Presentation on theme: "Functions, Graphs, and Limits"— Presentation transcript:
1Functions, Graphs, and Limits CALCULUS IChapter 1Functions, Graphs, and LimitsMr. Saâd BELKOUCH
2FunctionsGraph of a FunctionLinear FunctionsLimitsOne-Sided Limits and Continuity
3Section 1: FunctionsA function is a rule that assigns to each object in a set A exactly one object in a set B. The set A is called the domain of the function, and the set of assigned objects in B is called the range.A function is denoted by a letter such as f, g, h…etcThe value that the function f assigns to the number x in the domain is then denoted by f(x)f(x) = 2x3 + 7
4It may help to think of such a function as a "mapping" from numbers in A to numbers in B, or as a "machine" that takes a given number from A and converts it into a number in B through a process indicated by the functional rule.it is important to remember that a function assigns one and only one number in the range (output) to each number in the domain (input)
5A functional relationship may be represented by an equation y = f(x) Example:Find f(3) if f(x) = x2 + 4Solution:f(3) = = 13A function assigns one and only one number in the range to each number in the domain.A functional relationship may be represented by an equation y = f(x)y is called the dependent variable and x the indepedent variable since the value of y depends on that of x
6Domain of a functionIf a formula is used to define a function f, then we assume the domain of f to be the set of all numbers for which f(x) is defined. We refer to this as the natural domain of fExample: f (x) = 1 / ( x - 1)x can take any real number except 1 since x = 1 would make the denominator equal to zero and the division by zero is not allowed in mathematics. Hence the domain in interval notation is given by: (-infinity , 1) U (1 , +infinity)
7Find the domain and range of each of these functions. f(x) = Solution Example:Find the domain and range of each of these functions.f(x) =SolutionSince division by any number other than 0 is possible, the domain of f is the set of all numbers x such that x - 3 ≠0; that is, x ≠3. The range of f is the set of all numbers y except 0, since for any y≠ 0, there is an x such thaty = 1/x-3 in particular, x= 3+
8Example: Find the domain and range of this function. g(x) = Solution Since negative numbers do not have real square roots,g(t) can be evaluated only when t- 2 ≥0, so the domain of g is the set of all numbers t such that t≥2 . The range of g is the set of all nonnegative numbers, if y≥0, there is a t such that y = ; namely, t =
9Composition of functions Given functions f(u) and g(x), the composition f(g(x)) is the function of x formed by substituting u = g(x) for u in the formula for f(u).In other words, it is a combination of two functions, where you apply the first function, get an answer, and then fill that answer into the second function.Note that the composite function f(g(x)) "makes sense" only if the domain of f contains the range of g
10Here are two simple functions, which we'll label f and g: f(x) = 4x2 - 1 g(x) = 3x The composite function value we want is f( g(2) )First work out g(2) = 3(2) + 2 = 8 Then work out f(8) = 4(8)2 - 1 = 4(64) - 1 = 255 So f( g(2) ) = 255Notice that you do the inside function first. Then you fill that answer into the outside function.
11The Difference Quotient A difference quotient is an expression of the general formwhere f is a given function of x and h is a numberThe difference quotient is used in the definition of the derivative
12Section 2: The graph of a function The graph of a function is a diagram that exhibits a relationship between two sets of numbers as a set of points (or plot) having coordinates determined by the function.Graphs have visual impact. They also reveal information that may not be evident from verbal or algebraic descriptions.
13Rectangular coordinate system A rectangular coordinate system (or Cartesian coordinate system) is defined by an ordered pair of perpendicular lines (coordinate axes), generally a single unit of length for both axes, and an orientation for each axis.The coordinate axes separate the plane into four parts called quadrantsAny point P in the plane can be associated with a unique ordered pair of numbers (a, b) called the coordinates of P.a is the x coordinate (or abscissa) and b is called the y coordinate (or ordinate).
15The distance formulaThe distance between the points P(Xl, Yl) and Q(X2, Y2) is given byTo represent a function Y = f(x) geometrically as a graph, we plot values of the independent variable X on the (horizontal) X axis and values of the dependent variable Y on the (vertical) y axis.
16Example: Find the distance between the points P(-2, 5) and Q(4, -1). SolutionIn the distance formula, we have Xl = -2, Y1 = 5, X2 = 4, and Y2 = -1, so the distance between P and Q may be found as follows:D = =
17The graph of a functionThe graph of a function f consists of all points (x, y) where x is in the domain of f and y = f(x); that is, all points of the form (x,f(x)).
19Y and x interceptsThe points (if any) where a graph crosses the x axis are called x intercepts, and similarly, a y intercept is a point where the graph crosses the y axis.To find any x intercept, set y = 0 and solve for x. To find any Y intercept, set x = 0 and solve for y.
20Y and x intercepts Solution Graph the function f(x) = - x2 + x + 2. Include all x and y intercepts.SolutionThe y intercept is f(O) = 2. To find the x intercepts, solve the equation: f(x) = O. Factoring, we find that- x2 + x + 2 = factor -(x + 1)(x - 2) = 0Note that: uv = 0, if and only if u = 0 or v = 0x = -1 and x = 2, thus, the x intercepts are: (-1, 0) and (2, 0).Next, make a table of values and plot the corresponding points (x, f(x)).4321-1-2-3x-10-4f(x)
21Here is the graph of the function f(x) = - x2 + x + 2 with the intercepts
22Graphing ParabolasIn general, the graph of y = Ax2 + Bx + C is a parabola as long as A #0. All parabolas have a "U shape," and the parabola y = Ax2 + Bx + C opens up if A > 0 and down if A < O. The "peak".To graph a reasonable parabola, we should know:The location of the vertex ( where x = -B/2A )Whether the parabola opens up (A > 0) or down (A < 0)Any interceptsExample: y = -x2 + x + 2
23Intersection of Graphs The values of x for 'which two functions f(x) and g(x) are equal', are the x coordinates of the points where their graphs intersect.
