Mathematical Preliminaries. Developed for the Members of Azera Global By: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.

Transcription

1 Mathematical Preliminaries Developed or the Members o Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.

2 Outline Chapter One, Sets: Slides: 3-27 Chapter Two, Introduction to unctions: Slides: Chapter Three, Logarithms: Slides: Chapter Four: Trigonometr Slides: Chapter Five: Intro to Vectors Slides: Chapter Si: Dierential Calculus: Slides: Chapter Seven: Partial Dierentiation: Slides: Chapter Eight: Integral Calculus: Slides:

3 Sets Developed or the Members o Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.

4 Outline Slide 5-Introduction: Part One Slide 6-Introduction: Part Two Slide 7-Properties: Part One Slide 8-Properties: Part Two Slide 9-Terminolog Slide 10-Venn Diagrams Slide 11-Properties and Notation: Part One Slide 12-Properties and Notation: Part Two Slide 13-Properties and Notation: Part Three Slide 14-Equivalence: Part One Slide 15-Equivalence: Part Two Slide 16-Power Sets Slide 17-Tuples Slide 18-Cartesian Products Slide 19-Notation and Quantiiers Slide 20-Set Operations Slide 21-Set Operations: Unions Slide 22-Set Operations: Intersections Slide 23-Disjoint Sets Slide 24-Set Dierences Slide 25-Set Complements Slide 26-Generalized Unions Slide 27-Generalized Intersections

5 Introduction: Part One We have all implicitl dealt with sets Integers Z, rationals Q, naturals N, reals R, etc. We will develop more ull The deinitions o sets The properties o sets The operations on sets Deinition: A set is an unordered collection o unique objects Sets are undamental discrete structures and or the basis o more comple discrete structures like graphs

6 Introduction: Part Two The objects in a set are called elements or members o a set. A set is said to contain its elements Notation, or a set A: A: is an element o A A: is not an element o A

7 Properties: Part One Two sets, A and B, are equal is the contain the same elements. We write A=B. Eample: {2,3,5,7}={3,2,7,5}, because a set is unordered Also, {2,3,5,7}={2,2,3,5,3,7} because a set contains unique elements However, {2,3,5,7} {2,3}

8 Properties: Part Two A multi-set is a set where ou speci the number o occurrences o each element: {m 1 a 1,m 2 a 2,,m r a r } is a set where m 1 occurs a 1 times m 2 occurs a 2 times m r occurs a r times In Databases, we distinguish A set: elements cannot be repeated A bag: elements can be repeated

9 Terminolog The set-builder notation O={ Z =2k or some kz} reads: O is the set that contains all such that is an integer and is even A set is deined in intension when ou give its set-builder notation O={ Z 08 =2k or some k Z } A set is deined in etension when ou enumerate all the elements: O={0,2,4,6,8}

10 Venn Diagram: A set can be represented graphicall using a Venn Diagram U A B a z C

11 Properties and Notation: Part One A set that has no elements is called the empt set or null set and is denoted A set that has one element is called a singleton set. For eample: {a}, with brackets, is a singleton set a, without brackets, is an element o the set {a} Note the subtlet in {} The let-hand side is the empt set The right hand-side is a singleton set, and a set containing a set

12 Properties and Notation: Part Two For an set S S and S S A is said to be a subset o B, and we write A B, i and onl i ever element o A is also an element o B That is, we have the equivalence: A B A B

13 Properties and Notation: Part Three A set A that is a subset o a set B is called a proper subset i A B. That is there is an element B such that A We write: A B, I there are eactl n distinct elements in a set S, with n a nonnegative integer, we sa that: S is a inite set, and The cardinalit o S is n. Notation: S = n. A set that is not inite is said to be ininite

14 Equivalence: Part One To show that a set is a subset o, proper subset o, or equal to another set. To prove that A is a subset o B, use the equivalence discussed earlier A B A B To prove that A B it is enough to show that or an arbitrar nonspeciic element, A implies that is also in B. To prove that A is a proper subset o B, ou must prove A is a subset o B and B A

15 Equivalence: Part Two To show that two sets are equal, it is suicient to show independentl much like a biconditional that A B and B A Logicall speaking, ou must show the ollowing quantiied statements: A B B A

16 Power Set The power set o a set S, denoted PS, is the set o all subsets o S. Eamples Let A={a,b,c}, PA={,{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}} Let A={{a,b},c}, PA={,{{a,b}},{c},{{a,b},c}} Note: the empt set and the set itsel are alwas elements o the power set. The power set is a undamental combinatorial object useul when considering all possible combinations o elements o a set Let S be a set such that S =n, then PS = 2 n

