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CALCULUS I Chapter 1 Functions, Graphs, and Limits Mr. Saâd BELKOUCH.

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1 CALCULUS I Chapter 1 Functions, Graphs, and Limits Mr. Saâd BELKOUCH

2  Functions  Graph of a Function  Linear Functions  Limits  One-Sided Limits and Continuity 2

3 Section 1: Functions  A 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…etc  The value that the function f assigns to the number x in the domain is then denoted by f(x)  f(x) = 2x

4  It 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) 4

5 Example:  Find f(3) if f(x) = x  Solution: f(3) = = 13  A 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 5

6 Domain of a function  If 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 f  Example: 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) 6

7 Example:  Find the domain and range of each of these functions. f(x) =  Solution Since 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 that y = 1/x-3 in particular, x= 3+ 7

8 Example:  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 = +2. 8

9 Composition 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 9

10  Here are two simple functions, which we'll label f and g: f(x) = 4x g(x) = 3x + 2 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) = 4(64) - 1 = 255 So f( g(2) ) = 255 Notice that you do the inside function first. Then you fill that answer into the outside function. 10

11 The Difference Quotient  A difference quotient is an expression of the general form where f is a given function of x and h is a number  The difference quotient is used in the definition of the derivative 11

12 Section 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. 12

13 Rectangular 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 quadrants  Any 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). 13

14 14

15 The distance formula  The distance between the points P(Xl, Yl) and Q(X2, Y2) is given by  To 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. 15

16 Example:  Find the distance between the points P(-2, 5) and Q(4, -1).  Solution  In 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 = = 16

17 The graph of a function  The 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)). 17

18 Example of a graph of a function 18

19 Y and x intercepts  The 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. 19

20 Y and x intercepts  Graph the function f(x) = - x 2 + x + 2. Include all x and y intercepts.  Solution  The y intercept is f(O) = 2. To find the x intercepts, solve the equation: f(x) = O. Factoring, we find that - x 2 + x + 2 = 0 factor -(x + 1)(x - 2) = 0  Note that: uv = 0, if and only if u = 0 or v = 0  x = -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)) x f(x)

21 Here is the graph of the function f(x) = - x 2 + x + 2 with the intercepts 21

22 Graphing Parabolas  In general, the graph of y = Ax 2 + Bx + C is a parabola as long as A #0. All parabolas have a "U shape," and the parabola y = Ax 2 + 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 intercepts Example: y = -x 2 + x

23 Intersection 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. 23

24 Example:  Find all points of intersection of the graphs of f(x) = x and g(x) = x 2  Solution  You must solve the equation x 2 = x. Rewrite the equation as x 2 - x = 0 which leads to x(x - 1), so the solutions are 0 and 1 24

25 Power Functions, Polynomials, and Rational Functions  A polynomial is a function of the form p(x) = a n.x n " + a n-1 x x-1 +…………+ a l x + a o where n is a nonnegative integer and a o, a 1, ……….., a n are constants. The integer n (when not equal to zero) is called the degree of the polynomial A quotient of two polynomials p(x) and q(x) is called a rational function. 25

26 Power Functions, Polynomials, and Rational Functions ExplanationExample Polynomial Since all of the variables have integer exponents that are positive this is a polynomial. x 2 + 2x +5 Since all of the variables have integer exponents that are positive this is a polynomial. (x 7 + 2x 4 - 5) * 3x Not a polynomial because a term has a negative exponent 5x Not a polynomial because a term has a fraction exponent 3x ½ +2 Not 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. (2x 2 +3x) ÷ (x) 26

27 Vertical line test  A curve is the graph of a function if and only if no vertical line intersects the curve more than once. 27

28 Section 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+b where m and b are constants. 28

29 The Slope of a Line  The slope of the nonvertical line passing through the points (Xl, YI) and (X2, Y2) is given by the formula Slope = (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 b  We say that y=mx+b is the equation of the line whose slope is m and whose Y intercept is (0, b). 29

30 The Slope of a Line  Find the slope of the line joining the points (-2, 5) and (3, -1).  Solution: Slope = = = 30

31 The Point-Slope Form  The equation y - y o = m(x - xo) is an equation of the line that passes through the point (xo,yo) and that has slope equal to m. 31

32 The Point-Slope Form  Find the equation of the line that passes through the point (5,1) with a slope of ½  Solution  Use the formula y-y 0 =m(x-x 0 ) with (x 0,y 0 )=(5,1) and m = ½ to get y - 1 = ½(x - 5)  which you can rewrite as y = x - 32

33 Parallel, perpendicular lines  Let m 1 and m 2 be the slopes of the nonvertical lines L 1, and L 2. Then L 1 and L 2 are parallel if and only if m 1 =m 2  L 1 and L 2 are perpendicular if and only if m 2 = -1/ m 1 33

34 Example:  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 L 2 perpendicular to L through Q(2, -3).  Solution By 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 -4/3 1) Any line parallel to L must also have slope m = - 4/3 and the required line L 1 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 L 2, contains Q(2, -3), we have y + 3 = 3/4*(x-2) Which means: y = x - 34

35 Section 4: Limits  Let 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).  In both cases as x approaches 1, f(x) approaches 4. Intuitively, we say that lim x → 1 f(x) = 4. 35

36 The limit of a Function  If 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 writing Lim f(x) = L X  C Note: the limit of a constant is the constant itself, and the limit of f(x) = x as x approaches c is c. 36

37 Algebraic Properties of Limits 37

38 Limits of Two linear Functions  For any constant k, = k and That is, the limit of a constant is the constant itself, and the limit of f(x) = x as x approaches c is c. 38

39 39

40 Limits of Polynomials and Rational Functions  If p(x) and q(x) are polynomials, then lim(x → c) p(x) = p(c) And = Note: q( c ) must be different from zero 40

41 Limits at infinity  If the values of the function f(x) approach the number L as x increases without bound, we write  Similarly, we write when the functional values f(x) approach the number M as x decreases without bound. 41

42 42

43 Reciprocal Power Rules  If A and k are constants with k > 0 and x k is defined for all x, then: 43

44  We say that is an infinite limit if f(x) increases or decreases without bound as X  C.  We write if f(x) increases without bound as X  C if f(x) decreases without bound as X  C 44

45 Section 5: One-Sided Limits and Continuity  The graph of a continuous function can be drawn without lifting the pencil from the paper  A function is not continuous where its graph has a "hole or gap" 45

46 One-Sided Limits  If f(x) approaches L as x tends toward c from the left (x < c), we write  Likewise, if f(x) approaches M as x tends toward c from the right (c < x), then 46

47 Existence of a limit  The two-sided limit exists if and only if the two one-sided limits and both exist and are equal, and then = = 47

48 Continuity  A 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 48

49 Continuity 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 49

50 END OF CHAPTER I 50


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