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1 Chapter One PREREQUISITES FOR CALCULUS Sets and Intervals: Set: is a collection of things under certain conditions. Interval: is a set of all real numbers between two points on the real number line. (it is a subset of real numbers) 1. Open interval: is a set of all real numbers between A&B excluded (A&B are not elements in the set). {x: A < x < B} or (A, B) 2. Closed interval: is a set of all real numbers between A&B included (A&B are elements in the set). {x: A ≤ x ≤ B} or [A, B] 3.Half-Open interval (Half-Close): is a set of all real numbers between A & B with one of the end-points as an element in the set. a) (A, B]= {x: A < x ≤ B} b) [A, B)= {x: A ≤ x < B} Union and Intersections of intervals: if A and B are two sets then: The union is the set whose numbers belong to A or B (or both) and denoted by (A B), and intersection is the set whose numbers belong to both A and B and denoted by (A∩B). 2 Inequalities: Rules for Inequalities Let a, b, and c are real numbers, then: 1. if a < b and b < c then a < c 2. a < b a + c < b + c 3. a < b a - c < b - c 4. a < b and c > o ac < bc (c is positive) 5. a < b and c < o ac > bc (c is negative) special case a < b -a > -b 6. if a < b and c < d then a+c < b+d 7. a > 0 1 0 a 8. If a and b are both positive or both negative, and a < b 1 1 a b Analytical Geometry (Coordinate in the Plane) Each point in the plane can be represented with a pair of real numbers (a,b), the number a is the horizontal distance from the origin to point P, while b is the vertical distance from the origin to point P. The origin divides the x-axis into positive xaxis to the right and the negative x-axis to the left, also, the origin divides the y-axis into positive y-axis upward and the negative y-axis downward. The axes divide the plane into four regions called quadrants, numbered I, II, III and IV. 3 Distance between Points and (Mid-Point Formula): Distance between points in the plane is calculated with a formula that comes from Pythagorean Theorem: Distance Formula for Points in the Plane yo The distance between P(x1,y1) and Q(x2,y2) d (x) 2 (y ) 2 ( x2 x1 ) 2 ( y2 y1 ) 2 and the mid-point formula: xo x1 x2 ; 2 yo y1 y2 2 Slope and Equation of Line DEFENITION: Slope The constant m rise y y2 y1 run x x2 x1 is the slope of nonvertical line P1P2 m1 1 , m2 m2 1 m1 To see this, notice by inspecting similar triangles that m1=a/h and m2=-h/a. Hence m1. m2= (a/h). (-h/a)= -1 xo 4 Point-Slope Equation: We can write an equation for a non-vertical straight line L if we know its slope m and the coordinate P(x,y) of one point P1(x1,y1) on it. If P(x,y) is any other point on L, then we can use two points P1 and P to compute the P1(x1,y1) slope, m y y1 x x1 So that y - y1 =m (x - x1) or y = y1 + m (x - x1) The equation y = y1 + m (x - x1) is the point-slope equation of the line that passes through the point P1(x1,y1) and has slope m. The Distance from a Point to a Line: d To calculate the distance from a certain point P(x1,y1) to a line L: P(x1,y1) Q(x2,y2) 1. Find an equation for the line L` that pass L` through a point P(x1,y1) and perpendicular to the line L. 2. Find the point Q1(x2,y2) where L` meet with L. 3. Calculate the distance between P and Q. 4. to find d use the distance between two points formula. Or you can use the following formula: The distance (d) between the line L is Ax + By + C= 0 and the point P(x1,y1) d Ax1 By1 C A2 B 2 5 Angles between Two Lines: L1,m1 y To find the angle between the lines L1 and L2 which have slopes m1 and m2 respectively, draw horizontal line passes through the point of their intersections as shown. L2,m2 So Where: m1=tan tan tan = tan(-) tan tan 1 tan . tan x 0 m2=tan m1 m2 1 m1m2 Functions DEFINITION: Function A function from a set D (domain) to a set R (range) is a rule that assigns to unique (single) element f(x) R to each element x D f: x f(x) it means that f sends x to f(x) 1 it means that f sends x to x2 f: x f ( x) y 1 x2 R The set of x is called the "Domain" of the function (Df). The set of y is called the "Range" of the function (Rf). x & y are variables. x is independent variable. y is dependent variable. Domain (Df): is the set of all possible inputs (x-values). Range (Rf): is the set of all possible outputs (y-values). To find Domain (Df) and the Range (Rf) the following points must be noticed: 1. The denominator in a function must not equal zero (never divide by zero). 