17 Tuples Sometimes we need to consider ordered collections o objects The ordered n-tuple a 1,a 2,,a n is the ordered collection with the element a i being the i-th element or i=1,2,,n A 2-tuple n=2 is called an ordered pair

18 Cartesian Product Let A and B be two sets. The Cartesian product o A and B, denoted AB, is the set o all ordered pairs a,b where aa and bb AB = { a,b aa b B } The Cartesian product is also known as the cross product A subset o a Cartesian product, R AB is called a relation. Note: AB BA unless A= or B= or A=B Cartesian Products can be generalized or an n-tuple The Cartesian product o n sets, A 1,A 2,, A n, denoted A 1 A 2 A n, is A 1 A 2 A n ={ a 1,a 2,,a n a i A i or i=1,2,,n}

19 Notation with Quantiiers Whenever we wrote P or P, we speciied the universe o discourse using eplicit English language Now we can simpli things using set notation! Eample R 2 0 Z 2 =1 Also miing quantiiers: a,b,c R C a 2 +b+c=0

20 Set Operations Arithmetic operators +,-,, and set operators eist and act on two sets to give us new sets Union Intersection Set dierence Set complement Generalized union Generalized intersection

21 Set Operators: Union The union o two sets A and B is the set that contains all elements in A, B, r both. We write: AB = { A B } U A B

22 Set Operators: Intersection The intersection o two sets A and B is the set that contains all elements that are element o both A and B. We write: A B = { A B } U A B

23 Disjoint Sets Two sets are said to be disjoint i their intersection is the empt set: A B = U A B

24 Set Dierence The dierence o two sets A and B, denoted A\B or A B, is the set containing those elements that are in A but not in B A - B = { A B } U A B

25 Set Complement Deinition: The complement o a set A, denoted A, consists o all elements not in A. That is the dierence o the universal set and U: U\A A= A = { A } U A A

26 Generalized Union The union o a collection o sets is the set that contains those elements that are members o at least one set in the collection n A i = A 1 A 2 A n i= 1

27 Generalized Intersection The intersection o a collection o sets is the set that contains those elements that are members o ever set in the collection n A i = A 1 A 2 A n i=1

28 Chapter Two: Introduction to Functions Developed or Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.

29 Outline Slide 30-Relations and Functions Slide 31-Introduction to Functions Slide 32-Tables and Graphs Slide 33-Function Notation Slide 34-Linear Functions Slide 35-The Co-ordinate Plane Slide 36-Graphing a Function

30 Relations and Functions Relation A relation is an set o ordered pairs. A special kind o relation, called a unction, is ver important in mathematics and its applications. Function A unction is a relation in which, or each value o the irst component o the ordered pairs, there is eactl one value o the second component. In a relation, the set o all values o the independent variable is the domain. The set o all values o the dependent variable is the range

31 Introduction to Functions F G F is a unction. G is not a unction.

32 Tables and Graphs O Table o the unction, F Graph o the unction, F

33 Function Notation When a unction is deined with a rule or an equation using and or the independent and dependent variables, we sa is a unction o to emphasize that depends on. We use the notation =, called unction notation, to epress this and read, as o. The letter stands or unction. For eample, i = 5 2, we can name this unction and write = 5 2. Note that is just another name or the dependent variable.

34 Linear Function A unction that can be deined b = a + b, or real numbers a and b is a linear unction. The value o a is the slope o m o the graph o the unction. Beore we can draw a graph o our unction we must look at the co-ordinate plane or the Cartesian Co-ordinate plane.

35 The Co-ordinate Plane A unction that can be deined b = a + b, The plane o the grid is called the coordinate plane. The horizontal number line is called the. -ais The vertical number line is called the. - ais The point o intersection o the two aes is called the origin

36 Graphing a Function An ordered pair o real numbers, called coordinates o a point, locates a point in the coordinate plane. Each ordered pair corresponds to EXACTLY one point in the coordinate plane. The point in the coordinate plane is called the graph o the ordered pair. Locating a point on the coordinate plane is called graphing the ordered pair.

