Chapter 1-The Basics Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved – Natural numbers. {1, 2, 3, …} – Integers. {…,-3, -2, -1, 0, 1, 2, 3, …} – Rational numbers. { p/q : p,q , q ≠0}

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Sets of Real Numbers EXAMPLE: Sketch the sets S = {s : 1 < s < 4} and T = {t : -2 ≤ t < 3}. EXAMPLE: Sketch the set U = {u : 2u + 5 > 3u – 9}.

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Sets of Real Numbers DEFINITION: If x is a real number, then the expression |x|, called the absolute value of x, represents the linear distance from x to 0 on the number line. If x  0, then |x| = x. If x < 0, then |x| = -x. EXAMPLE: Sketch the set V = {x : |x – 4|≤ 3}.

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Sets of Real Numbers DEFINITION: The distance between two real numbers x and y is either x-y or y-x, whichever is nonnegative. A convenient way to say this is that the distance between x and y is |x-y|

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Intervals EXAMPLE: Write the set as a closed interval [a,b]. EXAMPLE: Write the interval (3,11) in the form..

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Intervals EXAMPLE: Solve the inequality EXAMPLE: Let Describe the intersection A B as the union of intervals.

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Triangle Inequality |a+b| ≤ |a| + |b| EXAMPLE: Suppose the distance of a to b is less than 3 and the distance of b to 2 is less than 6. Estimate the distance of a to 2.

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Approximation DEFINITION: If the absolute error |x-a| is less than or equal to 5X10 -(q+1), then a approximates (or agrees with) x to q decimal places. EXAMPLE: Does 22/7 approximate  to as many decimal places as does 3.14?

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Floating Point Representation DEFINITION: The floating point representation of a nonzero real number x is where p is an integer and a 1, a 2, … are natural numbers from 0 to 9 with a 1  0.

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Floating Point Representation EXAMPLE: Suppose that x = 0.412 X 0.300 – 0.617 – 0.200. A calculator that displays three significant digits is used to evaluate the products before subtraction. What relative error results?

Chapter 1-The Basics 1.1 Number Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. What interval does the set {x : |x-2|< 9} represent? 2. Write the interval (-5, 1) in the form {x : |x-c|< r}. 3. Write {x : |x + 8| > 7} as the union of two intervals. 4. For which numbers b is |-5+b| less than 5 + |b|? 5. What is the smallest number that approximates 0.997 to two decimal places? The largest?

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The set of all ordered pairs of real numbers is called the Cartesian plane and is denoted EXAMPLE: Sketch or graph all the points (x,y) in the plane that satisfy the equation y = 2x.

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Distance Formula and Circles Let P 1 = (a 1, b 1 ) and P 2 = (a 2, b 2 ) be points in the plane. The shortest distance between P 1 and P 2 is the length of line segment, which we denote by

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Distance Formula and Circles EXAMPLE: Calculate the distance between (2, 6) and (-4, 8). EXAMPLE: Graph the set of points satisfying (x - 4) 2 + (y - 2) 2 = 9

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Equation of a Circle An equation of the form with r > 0 has a graph that is a circle of radius r and center (h, k).

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Method of Completing the Square

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Method of Completing the Square EXAMPLE: Apply the method of completing the square to 3x 2 + 18x + 16 and to 5 - 4x - 4x 2.

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Parabolas, Ellipses, and Hyperbolas Ellipse: Insert Figure 9(?)

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Parabolas, Ellipses, and Hyperbolas Hyperbola

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Parabolas, Ellipses, and Hyperbolas Parabola: y = Ax 2 + Bx + C, A  0. THEOREM: The vertical line x = -B/(2A) is an axis of symmetry of the parabola y = Ax 2 + Bx + C.

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Regions in the Plane EXAMPLE: Sketch S={(x,y): y > 2x} EXAMPLE: Sketch G = {(x,y): |y|>2 and |x|≤ 5}

Chapter 1-The Basics 1.2 Planar Coordinates and Graphing in the Plane Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. What is the distance between the points (1,-2) and (-1,-3)? 2. Write the equation of a circle passing through the origin and with center 3 units above the origin. 3. Use the method of completing the square to express x 2 - 12x as the difference of squares. 4. Match the descriptions point, circle, parabola, hyperbola, and ellipse to the Cartesian equations 2x 2 -2x-4y 2 -3y = 25/2, 2x 2 -2x+4y 2 +3y-1/2 = 0, 3x 2 -3x+3y 2 +4y+2 = 0, 3x 2 - 3x+3y 2 +4y+25/12 = 0, and x 2 + 4x + y = 3. 5. What is the Cartesian equation of the parabola that opens downward and has its vertex at (-2, -5)?

