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

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


– Natural numbers. {1, 2, 3, …}

– Integers. {…,-3, -2, -1, 0, 1, 2, 3, …}

– Rational numbers. { p/q : p,q , q ≠0}



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}.




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}.




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|



Intervals



Intervals

EXAMPLE: Write the set

as a closed interval [a,b].

EXAMPLE: Write the interval (3,11) in the form ..



Intervals



Intervals

EXAMPLE: Solve the inequality

EXAMPLE: Let

Describe the intersection A B as the union of intervals.



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.



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?



Floating Point Representation

DEFINITION: The floating point representation of a nonzero real number x is

where p is an integer and a1, a2, … are natural numbers from 0 to 9 with a1 0.



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?



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


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.



The Distance Formula and Circles

Let P1 = (a1, b1) and P2 = (a2, b2) be points in the plane. The shortest distance between P1 and P2 is the length of line segment , which we denote by



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



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).



The Method of Completing the Square



The Method of Completing the Square

EXAMPLE: Apply the method of completing the square to 3x2 + 18x + 16 and to 5 - 4x - 4x2.



Parabolas, Ellipses, and Hyperbolas

Ellipse:

Insert Figure 9(?)




Hyperbola




Parabola: y = Ax2 + Bx + C, A 0.

THEOREM: The vertical line x = -B/(2A) is an axis of symmetry of the parabola y = Ax2 + Bx + C.



Regions in the Plane

EXAMPLE: Sketch S={(x,y): y > 2x}

EXAMPLE: Sketch G = {(x,y): |y|>2 and |x|≤ 5}



Quick Quiz1. 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 x2 -12x as the difference of squares.4. Match the descriptions point, circle, parabola, hyperbola, and ellipse to the Cartesian equations 2x2-2x-4y2-3y = 25/2, 2x2 -2x+4y2+3y-1/2 = 0, 3x2-3x+3y2+4y+2 = 0, 3x2-3x+3y2+4y+25/12 = 0, and x2 + 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


Slopes



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.



Slopes

EXAMPLE: Verify that the line through (3,-7) and (4,6) is parallel to the line through (2,5) and (4,31).



Equations of Lines

Point-Slope Form of a Line



Equations of Lines

The Two-Point Form of a Line



Equations of Lines

EXAMPLE: Write the equation of the line passing through points (2,1) and (-4,3)



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 pointexists).



Equations of Lines

The Slope-Intercept Form of a Line



Equations of Lines

EXAMPLE: What is the y-intercept of the line that passes through the points (-1,-4) and (4,6)?



Equations of Lines

The Intercept Form of a Line



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.



Least Squares Lines

THEOREM: Given the N+1 data points (x0, y0), (x0, y0),…, (x0, y0), the least squares line through (x0, y0) is given by y = m(x - x0) + y0 where



Quick Quiz1. 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


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:ST.



EXAMPLE: Let

What is the domain of F?

Examples of Functions of a Real Variable



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.



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 x2 + y2 = 13 the graph of a function?



Sequences

DEFINITION: A function whose domain is the set of positive integers is called an infinite sequence.

EXAMPLE: The Fibonacci sequence fn is defined by fn+2 = fn+1 + fn for n1. The values f1 and f2

are both initialized to be 1. What is f7?



Quick Quiz1. True or false: a is in the domain of a real-valued function f of a real variable if and only if the verticalline 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


Arithmetic Operations

Let c be a constant. Suppose f and g are functions with the same domain S. For s S,



Arithmetic Operations

EXAMPLE: Let f(x)=2x and g(x)=x3. Calculate f + g, f - g, f g, and f=g.



Polynomial Functions

DEFINITION: A polynomial function p is a function of the form

where N is a nonnegative integer and aN ≠ 0.



Composition of Functions

(g f)(x) = g(f(x))

EXAMPLE: Let f(x) = x2 + 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=(t2 + 4), how can we find two functions v and w for which u = v w?



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.



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.



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.



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.



Vertical and Horizontal Translations

EXAMPLE: Describe the relationship between the graph of y = x2+6x+13 and the parabola y = x2.



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?



Pairing functions-Parametric CurvesSuppose 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.



Pairing functions-Parametric Curves

EXAMPLE: A particle moves in the xy-plane with coordinates given by

Describe the particle's path C.



Quick Quiz1. 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+x2) is invertible. What is f-1 (x)?3. How do the graphs of f (x) = x3 + 2 and g (x) = (x - 1)3 + 4 compare?4. Does either of f (x) = (x2 +1)/(3x4 +5) and g (x) = x3/(3x4 +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


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.



Sine and Cosine Functions



Other Trigonometric Functions

EXAMPLE: Compute all the trigonometric functions for the angle =11/4.



Trigonometric Identities



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...

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