24Example:Find all points of intersection of the graphs of f(x) = x and g(x) = x2SolutionYou must solve the equation x2 = x. Rewrite the equation as x2 - x = 0 which leads to x(x - 1), so the solutions are 0 and 1
25Power Functions, Polynomials, and Rational Functions A polynomial is a function of the formp(x) = an.xn" + an-1xx-1 +…………+ alx + aowhere n is a nonnegative integer and ao, a1, ……….., an are constants. The integer n (when not equal to zero) is called the degree of the polynomialA quotient of two polynomials p(x) and q(x) is called a rational function.
26Power Functions, Polynomials, and Rational Functions ExplanationExample PolynomialSince all of the variables have integer exponents that are positive this is a polynomial.x2 + 2x +5(x7 + 2x4 - 5) * 3xNot a polynomial because a term has a negative exponent5x-2 +1Not a polynomial because a term has a fraction exponent3x½ +2Not a polynomial because of the division(5x +1) ÷ (3x)Is actually a polynomial because it's possible to simplify this to 2x + 3, which of course satisfies the requirements of a polynomial.(2x2 +3x) ÷ (x)
27Vertical line testA curve is the graph of a function if and only if no vertical line intersects the curve more than once.
28Section 3: Linear Functions A linear function is a function that changes at a constant rate with respect to its independent variable.The graph of a linear function is a straight line.The equation of a linear function can be written in the form y = mx+bwhere m and b are constants.
29The Slope of a LineThe slope of the nonvertical line passing through the points (Xl, YI) and (X2, Y2) is given by the formulaSlope = (Y2 – Y1) / (X2 – X1)Since the graph of a linear function (Y = mx + b) is represented by a line it has only one Y intercept which is bWe say that y=mx+b is the equation of the line whose slope is m and whose Y intercept is (0, b).
30The Slope of a LineFind the slope of the line joining the points (-2, 5) and (3, -1).Solution: Slope = = =
31The Point-Slope Form The equation y - yo = m(x - xo) is an equation of the line that passes through the point (xo,yo) and that has slope equal to m.
32The Point-Slope FormFind the equation of the line that passes through the point (5,1) with a slope of ½SolutionUse the formula y-y0=m(x-x0) with (x0,y0)=(5,1) and m = ½ to get y - 1 = ½(x - 5)which you can rewrite as y = x -
33Parallel, perpendicular lines Let m1and m2 be the slopes of the nonvertical lines L1, and L2. Then L1 and L2 are parallel if and only if m1 =m2L1 and L2 are perpendicular if and only if m2 = -1/ m1
34Example: Let L be the line 4x + 3y = 3. 1) Find the equation of a line L, parallel to L through P(-1, 4).2) Find the equation of a line L2 perpendicular to L through Q(2, -3).SolutionBy rewriting the equation 4x + 3y = 3 in the slope-intercept form we get:y = -4/3 x + 1 we can deduct that the slope of L is -4/31) Any line parallel to L must also have slope m = - 4/3 and the required line L1 contains P(-1, 4), so y-4 = -4/3 (x + 1) thus : y = - x +2 ) A line perpendicular to L must have slope m = , Since the required line L2, contains Q(2, -3), we have y + 3 = 3/4*(x-2)Which means: y = x -
35Section 4: LimitsLet f(x) = 2 x + 2 and compute f(x) as x takes values closer to 1. We first consider values of x approaching 1 from the left (x < 1), then we consider x approaching 1 from the right (x > 1).In both cases as x approaches 1, f(x) approaches 4. Intuitively, we say that limx→1 f(x) = 4.
36The limit of a FunctionIf f(x) gets closer and closer to a number L as x gets closer and closer to c from both sides, then L is the limit of f(x) as x approaches c. The behavior is expressed by writingLim f(x) = LXCNote: the limit of a constant is the constant itself, and the limit of f(x) = x as x approaches c is c.
40Limits of Polynomials and Rational Functions If p(x) and q(x) are polynomials, thenlim(x→c) p(x) = p(c)And=Note: q( c ) must be different from zero
41Limits at infinityIf the values of the function f(x) approach the number L as x increases without bound, we writeSimilarly, we writewhen the functional values f(x) approach the number M as x decreases without bound.
43Reciprocal Power Rules If A and k are constants with k > 0 and xk is defined for all x, then:
44We say that is an infinite limit if f(x) increases or decreases without bound as X C. We writeif f(x) increases without bound as X Cif f(x) decreases without bound as X C
45Section 5: One-Sided Limits and Continuity The graph of a continuous function can be drawn without lifting the pencil from the paperA function is not continuous where its graph has a "hole or gap"
46One-Sided LimitsIf f(x) approaches L as x tends toward c from the left (x < c), we writeLikewise, if f(x) approaches M as x tends toward c from the right (c < x), then
47=Existence of a limitThe two-sided limit exists if and only if the two one-sided limits and both exist and are equal, and then=
48ContinuityA function is said to be continuous at c if all three of these conditions are satisfied:a is defined.b exists.c =If f(x) is not continuous at c, it is said to have a discontinuity there
49Continuity on an interval A function f(x) is said to be continuous on an open interval a < x < b if it is continuous at each point x = c in that interval.Moreover, f is continuous on the closed interval a ≤ x ≤ b if it is continuous on the open interval a < x < b and:and