Mathematics. Course Summary. hsn.uk.net. Advanced Higher HSN28000

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1 hsn.uk.net Advanced Higher Mathematics UNITS, AND 3 HSN8000 This document was produced specially for the HSN.uk.net website, and we require that any copies or derivative works attribute the work to Higher Still Notes. For more details about the copyright on these notes, please see

2 Contents Binomial Theorem and Partial Fractions Binomial Theorem Partial Fractions Matrices Gaussian Elimination Matrix Algebra 3 Transformations of the Plane 4 Sequences and Series 5 Arithmetic Sequences 5 Geometric Sequences 5 3 Arithmetic Series 5 4 Geometric Series 5 5 Maclaurin Series 5 Iterative Schemes 7 Recurrence Relations 7 Complex Numbers 8 Introduction 8 Algebraic Operations 8 3 Polar Form 9 4 Geometric Interpretations 9 5 Fundamental Theorem of Algebra 0 6 De Moivre s Theorem 0 Methods of Proof Functions Critical and Stationary Points Derivative Tests 3 Curve Sketching 3 Vectors 4 Direction Ratios and Cosines 4 Scalar and Vector Product 4 3 Equations of Lines 5 4 Equations of Planes 5 5 Intersections 6 Differentiation 8 Rules 8 Standard Derivatives 8 3 Inverse Differentiation 8 - ii - HSN000

3 4 Implicit Differentiation 9 5 Parametric Differentiation 9 Integration 0 Standard Integrals 0 Integration by Substitution 0 3 Areas 4 Volumes of Solids of Revolution 5 Partial fractions and Integration 6 Integration by Parts Differential Equations Separable Differential Equations First Order Linear Differential Equations 3 Second Order Homogeneous Linear Differential Equations 4 Second Order Non-Homogeneous Differential Equations 3 - iii -

4 Binomial Theorem and Partial Fractions Binomial Theorem n n n! Cr = r = r! ( n r)! For n, the Binomial Theorem states: n n n n n ( x + y) = 0 x x y x y 3 x y + + n y The rth term of ( x + y ) n is given by: Partial Fractions L. n n n n n 3 3 n n r x n r f ( x ) = A + B x a x + b x a x + b ( )( ) ( ) ( ) f ( x ) = A + B ( x a) ( x a) ( x a) y f ( x ) = A + B + C x a x b x c x a x b x c ( )( )( ) ( ) ( ) ( ) f ( x ) A B C x a x b = x a + x b + x b ( )( ) ( ) ( ) ( ) r. f ( x ) = A + B + C ( x a) ( x a) ( x a) ( x a) 3 3 f ( x) A Bx + C = + x a x + bx + c x a x + bx + c ( )( ) ( ) ( ) If the numerator is of a higher (or equal) degree than the denominator, then algebraic long division should be used first to obtain a proper rational function. Page HSN000

5 Matrices Gaussian Elimination Elementary row operations are used to turn the matrix into upper-triangular form. Geometric interpretation If one row s results are all zeros then the three planes do not intersect at a point but they intersect in a line. If one row has zeros in the x, y and z places but a non-zero number on the right of the line then there is no solution and the planes do not intersect. Ill-conditioning This occurs when a small difference in the coefficients of the equations causes a large change in the solution. Matrix Algebra Addition and Subtraction Matrices can only be added or subtracted if they have the same order. Add or subtract entry-wise. Multiplication by a Scalar Multiply each element by the scalar. Matrix Multiplication Multiply row in first matrix by column in the second matrix, then multiply row in first matrix by column in second matrix etc. EXAMPLE 3 4 ( 4) + ( 3 0) + ( ) ( ) + ( 3 ) + ( 3) 0 0 = ( 4) ( 0) ( 0 ) ( ) ( ) ( 0 3) = 4 5. Page HSN000

