Chapter 1

February 14 2016

- mathematics is the stage for many of the highest achievements of human imagination - the essence of calculus does not lie in theorems and how to prove them, but rather the tools and how to use them - calculus is a problem solving art of immense power which is indispensable in all the quantitative sciences - the main theme is what calculus is good for, what it enables us to do and understand, and not what its logical nature is as seen from the specialized (and limited) point of view of the modern pure mathematician Chapter 1 Numbers Functions and Graphs - the world is dominated by motion and change - math is the most natural medium of communicating and understanding this motion and change - calculus is the branch of math where the primary purpose is the study of motion and change - the methods and applications of calculus constitute one of the greatest intellectual achievements of civilization - main objects of study in calculus are functions - functions are rules that tell us how one variable quantity depends on another - this is the master concept of the exact sciences - it offers us the prospect of understanding and correlating natural phenomena by means of mathematical machinery of great and sometimes mysterious power - this chapter focuses on what a function is 1.2 The Real Line - the real number system is quite important - other systems important later, such as complex numbers and multi-dimensional vectors - important subsets of real numbers: - positive integers 1, 2, 3, 4, 5 ... - integers -3, -2, -1, 0, 1, 2, 3 ... - rational numbers - can be represented as a fractions of integers - irrationals - not rational, cannot be represented as fractions of integers (sqrt 2, sqrt 3, pi) - the real line (a coordinate system) - arbitrary location for 0 and 1, then all real numbers can be placed by this scale - allows geometric representation of, and solutions to Inequalities, Absolute Values and Intervals Inequalities Main rules are: 1) if a > 0 and b < c, then ab < ac 2) if a < 0 and b < c, then ab > ac 3) if a < b, then a + c < b + c for any real number c - inequality is preserved on multiplication with a positive number - rearrange a > b to a - b > 0 by 3) for convenience Absolute Values Main rules are: - |ab| = |a| |b| - |a + b| <= |a| + |b| - geometrically means the distance from the origin 0 - rearrange |x + 2| = 3 to |x - (-2)| = 3 for geometric interpretation Intervals - a <= x <= b is equivalent to interval (or set) [a,b] - geometrically a segment - sets of numbers described by Inequalities and Absolute Values are often intervals - rearrange x3 > x to x(x + 1)(x - 1) > 0 - leftside = 0 at three points, divides real line into four intervals, test out to see which intervals satisfy x3 > x. so solution is 2 intervals (or sets) 1.3 The Coordinate Plane - coordinate plane - pairs of real numbers to define points in a plane - the definition is unique The Distance Formula - by Pythagorean Theorem, for right angle triangles a2 + b2 = c2 - distance d between two points is: d = sqrt ( (x1 - x2)2 + (y1 - y2)2 ) The Midpoint Formula - the midpoint of a line segment is simply average x, average y - by placing any triangle along x-axis, can show the midpoints are parallel to the side - a way to use coordinate system geometry to simplify algebra 1.4 Slopes and Equations of Straight Lines - the slope exists for any non-vertical line - m = (y2 - y1) / (x2 - x1) - ratio of height to base of right angle triangle - if x2 - x1 - 1 then m = y2 - y1 - the change in y as x changes by 1 - m > 0 rises to right - m < 0 falls to right - m = 0 horizontal - |m| is the steepness of the line Equations of a Line - the point-slope equation: - if point (x0, y0) is on the line and its slope is m, then (x, y) is on the line if its slope with (x0, y0) is m: - (y - y0) / (x - x0) = m - or, y - y0 = m (x - x0) - if the point is (0, b) where the line crosses the y-axis: - then y = mx + b - the slope-intercept equation - can tell at a glance the location and direction of the line - 6x - 2y - 4 = 0 can be rearranged to y = 3x - 2 - crosses (0, -2) with slpe 3 - general linear equation - Ax + By + C = 0 - y = - A/B x - C/B Parallel and Perpendicular Lines - parallel m1 = m2 - perpendicular m1m2 = -1 or m1 / 1 = 1 / -m2 - again useful for proving geometric theorems by algebra, e.g. proving if diagonals of a rectangle are perpendicular, then it is a square 1.5 Circles and Parabolas - the coordinate plan, also the Cartesian (Descartes) plane - goal is to exploit the correspondence between points and their coordinates to study geometric problems - especially the properties of curves - with the tools of algebra - generally, geometry is visual and intuitive, while algebra is rich in computational machinery, and each can serve the other in many fruitful ways - an equation F(x,y) = 0 usually determines a curve (its graph) which consists of all points P (x, y) whose coordinates satisfy the given equation - straight lines are the simplest curves Circles - The Distance Formula is useful in finding the equation of a curve whose geometric definition depends on one or more distances - circle a simple example, a constant distance from (h, k) - sqrt ( (x - h)2 + (y - k)2 ) = r - if the center is origin then (0, 0), then x2 + y2 = r - expanded, can be non-intuitive x2 + y2 + Ax + By + C = 0, reverse by completing the square (x + a)2 = x2 + 2ax + a2 - geometry ~ algebra - proving angle inscribed in a semicircle is a right angle by using m1m2 = -1 Parabolas - curve of