37 Chapter Three: Logarithmic Functions Developed or Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.

38 Outline Slide 39-Deinition: Logarithmic Functions Slide 40-Properties o Logarithms Slide 41-Properties o Natural Logarithms Slide 42-Properties o Natural Logarithms Slide 43-Characteristics o =log b Slide 44-Domain o Logarithmic Functions

39 Deinition: Logarithmic Function For > 0 and b > 0, b = 1, = log b is equivalent to b =. The unction = log b is the logarithmic unction with base b. Eponent Eponent Logarithmic orm: = log b Eponential Form: b =. Base Base

40 Properties o Logarithms For > 0 and b 1, log b b = The logarithm with base b o b raised to a power equals that power. b log b = b raised to the logarithm with base b o a number equals that number. General Properties: Common Logarithms 1. log b 1 = 0 1. log 1 = 0 2. log b b = 1 2. log 10 = 1 3. log b b = 0 3. log 10 = 4. b log b = log =

41 Properties o Natural Logarithms General Properties Natural Logarithms 1. log b 1 = 0 1. ln 1 = 0 2. log b b = 1 2. ln e = 1 3. log b b = 0 3. ln e = 4. b log b = 4. e ln = The unction =e has an inverse called the Natural Logarithmic Function. Y=ln

42 Properties o Natural Logarithms =e and =ln are inverses o each other!

43 Characteristics o = log b The -intercept is 1. There is no -intercept. The -ais is a vertical asmptote. = 0 I 0 < b < 1, the unction is decreasing. I b > 1, the unction is increasing. The graph is smooth and continuous. It has no sharp corners or edges = log b 0<b< = log b b>1

44 Domain o Logarithmic Functions Because the logarithmic unction is the inverse o the eponential unction, its domain and range are the reversed. c log b The domain is { > 0 } and the range will be all real numbers. For variations o the basic graph, sa the domain will consist o all or which + c > 0.

45 Chapter Four: Trigonometr Developed or Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.

46 Outline Slide 47-Right Triangle Trigonometr Slide 48-Right Triangle Trigonometr Slide 49-Trigonometric Ratios Slide 50-Reciprocal Functions Slide 51-Important Trigonometric Identities

47 Right Triangle Trigonometr Trigonometr is based upon ratios o the sides o right triangles. The si trigonometric unctions o a right triangle, with an acute angle, are deined b ratios o two sides o the triangle. The sides o the right triangle are: opposite adjacent hpotenuse hp θ adj opp

48 Right Triangle Trigonometr The hpotenuse is the longest side and is alwas opposite the right angle. The opposite and adjacent sides reer to another angle, other than the 90 o.

49 Trigonometric Functions sine, cosine, tangent, cotangent, secant, and cosecant. hp opp θ adj opp sin θ = cos θ = adj tan θ = hp hp hp hp csc = opp sec = cot = adj opp adj adj opp

50 Reciprocal Functions sin = 1/csc cos = 1/sec tan = 1/cot csc = 1/sin sec = 1/cos cot = 1/tan

51 Important Trigonometric Identities Reciprocal Identities sin = 1/csc cos = 1/sec tan = 1/cot cot = 1/tan sec = 1/cos csc = 1/sin Co unction Identities sin = cos90 cos = sin90 sin = cos π/2 cos = sin π/2 tan = cot90 cot = tan90 tan = cot π/2 cot = tan π/2 sec = csc90 csc = sec90 sec = csc π/2 csc = sec π/2 Quotient Identities tan = sin /cos Pthagorean Identities cot = cos /sin sin 2 + cos 2 = 1 tan = sec 2 cot = csc 2

52 Chapter Five Introduction to Vectors Developed or Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.

53 Outline Slide 54-Deinition Slide 55-Unit Vector: Part One Slide 56-Unit Vector: Part Two Slide 57-Coordinate Sstems Slide 58-Polar Coordinate Sstems Slide 59-Polar to Cartesian Coordinates Slide 60-Vector Addition Slide 61-Vector Multiplication: Part One Slide 62-Vector Multiplication: Part Two Slide 63-Vector Multiplication: Part Three A

54 Deinition Vector analsis is a mathematical tool with which electromagnetic EM concepts are most convenientl epressed and best comprehended. A quantit is called a scalar i it has onl magnitude e.g., mass, temperature, electric potential, population. A quantit is called a vector i it has both magnitude and direction e.g., velocit, orce, electric ield intensit. The magnitude o a vector or A A is a scalar written as A A

55 Unit Vector: Part One A unit vector along is deined as a vector whose magnitude is unit that is,1 and its direction is along e A e A A A e A 1 A A Thus: A Ae A which completel speciies direction e A A in terms o A and its