6 2. The values under even roots must be positive. Absolute Value Function: it is defined as: x if y x x2 x if x0 x0 Absolute Value Properties 1. |-a | = | a| A number and its additive inverse or negative have the same absolute value. 2. |ab | = | a||b | The absolute value of a product is the product of the absolute values. 3. a a b b The absolute value of a quotient is the quotient of the absolute values. 4. |a +b| ≤ |a | +| b| The triangle inequality. The absolute value of the sum of two numbers is less than or equal to the sum of their absolute values. Absolute Values and Intervals If a is any positive number, then 5. | x | = a if and only if x = ±a 6. | x | < a if and only if -a < x < a 7. | x | > a if and only if x > a or x < - a 8. | x | ≤ a if and only if -a ≤ x ≤ a 9. | x | ≥ a if and only if x ≥ a or x ≤ - a The Greatest Integer Function (Stepped Function): The function whose values at any number x is the greatest integer less than or equal to x is called greatest integer function. It is denoted x , or in some books [x] or [[x]] or int x The greatest integer function: [x] ≤ x <[x] +1 7 Properties of greatest integer value: 1. [[[[x]]]] = [x] 2. [x+n] =[x] + n where n is integer 3. [x-n] =[x] - n where n is integer 4. -[x] ≠ [-x] Function Defined in Pieces: While some functions defined by single formulas, others are defined by applying different formulas to different parts of their domains. One example is the absolute value function x if y x x2 x if x0 x0 Sums, Difference, Product and Quotients of Functions: Definition: If f and g are functions, then we define the functions (f+g)(x)= f(x)+g(x) Sum Difference (f-g)(x)= f(x)-g(x) or (g-f)(x)= g(x)-f(x) Product (f.g)(x)= f(x).g(x) Quotient (f/g)(x)= f(x)/g(x); where g(x) ≠0 …(1) or (g/f)(x)= g(x)/f(x); where f(x) ≠0 …(2) are also functions of (x), defined for any value of x that lies in both Df and Dg ( x D f Dg ), except the points which g(x) =0 in eq.(1) or f(x) =0 in eq.(2) Composition of Functions: Definition: If f and g are functions, the composite f o g "f composed with g" or g o f "g composed with f" are defined by: (f o g)(x) =f (g(x)) and (g o f)(x) =g (f(x)) respectively. 8 Graph of Functions (Graph of Curves): To graph the curve of a function, we can follow the following steps: 1. Find the domain and range of the function. 2. Check the symmetry of the function 3. Find (if any found) points of intersection with x-axis and y-axis. 4. Choose some another points on the curve. 5. Draw s smooth line through the above points. Symmetry Tests for Graphs: If f(x,y) = 0 is any function then: 1. Symmetry about x-axis: If f(x,-y)= f(x,y) 2. Symmetry about y-axis: If f(-x,y)=f(x,y) It is called an even function. 3. Symmetry about the origin: If f(-x,-y)=f(x,y) It is called an odd function DEFINITIONS Even Function, Odd Function A function y = f(x) is an even function of x if f(-x) = f(x) symmetry about y-axis odd function of x if f(-x) = -f(x) symmetry about origin for every x in the function's domain. Shifting, Shrinking and Stretching: Shift formulas: (for c > 0) Vertical shifts y= f(x)+c or y-c= f(x) y= f(x)-c or y+c= f(x) Horizontal shifts y= f(x+c) y= f(x-c) shifts the graph of f up by c units. shifts the graph of f down by c units. shifts the graph of f left by c units. shifts the graph of f right by c units. 9 Shrinking, Stretching and Reflecting Formulas: (for c > 1) y=c f(x) Stretches the graph of f c units along y-axis. y 1 f ( x) c Shrinks the graph of f c units along y-axis. y= f(cx) Shrinks the graph of f c units along x-axis. x y f( ) c (for c = -1) y=- f(x) y= f(-x) Stretches the graph of f c units along x-axis. Reflects the graph of f across the x-axis. Reflects the graph of f across the y-axis. Trigonometric Functions We measure angles in degrees, but in calculus it is usually best to use radians. 1. Degree measure: One degree (1o) is the measure of an angle generated by 1/360 of revolution. 