6 Determinants The determinant of a matrix a b A = c d is det A = ad bc. The determinant of a 3 3 matrix a b c A = d e f is g h i e f d f d e det A= a + b g i + c h i g h ( ) ( ) ( ) = aei fh+ bdi fg + cdh eg Inverses For an inverse of a matrix to exist the matrix must satisfy two conditions: the matrix must be square; the determinant of the matrix must be non-zero. The inverse of a matrix a b A = c d is d b A = det A c a. The inverse of a 3 3 matrix is found by starting with the augmented matrix ( A I ) and then using elementary row operations to turn it into ( I A ). Transpose of a Matrix The transpose of a matrix is found by interchanging the rows and columns. EXAMPLE 3 = = If A T 4 then A Page 3 HSN000

7 3 Transformations of the Plane Rotation Rotation through angle θ in the anticlockwise direction about the origin is cosθ sinθ represented by sinθ cosθ. EXAMPLE Rotation by π cos π is given by sin π 0 0. sin π cos π = Reflection Reflection in the line which makes an angle of θ with the x-axis is given by cosθ sinθ sinθ cosθ. EXAMPLE Reflection in the line y = x is given by Scaling Scaling is given by π π ( 4) ( 4) ( π) π ( ) cos sin 0 = 0. sin cos 4 4 λ 0 0 μ. This has the effect of multiplying the x-coordinate by λ and the y-coordinate by μ. Page 4 HSN000

8 Sequences and Series Arithmetic Sequences An arithmetic sequence has the form a, a+ d, a+ d, a+ 3 d, K where a is the first term and d is the common difference. The nth term is given by a+ ( n ) d. Geometric Sequences A geometric sequence has the form a ar ar 3,,,, ar K where a is the first term and r is the common ratio. The nth term is given by 3 Arithmetic Series n ar. ( ) The sum to n terms is given by Sn = a+ ( n ) d. n 4 Geometric Series The sum to n terms is given by S n a = n ( r ) r Provided < r <, the sum to infinity is given by. a S =. r 5 Maclaurin Series The method is to repeatedly differentiate the function and to substitute x = 0 into each answer. The Maclaurin series is then given by r= 0 r ( r ) x x x ( 3 ( ) ( ) ( ) ( ) ) x 3 ( 4) x 4 f 0 = f 0 + f 0 + f 0 + f ( 0) + f ( 0 ) + L r!!! 3! 4! Page 5 HSN000

9 Standard Maclaurin Series r x x x x x e x r= 0 r 3 4 = = L!! 3! 4! 3 4 x x x ln( + x) = x + +L x x x x = + +L! 4! 6! cos x x x sin x = x + +L 3! 5! 7! These can be used to find other Maclaurin series. e EXAMPLE 3 4 ( ) ( ) ( ) = + ( x ) L! 3! 4! x x x = + x L! 3! 4! x x x x Page 6 HSN000

10 Iterative Schemes Recurrence Relations A recurrence relation u = aun + b converges to a limit if < a <. The limit is given by Fixed point n+ b l =. a For a recurrence relation un+ = aun + b the corresponding equation is x = ax + b. The solution of the equation is the same as the limit and is called the fixed point. Calculating roots To calculate a root using an iterative scheme you must decide whether the scheme converges or diverges. Convergence is decided by differentiating and substituting in the approximate root. If the answer is between and, the scheme will converge. The closer the value is to zero the quicker the scheme will converge. Page 7 HSN000

11 Complex Numbers Introduction Definition A complex number hsa the form z = a+ bi where a and b are real numbers and i =. a is called the real part (Rez Argand diagram = a) and b is the imaginary part (Im z = b). A complex number z = a+ bi can be plotted on an Argand diagram. This is like a Cartesian coordinate diagram but with real (Re) and imaginary (Im) axes. Im b = + z a bi Algebraic Operations O a Re Adding and Subtracting Adding (or subtracting) complex numbers is done by adding (or subtracting) the real parts and the imaginary parts. Multiplying Multiplying two complex numbers is achieved by expanding brackets. Multiplying by i Multiplying a complex number by i will rotate it 90 anticlockwise about the origin on an Argand diagram. Complex Conjugate The complex conjugate of z is denoted z. If z = a+ bi then z = a bi. Geometrically this has the effect of reflecting z in the Re-axis. Page 8 HSN000

