Week 0 VK Room: M1.30 www.vincent-knight.com knightva@cf.ac.uk Last updated September 25, 2014 1 / 76
Overview Algebra Numbers Exponents Inequalities Graphs Linear Equations Quadratic Equations Complex Numbers Systems of Equations Induction Calculus Functions Differentiation Integration Probability Support material 2 / 76
Algebra Wikipedia: “Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.” 3 / 76
Numbers • Integers: Z = { . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . . } • Rationals: � { � ∃ p , q ∈ Z for which a = p } � Q = a � q • Real numbers: Z ⊂ Q ⊂ R 4 / 76
Exponents If a and b are any positive real numbers and x and y are any real numbers then: 1. a x a y = a x + y 2. a 0 = 1 3. a − x = 1 a x 4. ( a x ) y = a xy 5. a x b x = ( ab ) x 5 / 76
Inequalities When solving inequalities it is important to keep in mind whether or not the operation we are using is an increasing or a decreasing one. 10 10 f ( x ) =5 x g ( x ) = − 5 x +10 8 8 f ( b ) 6 6 f ( a ) f ( a ) 4 4 f ( b ) 2 2 0.5 1 1.5 2 0.5 1 1.5 2 a b a b a < b ⇒ f ( a ) < f ( b ) a < b ⇒ g ( a ) > g ( b ) 6 / 76
Coordinates in the plane The location of a point in a plane can be specified in terms of right handed cartesian axes: The point (1 . 3 , . 75) is plotted above. In general for a point P = ( x , y ), x / y is called the abscissa/ordinate of P . 7 / 76
Graphs If x and y connected by an equation, then this relation can be represented by a curve or curves in the ( x , y ) plance which is known as the graph of the equation. The equation y = x 3 is plotted above. 8 / 76
Graphs One particular type of graph is the graph of a line: y = mx + b • m is called the gradient of the line. • b is called the y-intercept of the line. 9 / 76
Exercise Find the equation for the line going through the points { ( . 5 , 3) , (4 , 1 . 1) } : 10 / 76
Solution General form of line y = mx + b through { ( x 1 , y 1 ) , ( x 2 , y 2 ) } can be obtained: } } y 1 = mx 1 + b ⇒ m ( x 1 − x 2 ) = y 1 − y 2 y 2 = mx 2 + b c ( a 2 − a 1 ) = a 2 b 1 − a 1 b 1 which gives: m = y 1 − y 2 x 1 − x 2 c = x 2 y 1 − x 1 y 2 x 2 − x 1 11 / 76
Solution So for ( x 1 , y 1 ) = ( . 5 , 3) and ( x 2 , y 2 ) = (4 , 1 . 1) we have: 1 . 9 m = − 3 . 5 ≈ − . 54 c = 11 . 45 3 . 5 ≈ 3 . 27 12 / 76
Exercise Where does the line y = − . 54 x + 3 . 27 intersect the y -axis and the x -axis? 13 / 76
Exercise Where does the line y = − . 54 x + 3 . 27 intersect the y -axis and the x -axis? This is equivalent to solving: y = − . 54 × 0 + 3 . 27 and 0 = − . 54 x + 3 . 27 13 / 76
Solving Linear Equations In general equations of the form: y = mx + b are solved by muliplying or adding various constants. 0 = − . 54 x + 3 . 27 ⇔ 0 − 3 . 27 = ( − . 54 x + 3 . 27) − 3 . 27 1 1 − 3 . 27 = − . 54 x ⇔ − 3 . 27 × − . 54 = . 54 x × − . 54 x ≈ 6 . 06 14 / 76
Solving Linear Equations In general equations of the form: y = mx + b are solved by muliplying or adding various constants. y = mx + b ⇔ y − b = ( mx + b ) − b y − b = mx ⇔ ( y − b ) × 1 m = mx × 1 m x = y − b m 15 / 76
Quadratic A “quadratic” is an expression of the form: ax 2 + bx + c • a is called the quadratic coefficient, • b is called the linear coefficient, • c is called the constant term or free term. 16 / 76
Quadratic 17 / 76
Solving a Quadratic Equation General solution of the equation: ax 2 + bx + c = 0 is given by: √ b 2 − 4 ac x = − b ± 2 a 18 / 76
Exercise Solve the equation: 3 x 2 − 3 2 x − 2 = 0 19 / 76
Solution From the previous formula we have: √( 3 ) 2 − 4 × 3 × ( − 2) √ 3 9 3 2 ± 4 + 24 2 ± 2 x = ⇔ x = 2 × 3 6 √ 9 + 96 √ 1 x = 3 ⇔ x = 1 105 2 12 ± 4 ± 6 12 20 / 76
Exercise Solve the equation: 4 x 2 − 2 x + 10 = 3 21 / 76
Solution From the previous formula we have: √ ⇔ x = 2 ± √− 108 2 2 − 4 × 4 × 7 x = 2 ± 2 × 4 8 √ ⇔ x = 2 ± i √ 3 × 36 i 2 108 x = 2 ± 8 8 √ √ x = 2 ± 6 i 3 = 1 4 ± 3 3 4 i 8 22 / 76
Very brief description of Complex Numbers i 2 = − 1 Complex numbers: C = { a + bi | a , b ∈ R } If z = a + ib : • a is the real part of z . • b is the imaginary part of z . 23 / 76
Solving Systems of Equations A system of equations is a collection of equations involving the same set of variables. For example: 3 x + 2 y =1 2 x − 2 y = − 2 Various techniques can be used to solve such a problem. 24 / 76
