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Transcript
MA108 CALCULUS WITH ALGEBRA I Tuesday, 9/11/12 Today: Reading (get the textbook!): Office hours end at 2:30 today only Questions on exercises assigned last time? Absolute Value Coordinate Geometry Introduction to functions Appendix B 1.1 Exercises (not to hand in): App. A: 1-11 odd, 47-53 odd App. B: 1-9 odd, 21-27 odd 1.1: 3a-e, 15, 25, 31, 33 Tuesday, 9/11/12, Slide #1 Absolute Value -x, if x ≥ 0, -x, if x < 0 |x| is never negative |x| can be thought of as the distance from x to 0 Examples: Definition: |x| = To say |x| < 5 means -5 < x < 5 To say |x| ¥ 10 means x ¥ 10 or x § -10 Tuesday, 9/11/12, Slide #2 Exercise If |x – 3| > 2, what’s true of x? A. x < 1 or x > 5 B. x = 5 C. 1 < x < 5 D. -5 < x < -1 E. x > 2 F. None of the above Tuesday, 9/11/12, Slide #3 Appendix B: Coordinate Geometry and Lines = 1: The set of all real numbers Corresponds to points on a line 2: The set of all ordered pairs, (x, y), of real numbers x and y Corresponds to points in the plane Tuesday, 9/11/12, Slide #4 Points and lines in the plane Distance between two points P1(x1, y1) and P2(x2, y2): Example: What is the distance from (-2,2) to (1,3)? Slope of line through P1 and P2: Example: What is the slope of the line through (-2,2) and(1,3)? Tuesday, 9/11/12, Slide #5 Equations of lines Point-slope form: Line with slope m, passing through point P1(x1,y1) has equation y – y1 = m (x – x1) Slope-intercept form: Line with slope m, and with y-intercept (0, b) has equation y=mx+b Exercise: Given the two points P1(2, -4) and P2(5, 0): Find an equation of the line through P1 and P2 Find the y-intercept of the line. Tuesday, 9/11/12, Slide #6 Lines as functions When we write y = m x + b, it’s natural to view y as a function of x, i.e., Start with input a number x Use the formula y = m x + b Compute output number y If we solve an equation for x instead of y, then we view x as a function of y. Usually we solve for y in terms of x: x is the input or “independent variable” y is the output or “dependent variable” We often give the function formula a name, like f(x) Tuesday, 9/11/12, Slide #7 Example of a linear function Define y = f(x) where f(x) = 7 x – 2 What is f(1), f(-1), f(2), f(-2)? The “graph” of a function f(x) is the graph of the equation y = f(x), i.e., in this example, y = 7 x – 2 What does this graph look like? Tuesday, 9/11/12, Slide #8 General Concept of Function A function is a rule which assigns to every element of an input set (the domain D) exactly one element of an output set (the range E). You can think of a function as a machine, taking in an input value and putting out an output value. Or think of it as arrows pointing from the domain to the range. Tuesday, 9/11/12, Slide #9 Two Examples from Daily Life 1. Body Mass calculator 2. Currency converter Tuesday, 9/11/12, Slide #10 Four Ways to Represent a Function Algebraically Visually : Draw a graph. Graph of above example? Numerically : Make a table of values. Example? Verbally : Find a formula. Example: y = x2 + 1 : Describe it in words. Example? Tuesday, 9/11/12, Slide #11 Some Ideas about Functions of Real Numbers If not otherwise stated, the domain of a function in calculus is the set of all real numbers that can be put into the function. 1 Example: What’s the domain of f ( x) = 2x + 4 Vertical Line Test for the graphs: A graph in the plane is a function of the horizontal variable if and only if each vertical line intersects the graph at most once. Example: What’s a common graph that is not the graph of a function? Tuesday, 9/11/12, Slide #12