12 Division of complex numbers Division of two complex numbers can be done using the following process.. Find the complex conjugate of the denominator.. Multiply top and bottom by the conjugate of the denominator. 3. Express answer in the form a + bi. EXAMPLE ( i) i i = = = = i. + i ( + i)( i) i 3 Polar Form The polar form of a complex number z = a+ bi is expressed as ( cosθ sinθ) z = r + i where the modulus r = a + b and the argument θ is the angle between the positive Re-axis and the line from the origin to z. Principle value of the argument The principle value of the argument is the value which lies between π and π. 4 Geometric Interpretations z = r represents a circle with centre ( 0,0 ) and radius r. z a = r represents a circle with centre a (in the complex plane) and radius r. z a > r represents the points outside the circle stated above. z a < r represents the points inside the circle stated above. z a z b = represents the points which lie on the perpendicular bisector of the line joining a and b. Page 9 HSN000

13 5 Fundamental Theorem of Algebra If a polynomial P( z ) has degree n and P( z ) = 0 then there are n solutions (e.g. a quadratic has two solutions, and a cubic has three solutions in ). To factorise a quadratic, use the quadratic formula or complete the square. To factorise a cubic: find one root by inspection, use algebraic long division to find the quadratic factor. Factorise the quadratic factor (if possible) using the quadratic formula or completing the square. Note that if z is a root then z is also a root. Multiplication and division in polar form ( θ θ ) sin( θ θ ) zz = rr cos + + i + When multiplying two complex numbers: multiply the moduli; add the arguments. Similarly when dividing two complex numbers: divide the moduli; subtract the arguments. 6 De Moivre s Theorem ( cos sin ) n n θ θ ( cos θ sin θ) z = r + i z = r n + i n. De Moivre s theorem for fractional powers p p q q p p ( r( cosθ isinθ) ) r cos( qθ) isin( qθ) ( ) + = +. Strategy for finding the cube roots of a complex number To find the cube roots follow this strategy: = +.. Write z in two more equivalent ways by adding π to the argument Write down the cube roots by taking the cube root of z (i.e. z ).. Write the number in polar form: z r( cosθ isinθ ) Page 0 HSN000

14 Methods of Proof Disproving a statement by giving a counter example If you can find one example that will disprove the statement, then it must be false. Proof by exhaustion If the statement is given for a small set of values, then showing that the statement is true for every value will be enough to prove it. Proof by contradiction This method assumes that the statement is false and then shows that this leads to something we know to be false (a contradiction). Therefore the original assumption must be false, which means that the statement is true. Proof by induction This is a two step process.. Show that the statement holds for some value of n (usually n =).. Assume that the statement holds for some constant n = k and using this assumption show that the statement holds for n = k +. Then the statement holds for all n. Direct Proofs No set rules just make logical deductions. Page HSN000

15 Functions Critical and Stationary Points A critical point exists where the derivative is zero or undefined. Stationary points can be: maximum turning points; minimum turning points; horizontal points of inflection. Derivative Tests First derivative test The first derivative test is the nature table method used in Higher. Second derivative test If a stationary point exists at x calculate f ( a). = a then the second derivative test is to If f ( a) < 0 then the point is a maximum turning point. If f ( a) > 0 then the point is a minimum turning point. The second derivative test may not always work; if f ( a) = 0 then you should revert to the first derivative test. Non-horizontal points of inflection A non-horizontal point of inflection exists at x = a when f ( a) = 0 but f ( a) 0 and f ( x) changes sign at x = a, i.e. the curve changes from being concave up to concave down or vice versa. Page HSN000

16 3 Curve Sketching. Identify x-axis and y-axis intercepts.. Identify turning points and their nature. 3. Consider the behaviour as x ±. This could be a horizontal asymptote or a slant asymptote (the equation of which is the polynomial part after algebraic long division). 4. Check where y is undefined (this means a vertical asymptote exists). 5. Sketch the curve showing asymptotes, turning points and x- and y-axis intercepts. Transformations of curves All transformations from Higher should be known. The modulus function To sketch the modulus of a function, take any part of the curve which is below the x-axis and reflect it in the x-axis. Page 3 HSN000