all points equally distant from a fixed point F (focus) and a fixed line d (directrix) - stated algebraically, if focus is (0, p) and line is y = -p - sqrt (x2 + (y - p)2) = y + p, or x2 = 4py - general form y = ax2 which conceal constant p but useful for visualizing graph - shifting coordinates (x - a) (y - b) can be expanded to general form y - a = (x - b)2 or y = ax2 + bx + c - so far used a static concept of the curve as a certain set of points or a geometric figure - often possible to adapt the dynamic point of view, a curve thought of as the path of a moving point, e.g. circle is path of point maintaining a fixed distance - has advantage of intuitive vividness, curve often called a locus, parabola is the locus of a point that moves while maintaining equal distances from a given point and given line 1.6 The Concept of a Function - functions are a vital concept - particularly for Calculus, most of our work will be concerned with developing machinery for the study of functions and applying this machinery to problems in science and geometry - consider the parabola y = x2 - previously thought of as a relation between the variable coordinates of a point (x, y) moving along a curve - now shift point of view. consider as a formula that provides a mechanism for calculation the numerical value of y when x is given - so, the value of y is said to depend on, or to be a function of, the value of x - the dependence is noted by y = f(x) f(x) = x2 - where f is a rule that squares whatever follows - general concept of a function: - assigns a single real number y to each x - y is the single result of any x - the function does not need to be algebraic, can be a verbal description, as long as it assigns all x's to a unique y - x belongs in the domain of allowed values, y the range - graphs allow us to see the function in its entirety over its domain and range - if the domain is unspecified, said to include all real numbers for which the formula makes sense - functions can be changed and decomposed g(f(x)), this will be useful 1.7 Types of Functions, Formulas from Geometry - in practice, functions often arise from algebraic relations between variables - so an equation involving x and y determines y as a function of x if the equation is equivalent to one that expresses y uniquely in terms of x - in cases where solving the equation results in more than one value of y e.g. y2 = x, then we can get two functions from this equation ( +sqrt(x) and -sqrt(x) ) Polynomials - functions where all components are non-negative integral exponents - p(x) = a0 + a1x + a2x2 ... - polynomials can be added, subtracted and multiplied together, the results are again polynomials Rational Functions - divisions are allowed, includes polynomials as a subset - ( a0 + a1x + a2x2 ... ) / ( b0 + b1x + b2x2 ...) Algebraic Functions - root extractions, y = sqrt (x) etc - defined later Transcendental Functions - any function that is not algebraic - trigonometric, inverse trigonometric, exponential and logarithmic - explained later 1.8 Graphs of Functions Polynomials - y = 1 horizontal line - y = x line slow 1 - y = xn, n even, a parabola - y = xn, n odd, left down right up parabola - y = 2x - 1, first degree, lines - y = 3x2 - 2x + 1, second degree, parabolas - the zero of a function is a root of f(x) = 0 - if the funciton has any, the values of x at which the graph crosses or touches the x-axis, the x-intercepts - y = ax2 + bx + c, the zeros, by the quadratic formula, are x = -b +/- sqrt( b2 - 4ac ) / 2a - 3 possibilities for zeros, 0, 1 or 2 zeros, of parabolas above, at, and below the x-axis - y = x3, third and above degree polynomials - show in factored form, e.g. of y = x3 - 3x, to find the zeros - y = x( x2 - 3) = x ( x + sqrt 3) ( x - sqrt 3 ) - these zeros divide the x-axis (which is like the real number line from before) into intervals - analyse intervals to see if hte function is positive or negative - can also see if functions rise or fall when x gets really large or really small Rational Functions - tend to have asymptotes, infinite discontinuities - simplest is 1/x - undefined at x = 0 - positive when x is positive - negative when x is negative - small when |x| is large - large when |x| is small - a straight line is called an asymptote of a curve if, as a point moves out along an extremity of the curve, the distance from this point to the line approaches 0 - x-axis, y-axis both asymptotes - x = 0 is an infinite discontinuity (approaches infinity) - y = x / (x - 1) - undefined at x = 1, also y = 1 - use intervals and approaching values to graph - y = x / (x2 - 3x + 2) - decompose to y = x / (x - 1) (x - 2) - infinite discontinuities at x = 1 and x = 2 - use intervals to graph - y = x + 1 / x - infinite discontinuity at x = 0 - x small, y gets large and second term dominates - x large, y gets large as first term dominates - y = x also an asymptote - y = ( x2 - 1 ) / ( x - 1 ) - cancels for ( x - 1 ), can become x + 1, so it is a line - but discontinuous at x = 1 (a point discontinuity) - so the two functions are not equal because their domains are different Algebraic Functions - y = sqrt x, y = sqrt ( 25 - x2 ) - positive above x-axis portions of parabolas and circles - we are less interested in graphs of high accuracy than in those that display broad features - where falling, rising, the presence of gaps, high and low points, the approximate shape - the primary aim of math is insight, graphs are invaluable aids for gaining visual insight into individual characteristics of functions