. To be perpendicular then the The slope of m2 has to be the negative reciprocal of m1.

Vertical Lines A vertical line is one that goes straight up and down, parallel to the y-axis of

the coordinate plane. All points on the line will have the same x-coordinate.

Example: x=2

Horizontal Lines The slope does not exist

This will be the y coordinate.

Example y = 2

Example: write a horizontal and a vertical line

equation for the point (2,5) Vertical Line : x = 2 Horizontal Line : y =5

Point Slope Equation

Y2-y1 = m(x2-x1)

Y = m(x-x1) + y1

is the point-slope equation of the line through the point (x1, y1) with slope m

Point slope example Write the point-slope equation for the line through the point (2,3) with

slope -3/2.

Slope-Intercept Equation The y-coordinate of the point where a line intersects the y-axis is the y-

intercept of the line. Similarly, the x-coordinate of the point where a line intercepts the x-axis is the x-intercept of the line.

A line with slope m and y-intercept b passes through (0, b)

Y = m(x-0) + b ; or more simply, y = mx + b

Slope Intercept Example Write the slope-intercept equation for the line through (-2, -1) and (3, 4)

Hint: Use the both formulas.

General Linear Equation Ax + By = C (A and B not both 0)

Find the slope and y-intercept of the line 8x + 5y = 20

Examples Write an equation for the line through the point (-1, 2) that is

Parallel to the line y = 3x - 4 Perpendicular to the line y = 3x - 4

Example The following table gives values for the linear function f(x) = mx +b

Determine m and b

x F(x)-1 14/31 -4/32 -13/3

Regression Example Predict the world population in the year 2010, and compare this prediction

with the Statistical Abstract prediction of 6812 million. Year Population (millions)

1980 44541985 48531990 52851995 56962003 63052004 63782005 6450

Homework Page 9; #1-37 odd