17 Vectors Direction Ratios and Cosines Direction Ratios If a vector is p = ai + bj + ck, then the direction ratio is a: b: c. Direction Cosines If a vector is p = ai + bj + ck then the direction cosines are a p, b p and c p. Scalar and Vector Product Scalar product The scalar product of p = ai + bj + ck and q = di + ej + fk is pq. = ad + be + cf. Properties pq. = 0 pand q are perpendicular. p. q + r = p. q + p. r. ( ) Vector product The vector product of p = ai + bj + ck and q = di + ej + fk is i j k p q = a b c = ( bf ec ) i + ( af dc ) j + ( ae db) k. d e f The vector product is only defined for vectors in three dimensions. Geometrically The magnitude of the vector product of a and b is given by a b = a b sinθ where θ is the angle between a and b. The vector product a b is perpendicular to both vectors a and b; its direction is determined by the right hand rule. Scalar triple product The scalar triple product is a. ( b ) c. Note that this is a scalar. Page 4 HSN000

18 3 Equations of Lines Vector form of the equation of a line The line through point A with direction d has equation where a = OA uuur, = x + y + z r = a + λd r i j k and λ is a real parameter. Parametric form of the equation of a line From the vector equation, x a d where A is the point (,, 3) = + λ, y a λd Symmetric form of the equation of a line = +, z = a3 + λd3 a a a and the direction is d = d i + d j + d k. 3 If x = a+ λd, y = a + λd, z = a3 + λd3 are the parametric equations of a line, then the symmetric equation is x a y a z a3 = = d d d3 This is obtained by rearranging the parametric equations for λ. 4 Equations of Planes Vector equation of a plane The vector equation of the plane containing point A with normal n is rn. = an. where a = OA uuur and r = xi + yj + zk. To find the vector equation of the plane containing the non-collinear points A, B and C, use the following process. uuur uuur. Calculate AB and AC, and call one of these a.. Calculate the normal as = uuur uuur n AB AC. 3. Using n and a write down rn. = an.. Page 5 HSN000

19 Cartesian equation of a plane The Cartesian equation of a plane containing the point A (,, 3) normal n= ni + n j + n k is 3 nx + ny+ nz 3 = d where d = an.. (This comes from the vector equation.) Parametric equation of a plane The parametric equation of a plane is a a a, with r = a+ λb+ μc where a is the position vector of a point in the plane, b and c are two noncollinear vectors parallel to the plane, and λ and μ are real parameters. 5 Intersections Intersection of, and the angle between two lines Two lines in three dimensions can be identical; be parallel; intersect; be skew (not intersect at all). The angle between two lines L and L is calculated by finding their directions d and d, and then using the rearranged form of the scalar product formula from Higher: d. d cos θ =. d d Intersection of, and angle between two planes Two planes in three dimensions can be coincident, i.e. the same plane; be parallel; intersect in a line. Note that parallel and coincident planes have normals that are proportional. The intersection of two planes can be found using Gaussian elimination or algebraic manipulation to solve their equations simultaneously. Page 6 HSN000

20 The angle between two planes is the angle between their normals. Given planes with normals m and n, the angle between them is calculated using:. cosθ = nm nm. Intersection of three planes Three planes in three dimensions can intersect at a point; intersect in a line; have no common point of intersection. Use Gaussian elimination to work out how the planes intersect: a unique solution means intersection at a point; a family of solutions means intersection in a line; no solution means no common point of intersection. Intersection of, and the angle between a line and a plane A point of intersection can found by substituting the equation of the line in parametric form into the equation of the plane and solving for λ. To find the angle between a line and a plane:. find n, the normal to the plane;. find the angle θ between the line and n; 3. the angle between the line and the plane is π θ. Page 7 HSN000