Elimination of Variables • Use first equation to obtain expression for first variable as a function of other variables. • Substitute and use second equation to obtain expression for second variable as a function of other variables. • etc... 25 / 76
Exercise Solve: 3 x + 2 y =1 2 x − 2 y = − 2 26 / 76
Solution First equation gives: 3 x + 2 y = 1 ⇒ x = 1 − 2 y 3 Substituting in to second equation gives: ( 1 − 2 y ) 2 − 2 y = − 2 3 which implies: y = 4 5 Substituting in to our expression for x we get: x = − 1 5 27 / 76
Shorthand notation • Summation: n ∑ a i = a 1 + a 2 + a 3 + · · · + a n i =1 • Multiplication: n ∏ a i = a 1 × a 2 × a 3 × · · · × a n i =1 28 / 76
Examples • Summation: 4 i × 2 i = 1 × 2+2 × 2 2 +3 3 +4 × 2 4 = 2+8+3 × 8+4 × 16 = 98 ∑ i =1 • Multiplication: 3 k 2 = 1 × 2 2 × 3 2 = 36 ∏ k =1 29 / 76
Proof by Induction Technique often used to prove algebraic relationships. Basic idea: • Prove that something is true at the start. • Prove that if something is true at point k then it is true at point k + 1. 0 k k+1 30 / 76
Proof by Induction Technique often used to prove algebraic relationships. Basic idea: • Prove that something is true at the start. • Prove that if something is true at point k then it is true at point k + 1. 0 k k+1 30 / 76
Exercise Prove that: n i = n ( n + 1) ∑ 2 i =0 31 / 76
Solution • True for n = 0?: 0 i = 0 and n ( n + 1) ∑ = 0 2 i =0 • If true for n = k , true for n = k + 1?: k +1 k i + k + 1 = k ( k + 1) + k + 1 = ( k + 1)( k + 2) ∑ ∑ i = 2 2 i =0 i =0 32 / 76
Calculus Wikipedia: “Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations” 33 / 76
D f E Functions A function f is a rule that assigns to each element x in a set D exactly one element, called f ( x ), in a set E . 34 / 76
Functions A function f is a rule that assigns to each element x in a set D exactly one element, called f ( x ), in a set E . D f E 34 / 76
Functions A function f is a rule that assigns to each element x in a set D exactly one element, called f ( x ), in a set E . D f E • We usually consider functions for which the sets D and E are sets of real numbers. • The set D is called the domain of the function. • The range of f is the set of all possible values of f ( x ) as x varies throughout the domain. • A symbol that represents an arbitrary number in the domain of a function f is call an independent variable. • A symbol that represents a number in the range of f is called a dependent variable. 34 / 76
Example The function f ( x ) = 3 x 3 − 4 x − 4 is plotted below: 35 / 76
Even and Odd Functions • If a function f satisfies f ( − x ) = f ( x ) for all x in its domain then f is called an even function: • If a function f satisfies f ( − x ) = − f ( x ) for all x in its domain then f is called an odd function: 36 / 76
Tangent Curves The tangent line to the curve y = f ( x ) at the point P = ( a , f ( a )) is the line through P with gradient: f ( x ) − f ( a ) m = lim x − a x → a 37 / 76
Tangent Curves 38 / 76
Tangent Curves 38 / 76
Tangent Curves 38 / 76
Tangent Curves 38 / 76
Tangent Curves 38 / 76
Derivative The derivative of a function f at a number a , denoted by f ′ ( a ) is: f ( a + h ) − f ( a ) f ′ ( a ) = lim h h → 0 39 / 76
Exercise Find the derivative of the function f ( x ) = x 2 − 3 x + 2 at the number a . 40 / 76
Solution f ( a + h ) − f ( a ) f ′ ( a ) = lim h h → 0 ( a + h ) 2 − 3( a + h ) + 2 a 2 − 3 a + 2 ( ) ( ) − − f ( a ) = lim h h → 0 a 2 + 2 ah + h 2 − 3 a − 3 h + 2 − a 2 + 3 a − 2 = lim h h → 0 2 ah + h 2 − 3 h = lim = lim h → 0 2 a + h − 3 h h → 0 =2 a − 3 41 / 76
Rules of Differentiation • The Power Rule: d dx ( x n ) = nx n − 1 • The Constant Multiple Rule: dx ( cf ( x )) = c d d dx ( f ( x )) • The Sum Rule: dx ( f ( x ) + g ( x )) = d d dx ( f ( x )) + d dx ( g ( x )) 42 / 76
Rules of Differentiation • The Product Rule: dx ( f ( x ) g ( x )) = g ( x ) d d dx ( f ( x )) + f ( x ) d dx ( g ( x )) • The Quotient Rule: ( f ( x ) = g ( x ) d dx ( f ( x )) − f ( x ) d dx ( g ( x )) d ) g ( x ) 2 dx g ( x ) 43 / 76
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