56 Unit Vector: Part Two A unit vector along is deined as a vector whose magnitude is unit that is,1 and its direction is along e A e A A A e A A A 1 which completel speciies direction e A Thus: A Ae A A in terms o A and its A vector A in Cartesian or rectangular coordinates ma be represented as A,A,Az Where: A e where A X, A, and A Z are called the components o A in the,, and z directions, respectivel; e,, and e e z are unit vectors in the, and z directions, respectivel. A e A z e z

57 Coordinate Sstems Common coordinate sstems are: Cartesian Polar Also called rectangular coordinate sstem - and - aes intersect at the origin Points are labeled,

58 Polar Coordinate Sstem Origin and reerence line are noted Point is distance r rom the origin in the direction o angle, ccw rom reerence line The reerence line is oten the -ais. Points are labeled r,

59 Polar to Cartesian Coordinates Based on orming a right triangle rom r and = r cos = r sin I the Cartesian coordinates are known: tan r 2 2

60 Vector Addition, Rules The three basic laws o algebra obeed b an given vector A, B, and C, are summarized as ollows: Commutative A B B A ka Ak Associative A B C A B C kla kla Distributive ka B ka kb where k and l are scalars

61 Vector Multiplication: Part One A B When two vectors and are multiplied, the result is either a scalar or a vector depending on how the are multiplied. The two tpes o vector multiplication: 1. Scalar or dot product: 2.Vector or cross product: A B A B The dot product o the two vectors A and B is deined geometricall as the product o the magnitude o B and The projection o A onto B or vice versa: A B ABcos where is the smaller angle between A and AB AB B

62 Vector Multiplication: Part Two The cross product o two vectors A and is deined as where A B ABsin ABe n is a unit vector normal to the plane containing B e n and. The direction o is determined using the righthand rule or the right-handed screw rule. e n B A e n Direction o and A B using a right-hand rule, b right-handed screw rule

63 Vector Multiplication: Part Three Note that the cross product has the ollowing basic properties: i It is not commutative: A B B A It is anticommutative: A B B A ii It is not associative: A B C A B C iii It is distributive: A B C A B A C iv A A 0 sin 0

64 Chapter Si Dierential Calculus Developed or Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E. 64

65 Outline Slide 66-Dierential Calculus Slide 67-Dierentiation and the Derivative Slide 68-Deinition o Derivative Slide 69-Various Smbols or Derivative Slide 70-Piecewise Linear Segments Slide 71-Eample o Simple Derivatives Slide 72-Chain Rule o Dierentiation Slide 73-Table o Derivative: Part One Slide 74-Table o Derivative: Part Two Slide 75-Higher Order Derivatives Slide 76-Application: Ma and Min Slide 77-Displacement and Velocit

66 Dierential Calculus The two basic orms o calculus are dierential calculus and integral calculus. This lecture will be devoted to the ormer. Integral Calculus will be presented in another lecture. 66

67 Dierentiation and the Derivative The stud o calculus begins with the basic deinition o a derivative. A derivative is obtained through the process o dierentiation, and the stud o all orms o dierentiation is collectivel reerred to as dierential calculus. I we begin with a unction and determine its derivative, we arrive at a new unction called the irst derivative. I we dierentiate the irst derivative, we arrive at a new unction called the second derivative, and so on. 67

68 Deinition o Derivative The derivative o a unction is the slope at a given point. 68

69 Various Smbols or the Derivative d d or ' or d d Deinition: d d lim 0 69

70 Piecewise Linear Segment, 2 2 d slope d ,

71 Eample o a Simple Derivative d d lim

72 Chain Rule o Dierentiation u u u d d u du ' u du d du d d where ' u d u du 72

73 Table o Derivatives: Part One ' Derivative Number a a ' D-1 u v u ' v ' D-2 u ' du d u du D-3 u d du d a 0 D-4 n n 0 n u n 0 n1 du nu d uv dv du u v d d u du dv v u v d d 2 v u e n 1 n D-5 D-6 D-7 D-8 u du e d D-9 73

74 Table o Derivatives:Part Two u a ln u log a u sin u cosu u du ln a a D-10 d 1 du u d D-11 du log a e u d D-12 1 du cosu d D-13 sin u du d D-14 tan u 2 sec du u d D-15 sin cos tan u u u 1 du 1 sin u 2 1 u d du 1 0 cos u 2 1 u d 1 du 1 tan u 2 1 u d 2 2 D-16 D-17 D-18 74