Chapter 1-The Basics 1.3 Lines and Their Slopes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Slopes THEOREM: Two lines, neither of which is horizontal or vertical, are mutually perpendicular if and only if their slopes are negative reciprocals. THEOREM: Two lines are parallel if and only if they have the same slope.

Chapter 1-The Basics 1.3 Lines and Their Slopes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Slopes EXAMPLE: Verify that the line through (3,-7) and (4,6) is parallel to the line through (2,5) and (4,31).

Chapter 1-The Basics 1.3 Lines and Their Slopes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Equations of Lines EXAMPLE: Write the equation of the line passing through points (2,1) and (-4,3)

Chapter 1-The Basics 1.3 Lines and Their Slopes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Equations of Lines DEFINITION: The x- intercept of a line is the x-coordinate of the point where the line intersects the x-axis (provided such a point exists). The y-intercept of a line is the y- coordinate of the point where the line intersects the y-axis (provided such a point exists).

Chapter 1-The Basics 1.3 Lines and Their Slopes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Equations of Lines EXAMPLE: What is the y-intercept of the line that passes through the points (-1,-4) and (4,6)?

Chapter 1-The Basics 1.3 Lines and Their Slopes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Equations of Lines EXAMPLE: Find the intercept equation of the line with the x-intercept 3 and the y- intercept -5. State the slope-intercept form as well.

Chapter 1-The Basics 1.3 Lines and Their Slopes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Least Squares Lines THEOREM: Given the N+1 data points (x 0, y 0 ), (x 0, y 0 ),…, (x 0, y 0 ), the least squares line through (x 0, y 0 ) is given by y = m(x - x 0 ) + y 0 where

Chapter 1-The Basics 1.3 Lines and Their Slopes Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. What is the slope of the line through the points (-1,-2) and (3, 8)? 2. What is the slope of the line described by the Cartesian equation 3x + 4y = 7? 3. What is the slope of the line that is perpendicular to the line described by the Cartesian equation 5x - 2y = 7? 4. What are the intercepts of the line described by the equation 5x – y/2 = 1? 5. What is the equation of the least squares line through the origin for the points (1, 2), (2, 5), and (3, 10)?

Chapter 1-The Basics 1.4 Functions and Their Graphs Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved DEFINITION: A function f on the set S with values in the set T is a rule that assigns to each element of S a unique element of T. We say “f maps S into T” and denote by f:S  T.

Chapter 1-The Basics 1.4 Functions and Their Graphs Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let What is the domain of F? Examples of Functions of a Real Variable

Chapter 1-The Basics 1.4 Functions and Their Graphs Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Piecewise-Defined Functions EXAMPLE: Schedule X from the 2008 U.S. income tax Form 1040 is reproduced below. This form helps determine the income tax T(x) of a single filer with taxable income x. Write T using mathematical notation.

Chapter 1-The Basics 1.4 Functions and Their Graphs Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Graphs of Functions DEFINITION: The graph of f is the set of all points (x, y) in the xy-plane for which x is in the domain of f and y = f(x). EXAMPLE: Is the graph of the equation x 2 + y 2 = 13 the graph of a function?

Chapter 1-The Basics 1.4 Functions and Their Graphs Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Sequences DEFINITION: A function whose domain is the set of positive integers is called an infinite sequence. EXAMPLE: The Fibonacci sequence f n is defined by f n+2 = f n+1 + f n for n  1. The values f 1 and f 2 are both initialized to be 1. What is f 7 ?

Chapter 1-The Basics 1.4 Functions and Their Graphs Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. True or false: a is in the domain of a real-valued function f of a real variable if and only if the vertical line x = a intersects the graph of f exactly once. 2. True or false: b is in the range of a real-valued function f of a real variable if and only if the horizontal line y = b intersects the graph of f at least once. 3. True or false: b is in the image of a real-valued function f of a real variable if and only if the horizontal line y = b intersects the graph of f exactly once. 4. What is the domain of the function defined by the expression:

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Arithmetic Operations Let c be a constant. Suppose f and g are functions with the same domain S. For s  S,

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Arithmetic Operations EXAMPLE: Let f(x)=2x and g(x)=x 3. Calculate f + g, f - g, f  g, and f=g.