2. Radian measure: The radian measure of the angle ACB at the center of the unit circle (circle with radius equals one unit) equals to the length of the arc that the angle cuts from the unit circle. If angle ACB cuts an arc A`B` from a second circle centered at C, then circular sector A`CB` will be similar to circular sector ACB. In particular, Length of arc A`B` Length of arc AB Radius of sec ond circle Radius of first circle In notations s r 1 s r or s=r To find the relation between degree measure and radian measure, you know that one circle equals 360o in degree and 2 in radians so: 2 radians= 360o radians= 180o 10 1o 180 rad 0.01745rad o 180 and 1rad 57 o17'44.8" The six basic trigonometric functions: sine: sin y , r cosecant: csc x r cosine: cos , tangent: tan secant: sec 1 r , sin y 1 r cos x sin y 1 cos x , cotangent: cot cos x tan sin y from Pythagoras theorem: x2+ y2 = r2 x2 y2 1 r2 cos2sin2 x2 y 2 1 r2 r2 true for all values of When we divided eq.(1) by cos2yields: cos 2 sin 2 1 2 2 cos cos cos 2 1tan2 sec2 And when we divided eq.(1) by sin2yields: cos 2 sin 2 1 2 2 sin sin sin 2 cot2 +1 csc2 Identities: - Periodicity: A function is periodic with period p if f(x+p)= f(x) for every value of x. cos(±2) =cos sin(±2) =sin tan(±2) =tan cot(±2) =cot sec(±2) =sec csc(±2) =csc - Symmetry: 11 Even functions Odd functions cos(-x)=cosx sec(-x)=secx sin(-x)=-sinx tan(-x)=-tanx cot(-x)=-cotx csc(-x)=-cscx - Shift formulas: sin(x + /2) = cos(x); cos(x + /2) = -sin(x) sin(x - /2) = -cos(x); cos(x - /2) = sin(x) - Addition formulas: cos(A+B)=cosA cosB – sinA sinB sin(A+B)=sinA cosB + cosA sinB - Double angle formulas: cos2=cos2sin2 sin2 2.sin .cos - Half angle formulas: cos 2 1 cos 2 2 sin 2 1 cos 2 2 Graph of trigonometric functions: 1. y = sin x Domain and Range of the function v Df=(-∞,∞) From Figure nearby, we conclude that: -r ≤ v ≤ r 1 v 1 r (divide the inequality by r) 1 sin 1 Rf=[-1,1] Symmetry: f(-x) = sin(-x) =-sinx = - f(x) ≠ f(x) u 12 So it is an odd function (it is symmetric about the origin). Additional points: 1 0 0 x y 0 3 -1 2 0 2. y = cos x Domain and Range of the function Df=(-∞,∞) From Figure, we conclude that: -r ≤ u ≤ r 1 (divide the inequality by r) u 1 r 1 cos 1 Rf=[-1,1] Symmetry: f(-x) = cos(-x) =cosx = f(x) ≠- f(x) So it is an even function (it is symmetric about the y-axis). Additional points: x y 3. y = tan x = sin x cos x Domain and Range of the function cosx ≠ 0 Df= R\{ x n 2 xn 2 ; n=±1, ±3, ±5 ; n=±1, ±3, ±5} Rf=(-∞,∞) Symmetry: f(-x) = tan(-x) = sin( x) sin x =-tanx ≠ f(x) cos( x) cos x 0 1 0 -1 3 0 2 1 13 =- f(x) So it is an odd function (it is symmetric about the origin). Asymptotes: To find vertical asymptote put the denominator equal to zero. xn cosx = 0 2 ; n=±1, ±3, ±5 Additional points: x y ±∞ 0 0 4. y = cot x = 0 3 ±∞ 2 0 cos x sin x Domain and Range of the function x=n ; n=0, ±1, ±2, ±3 sinx ≠ 0 Df= R\{x=n; n=0, ±1, ±2, ±3} Rf=(-∞,∞) Symmetry: f(-x) = cot(-x) = cos( x) cos x =-cotx ≠ f(x) sin( x) sin x =- f(x) So it is an odd function (it is symmetric about the origin). Asymptotes: To find vertical asymptote put the denominator equal to zero. sinx = 0 x≠n ; n=0, ±1, ±2, ±3 Additional points: x y 0 ±∞ 5. y = sec x = 0 ±∞ 3 0 1 cos x Domain and Range of the function 2 ±∞ 14 cosx ≠ 0 xn Df= R\{ x n 2 ; n=±1, ±3, ±5 ; n=±1, ±3, ±5} 2 From Figure, we conclude that: -r ≤ u ≤ r 1 (divide the inequality by r) u 1 r 1 cos 1 sec 1 sec 1 or cos 1 1 1 cos sec 1 Rf= R\(-1,1) Symmetry: f(-x) = sec(-x) = 1 1 =sec x = f(x) cos( x) cos x ≠- f(x) So it is an even function (it is symmetric about the y-axis). Asymptotes: To find vertical asymptote put the denominator equal to zero. cosx = 0 xn 2 ; n=±1, ±3, ±5 Additional points: x y 6. y = csc x = 0 1 ±∞ -1 3 ±∞ 2 1 sin 1 1 sin x Domain and Range of the function sinx ≠ 0 x≠n; n=0, ±1, ±2, ±3 Df= R\{x=n; n=0, ±1, ±2, ±3} From Figure, we conclude that: -r ≤ v ≤ r 1 (divide the inequality by r) v 1 r csc 1 1 sin 1 csc 1 or csc 1 1 1 sin 15 Rf= R\(-1,1) Symmetry: f(-x) = csc(-x) = 1 1 =-cscx ≠ f(x) sin( x) sin x =- f(x) So it is an odd function (it is symmetric about the origin). Asymptotes: To find vertical asymptote put the denominator equal to zero. sinx = 0 x=n ; n=0, ±1, ±2, ±3 Additional points: x y x3 x y x 0 ±∞ 1 ±∞ 3 -1 2 ±∞