1 § 1-1 Real Numbers, Inequalities, Lines, and Exponents The student will learn about: the Cartesian plane, straight lines, an application, integer exponents, and fractional exponents.

2 Introduction Quite simply, calculus is the study of rates of change. We will use calculus to analyze rates of inflation, rates of learning, rates of population growth, and rates of natural resource consumption, etc. 2 In this first section we will study linear relationships between two variables — that is, relationships that can be represented by lines.

3 The section starts using basic definitions of the real numbers, inequalities, set and interval notation. The student is responsible for knowing this material. 3

4 Cartesian Coordinate System Students should be familiar with the basic terminology of the Cartesian coordinate system.

5 Lines and Slopes The symbol ∆ (read “delta,” the Greek letter D) for mathematicians means “the change in.” For any two points (x 1, y 1 ) and (x 2, y 2 ) we define  x = x 2 – x 1. The change in x is the difference in the x-coordinates.  y = y 2 – y 1. The change in y is the difference in the y-coordinates. 5

6 Slope of a Line 1. Def: If P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) are two points on a line then the slope of that line is given by the formulas:

7 Slopes of Lines Discuss slope.

8 Example – FINDING SLOPES AND GRAPHING LINES Find the slope of the line through the following pair of points, and graph the line. (2, 1), (3, 4) Solution:

9 Slope-Intercept form of a Line Def: The equation y = mx + b is called the slope-intercept form of a line. The slope is m and the y intercept is b. We will use this form to graph a line on graph paper and on our calculator.

10 Slope-Intercept Form of a Line The slope-intercept form when graphing. y = 2x - 3 We first plot the y intercept (0, - 3). We finish by connecting those points to form the line. The slope is 2 = 2/1 so we then plot another point 2 units up and 1 unit over from the y – intercept.

11 Graphing: Calculator Sketch a graph of y = 2x – 1. By calculator 1. Turn your calculator on and in the y = window enter 2x – On the zoom screen choose “zoom standard” and touch “enter”.

12 Point-Slope Form of a Line Def: The equation y – y 1 = m (x – x 1 ) is called the point-slope form of a line. The slope is m and (x 1, y 1 ) is a point on the line. We will use this form to solve problems that ask for the equation of a line.

13 Example 3 – USING THE POINT-SLOPE FORM Find an equation of the line through (6, –2) with slope y – y 1 = m(x – x 1 ) Solution:

14 Special Cases A vertical line has an equation of x = a. x = 2 A horizontal line has an equation of y = b. y = 1

15 Equations of Lines There is one form that covers all lines, vertical and nonvertical. Any equation that can be written in this form is called a linear equation, and the variables are said to depend linearly on each other. We will sometimes use this form to write the final form of our line. Ax + By = C For constants A, B, C, with A and B not both zero.

16 X - Intercepts The x intercept is 3 The x intercept of f (x) occurs where and if the graph of the function crosses the x-axis. Algebraically it is the x value where f (x) is zero, or (x, 0). That is, to find the x intercept, let f (x) = 0 and solve for x. Remember f(x) is y! 2x + 3y = 6 2x + 3· 0 = 6 2x = 6 x = 3

17 Y - Intercept The y intercept is 2 That is, to find the y intercept let x = 0 and solve for f (x). 2x + 3y = 6 2· 0 + 3y = 6 3y = 6 The y intercept of f (x) occurs where and if the graph of the function crosses the y-axis. Algebraically it is the f (x) value where x is zero, or (0, f (x)). y = 2

18 Intercepts on a Calculator 1. Consider 2x + 3y = 6. First solve the equation for y. 2. Graph this equation on your calculator. 3. To find the x-intercept use the “zero” function under the “Calc” menu. 4. To find the y-intercept use the “value” function under the “Calc” menu with an x value of 0. Note: x-intercept is 3 or (3,0) and the y-intercept is 2 or (0,2). Sometimes you may also use Table to find both the x-intercepts and the y-intercept at the same time!

19 Review Equations of a Line GeneralAx + By = C Not of much use. Test answers. Slope-Intercept Formy = mx + b Graphing on a calculator. Point-slope formy – y 1 = m (x – x 1 ) Finding the equation of a line. Horizontal liney = b Vertical linex = a

20 Application Example Linear Depreciation. Office equipment was purchased for $20,000 and is assumed to have a scrap value of $2,000 after 10 years. If its value is depreciated linearly (for tax purposes) from $20,000 to $2,000: 1. Find the linear equation that relates value (V) in dollars to time (t) in years. We know two points. What are they? t = 0 and V = $20,000 (0, $20,000) and t = 10 and V = $2,000 (10, $2,000)

21 The slope is  V/  t or m = (2000 – 20000) / (10 – 0) = You may use either point. I will choose (10, 2000). Application Example Continued Linear Depreciation. Office equipment was purchased for $20,000 and is assumed to have a scrap value of $2,000 after 10 years. If its value is depreciated linearly for (tax purposes) from $20,000 to $2,000: So V = -1800t Points are (10, 2000) and (0, 20000). Substituting into V – V 1 = m (t – t 1 ), yields V – 2000 = (t – 10) y – y 1 = m (x – x 1 ) = t

22 Application Example Continued Linear Depreciation. Office equipment was purchased for $20,000 and is assumed to have a scrap value of $2,000 after 10 years. If its value is depreciated linearly for (tax purposes) from $20,000 to $2,000: V(6) = = 2. What would be the value of the equipment after 6 years? V = -1800t = $9,

23 Application Example Continued 3.Graph the equation V = -1800t ≤ t ≤ Write a verbal interpretation of the slope of the line found in the first part of this problem. 4. Use “value” to find V (6) on the calculator.

24 Positive Integer Exponents Definition: The 3 4 is an exponential expression. The base is 3 and the exponent or power is 4. It is an abbreviation for repeated multiplication. I.e. 3 4 = 3 · 3 · 3 · 3 = 81 More generally x n = x · x · · · x n

25 Exponential Properties 1.Exponential laws: x m x n = x m + n 1.Exponential laws: x m x n = x m + n x m / x n = x m – n 1.Exponential laws: x m x n = x m + n x m / x n = x m – n (x m ) n = x m n 1.Exponential laws: x m x n = x m + n x m / x n = x m – n (x m ) n = x m n (xy) n = x n y n 1.Exponential laws: x m x n = x m + n x m / x n = x m – n (x m ) n = x m n (xy) n = x n y n (x/y) n = x n / y n

26 Zero and Negative Exponents 1. By definition x 0 = 1

27 Rational Exponents 1. By definition  9 = 3 means the principal square root or the positive root if there are two. 4. On a calculator is x ^ ( m / n )

28 Summary. We did an applied problem involving a straight line graph and saw the meaning of the y-intercept and the slope of the graph. We learned about straight lines, slope, and the different forms for straight lines. We had a brief introduction to the Cartesian plane and graphing.

29 Summary. We learned fractional exponents. We learned about integer exponents positive, zero, and negative.

30 ASSIGNMENT §1.1 on my website 15, 16, 17, 18, 38, 39, 40, 41, 42.