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Polynomial Functions DEFINITION: A polynomial function p is a function of the form where N is a nonnegative integer and a N ≠ 0.

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Composition of Functions (g  f)(x) = g(f(x)) EXAMPLE: Let f(x) = x 2 + 1 and g(x) = 3x + 5. Calculate g  f and f  g. EXAMPLE: How can we write the function r(x) = (2x + 7) 3 as the composition of two functions? If u(t) = 3=(t 2 + 4), how can we find two functions v and w for which u = v  w?

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Functions DEFINITION: A function f : S  T is onto if for every t in T there is at least one s in S for which f(s) = t. DEFINITION: A function f : S  T is one-to-one if it takes different elements of S to different elements of T.

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Functions THEOREM: Suppose that a function f: S  T is both one-to-one and onto. Then there is a unique function f -1 :T  S such that f(f -1 (t))=t for all t in T and f -1 (f(s))=s for all s in S. EXAMPLE: The function on the reals defined by f(s)=2s+3 is one-to-one and onto. Find f -1.

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Graph of the Function THEOREM: If f: S  T is an invertible function, then the graph of f -1 : T  S is obtained by reflecting the graph of f through the line y = x.

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Vertical and Horizontal Translations THEOREM: The graph of x|  f(x+h) is obtained by shifting the graph of f horizontally by an amount h. The shift is to the left if h>0 and to the right if h<0. The graph of of x|  f(x)+k is obtained by shifting the graph of f vertically by an amount k. The shift is to the up if k>0 and to the down if k<0.

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Vertical and Horizontal Translations EXAMPLE: Describe the relationship between the graph of y = x 2 +6x+13 and the parabola y = x 2.

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Even and Odd functions A function is even if f(-x) = f(x) for every x in its domain. A function is odd if f(-x) = -f(x) for every x in its domain. EXAMPLE: What symmetries do the graphs of the following functions possess?

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Pairing functions-Parametric Curves Suppose C is a curve in the plane and I is an interval of real numbers. If  1 and  2 are functions with domain I, and if the plot of the points (  1 (t),  2 (t)) for t in I coincides with C, then C is said to be parameterized by the equations x=  1 (t) and y =  2 (t). These equations are called parametric equations for C. The variable t is said to be a parameter for the curve and I is the domain of the parameterization of C. We say that C is a parametric curve.

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Pairing functions-Parametric Curves EXAMPLE: A particle moves in the xy-plane with coordinates given by Describe the particle's path C.

Chapter 1-The Basics 1.5 Combining Functions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. If g (x) = (2x + 1) / (3x + 4) for x > 0 and f (x) = 1/x for x > 0, what (g  f) (x)? 2. The function f : [1,  )  (0, 1/2] defined by f (x) = x/(1+x 2 ) is invertible. What is f -1 (x)? 3. How do the graphs of f (x) = x 3 + 2 and g (x) = (x - 1) 3 + 4 compare? 4. Does either of f (x) = (x 2 +1)/(3x 4 +5) and g (x) = x 3 /(3x 4 +5) have a graph that is symmetric with respect to the y-axis? With respect to the origin? 5. Describe the parametric curve x = |t| + t, y = 2t, -1 ≤ t ≤ 1

Chapter 1-The Basics 1.6 Trigonometry Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Sine and Cosine Functions DEFINITION: Let A = (1, 0) denote the point of intersection of the unit circle with the positive x-axis. Let  be any real number. A unique point P = (x, y) on the unit circle is associated with  by rotating OA by  radians. The radius OP is called the terminal radius of , and P is called the terminal point corresponding to . The number y is called the sine of  and is written sin(  ). The number x is called the cosine of  and is written cos(  ). EXAMPLE: Compute the sine and cosine of  /3.

Chapter 1-The Basics 1.6 Trigonometry Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Other Trigonometric Functions EXAMPLE: Compute all the trigonometric functions for the angle  =11  /4.

Chapter 1-The Basics 1.6 Trigonometry Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz 1. What is the domain of the sine function? 2. What is the image of the cosine function? 3. Which trigonometric functions are even? Odd? 4. Simplify sin(2  )/sin(  ).

Chapter 1-The Basics Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

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Chapter 1-The Basics Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

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Presentation on theme: "Chapter 1-The Basics Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved."— Presentation transcript:

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