Understanding Lines and Slopes in Calculus: Concepts Essentials

1. Chapter 1Prerequisites for Calculus

2. Section 1.1 Lines • Calculus – 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.

3. 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)

4. 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 =

5. Slope Example Find the slope of the the two points (1,2) and (4,3)

6. Parallel and Perpendicular Lines • Parallel lines have the same slope

7. 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.

8. 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

9. 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

10. 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

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

12. 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

13. Slope Intercept Example • Write the slope-intercept equation for the line through (-2, -1) and (3, 4) • Hint: Use the both formulas.

14. General Linear Equation • Ax + By = C (A and B not both 0) • Find the slope and y-intercept of the line 8x + 5y = 20

15. 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

16. Example • The following table gives values for the linear function f(x) = mx +b • Determine m and b

17. Regression Example • Predict the world population in the year 2010, and compare this prediction with the Statistical Abstract prediction of 6812 million.

18. Homework • Page 9; #1-37 odd