75 Higher-Order Derivatives d d ' d d d 2 2 '' d d d 2 2 d d d d d d d d d d d d 75

76 Applications: Maima and Minima 1. Determine the derivative. 2. Set the derivative to 0 and solve or values that satis the equation. 3. Determine the second derivative. a I second derivative > 0, point is a minimum. b I second derivative < 0, point is a maimum. 76

77 Displacement, Velocit, Acceleration Displacement Velocit Acceleration a d v dt dv dt 2 d dt 2 77

78 Chapter Seven Partial Derivatives and Gradients Developed or Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.

79 Outline Slide 80- Deinition: Partial Derivative Slide 81-Total Dierential Slide 82-Eact and Ineact Dierentials Slide 83-Properties: Part One Slide 84-Properties: Part Two Slide 85-Directional Derivatives: Part One Slide 86-Directional Derivatives: Part Two Slide 87-Directional Derivatives: Part Three Slide 88-Directional Derivatives: Part Four Slide 89-Directional Derivatives: Part Five Slide 90-Directional Derivatives: Part Si Slide 91-Directional Derivatives: Part Seven Slide 92-The Gradient: Part One Slide 93-The Gradient: Part Two Slide 94-The Gradient: Part Three Slide 95-The Tangent Plane Slide 96-The Gradient: Summar

80 ,, lim,, lim 0 0 the partial derivative o, with respect to and are second partial derivatives o two-variable unction, Deinition: Partial Derivative

81 The total dierential and total derivative...,..., unction variable n - or is dierential total the 0, and 0 as ],, [ ],, [,,,,,, and n n n d d d d d d d d Total Dierential

82 I a unction can be obtained b directl integrating its total dierential, the dierential o unction is called eact dierential, whereas those that do not are ineact dierential. ineact dierential doesnot eist, unction 3 2 eact dierential, 1 1 d d d d d d Eact and Ineact Dierentials Properties o eact dierentials: B A B A B A d d B d A,,, and,,, 2 2

83 d z d z dz z z dz z d d z dz z d d z z z,,, Properties: Part One relation cclic 1 relation reciprocit 0 constant a is i 0 constant a is i ] [ 1 z z z z z z z z d dz z dz z z d d

84 The chain rule du d du d du d du d du d u du d du d du d d d d u u n n i n i i i i n... and,...,, variables man or, and, or Properties: Part Two dt t t F dt t dt t F t t t t F t F t t t F t t F t t F dt t t F,,, ], [, ], [ ], [,,,,,, 2 2 Partial Dierentiation o Integrals

85 Directional Derivatives: Part One Recall that, i z =,, then the partial derivatives and are deined as:, lim 0 0 h0 h,, h, lim 0 0 h0, h, h

86 Directional Derivatives: Part Two Suppose that we now wish to ind the rate o change o z at 0, 0 in the direction o an arbitrar unit vector u = <a, b>. To do this, we consider the surace S with equation z =, [the graph o ] and we let z 0 = 0, 0. Then, the point P 0, 0, z 0 lies on S.

87 Directional Derivatives: Part Three The vertical plane that passes through P in the direction o u intersects S in a curve C. The slope o the tangent line T to C at the point P is the rate o change o z in the direction o u.

88 Directional Derivatives: Part Four Now, let: Q,, z be another point on C. P, Q be the projections o P, Q on the -plane. Then the vector parallel to U. So: P ' Q ' hu For some scaler h. Thereore: PQ ' ' ha, hb 0 = ha 0 = hb is

89 Directional Derivatives: Part Five From: 0 = ha 0 = hb Then: z z z 0 h h 0 ha, 0 hb 0, 0 h In the limit as h 0, we obtain the rate o change o z in the direction o U. This is called the directional derivative o in the direction o U. D, u lim h ha, 0 hb 0, 0 h

90 Directional Derivatives: Part Si I we deine a unction g o the single variable h b D,, a, b u I we deine a unction g o the single variable h b: g h ha, hb 0 0 then, b the deinition o a derivative, we have the ollowing equation. g '0 g h g0 lim h0 h 0 ha, 0 hb 0, 0 lim h0 h D, u 0 0

91 Directional Derivatives: Part Seven Suppose the unit vector u makes an angle θ with the positive -ais, as shown. Then, we can write u = <cos θ, sin θ> and the directional derivative becomes: D,, cos, sin u Notice that the directional derivative can be written as the dot product o two vectors: D,, a, b u,,, a, b,,, u

92 The Gradient: Part One The irst vector in that dot product occurs not onl in computing directional derivatives but in man other contets as well. This directional derivative is called the Gradient o. The Gradient o is written as: which is read as del I is a unction o two variables and then the gradient o, is deined as: i j We can rewrite the epression or the directional derivative as: This epresses the directional derivative in the direction o u as the scalar projection o the gradient vector onto u.,,,, D,, u u

93 The Gradient: Part Two For unctions o three variables, we can deine directional derivatives in a similar manner. The directional derivative o at 0, 0, z 0 in the direction o a unit vector u = <a, b, c> is: D,, z u lim h ha, 0 hb, z0 hc 0, 0, z0 h Using vector notation we can rewrite the directional derivative as: 0 hu 0 Du 0 lim h0 h where: 0 = < 0, 0 > i n = 2 0 = < 0, 0, z 0 > i n = 3

94 The Gradient: Part Three For a unction o three variables, the gradient vector, denoted b or grad, is:,, z And is written as:,, z,,, z,,,, z z,, z i j k z The directional derivative can be rewritten as: D,, z,, z u u The maimum value o the directional derivative D u is: and it occurs when u has the same direction as the gradient vector

95 Tangent Plane Suppose S is a surace with equation F,, z that is, it is a level surace o a unction F o three variables. Then, let P 0, 0, z 0 be a point on S. Then, let C be an curve that lies on the surace S and passes through the point P. The curve C is described b a continuous vector unction rt = <t, t, zt> The gradient vector at P F 0, 0, z0 is perpendicular to the tangent vector r t 0 and to an curve C on S that passes through P. Thus the direction o the normal line is given b the gradient vector. F 0, 0, z0

96 Summar o Gradient We now summarize the was in which the gradient vector is signiicant. For a unction o three variables and a point P 0, 0, z 0 in its domain we know that the gradient vector 0, 0, z0 gives the direction o astest increase o. On the other hand, we know that 0, 0, z0 is orthogonal to the level surace S o through P. So, it seems reasonable that, i we move in the perpendicular direction, we get the maimum increase.

97 Chapter Eight: Integral Calculus Developed or Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.

98 Outline Slide 99- Integral Calculus Slide 100-Total Dierential Slide 101-Anti-Derivative Slide 102-Indeinate and Deinite Integral Slide 103-Deinite Integral: Area under the Curve Slide 104-Guidelines Slide 105-Tabulation o Integrals Slide 106-Common Integrals: Part One Slide 107-Common Integrals: Part Two Slide 108-Displacement, Velocit, Acceleration 98

99 Integral Calculus The basic concepts o dierential calculus were covered in the preceding presentation. This presentation will be devoted to integral calculus, which is the other broad area o calculus. 99

100 Anti-Derivatives An anti-derivative o a unction is a new unction F such that df d 100

101 Indeinite and Deinite Integrals Indeinite d Deinite 1 2 d 101

102 Deinite Integral/ Area Under the Curve K Approimate Area k k a b Eact Area as Deinite Integral b d lim a d k k 102

103 Deinite Integral with Variable Upper Limit a d More proper orm with dumm variable u du a 103

104 Guidelines I is a non-zero constant, integral is either increasing or decreasing linearl. I segment is triangular, integral is increasing or decreasing as a parabola. I =0, integral remains at previous level. Integral moves up or down rom previous level; i.e., no sudden jumps. Beginning and end points are good reerence levels. 104

105 Tabulation o Integrals F d I b a d b I F F b F a a 105

106 Common Integrals: Part One n F d Integral Number a af I-1 u v u d v d I-2 a a I-3 n1 n1 I-4 n 1 a e a e I-5 a 1 ln I-6 sin a 1 cos a I-7 cos a a 1 sin a a I-8 2 sin a 1 1 sin 2 a 2 4a I-9 106

107 Common Integrals: Part Two 2 cos a 1 1 sin 2 a 2 4a I-10 sin a 1 sin a cos a 2 a a I-11 cos a 1 cos a sin a 2 a a I-12 sin a cos a 1 sin 2 a 2a I-13 sin a cos b cos a b cos a b 2 2 or a b 2 a b 2 a b I-14 a e ln 1 2 a b e a a 2 a 1 I-15 ln 1 1 tan 1 a ab b I-16 I

108 Displacement, Velocit, Acceleration a a t v v t velocit in meters/second m/s t displacement in meters m 2 2 acceleration in meters/second m/s dv dv dt a t dt dt dv at dt dv a t dt v a t dt C 1 dv v d dt vt d d dt v t dt dt v t dt C 2 108