21 Differentiation Rules Product Rule ( fg ) = f g + fg. Quotient Rule f fg fg' =. g g Chain Rule d ( f ( g( x) )) = f ( g( x) ). g ( x). dx Trigonometric Definitions sec x = cos x cosecx = sin x cot x = tan x Standard Derivatives d x ( e ) e dx d ( sin x ) = dx x d ( sin a x ) = dx a x 3 Inverse Differentiation x = ( ln x ) d ( cos x ) = dx x d ( cos x a ) = dx a x d If f ( x ) has inverse f ( x ) then f ( x ) = dx f f ( x ) dy =. dx dx dy d = dx x d ( tan x ) = dx + x d ( tan a x ) = dx a + x ( ). Page 8 HSN000

22 4 Implicit Differentiation d d dy =. dx dy dx. This is done using the chain rule: f ( y) f ( y) EXAMPLE Given x + xy+ y =, find dy dx. Differentiating throughout with respect to x: dy ( x ) + ( xy) + ( y ) = () d d d d dx dx dy dx dx dy dy x + y + x + y = 0 dx dx dy ( x + y) = x y dx dy x + y =. dx x + y 5 Parametric Differentiation dy dy If x = f () t and y = f () t then = dt. dx dx dt EXAMPLE A curve is defined parametrcally by x = t +, y = t 3. Find dy dx in terms of t. dy dy 6 = dt t = = 3 t. dx dx dt Page 9 HSN000

23 Integration Standard Integrals n+ n x x dx = + c n + ( ) ( + ) n+ n ax b ax + b dx = + c a( n+ ) sin sin x dx = cos x + c cos x dx = sin x + c ( ax + b) dx = a cos( ax + b) + c cos( ) a sin( ) sec x dx = tan x + c ax + b dx = ax + b + c x + x = + dx sin x c dx tan x c = + dx = sin ( x ) + a c a x dx = tan ( x ) + a a c a x = x ax+ b + e dx e + c = ax b e dx a e + c x x dx = ln x + c dx = a ln ax + b + c ax + b Integration by Substitution To find f ( x) dx using the substitution u = u ( x ). find du and rearrange to get dx in terms of du ; dx. replace dx by this expression and replace u( x) with u; 3. integrate with respect to u. Page 0 HSN000

24 EXAMPLE Use the substitution = 8 du Given u = x 8, = x dx u x to find ( ) so du 3 x ( x 8) dx = x u x = 3 xu du = ( u+ 8) u du ( 8 ) ( 7 ) du dx = x. Then = u + u du 4 4 = u + u + c ( ) ( ) 7 4 x 3 x 8 dx. 3 3 = x x 8 + c. To find a definite integral, you should change the limits to be in terms of u. 3 Areas The area between two curves or between a curve and an axis can be calculated as in Higher. 4 Volumes of Solids of Revolution The solid formed by rotating a curve between x = a and x = b about the x-axis has volume V = b a π y dx 5 Partial fractions and Integration Partial fractions can be used to integration rational functions. EXAMPLE. dx = = ( + ) + ( + )( ) 4 dx + 4 ln x 3 ln x c. x x 3 x 3 x 6 Integration by Parts fg = fg f g Sometimes one application of integration by parts is not enough and a repeated application must be used. 3 Page HSN000

25 Differential Equations Separable Differential Equations dy dx If = f ( x) g( y) then g y ( ) dy = f ( x ) dx. A general solution has a +c and a particular solution can be found if initial conditions are given. First Order Linear Differential Equations These have the form dy P( x) y f ( x) dx + = and can be solved using the following process.. Identify P( x ) (the equation must be in the form above). P( x) dx. Calculate the integrating factor I( x) = e. d 3. Then ( I ( x ) y ) = I ( x ) f ( x ). dx 4. Integrate both sides to get I( x) y = ( I( x) f ( x) ) dx. 5. Rearrange to give y. If given initial conditions then a particular solution can be found. 3 Second Order Homogeneous Linear Differential Equations These have the form d y dy + + = 0 a b cy dx dx and can be solved using the following process.. Write down auxiliary equation: am + bm + c = 0.. Solve for m and write down the general solution as follows distinct real roots m and m : y = Ae + Be ; mx mx one repeated real root m: y = Ae + Bxe ; px complex roots p qi y = e Acosqx+ Bsinqx. mx mx ± : ( ) Page HSN000

26 4 Second Order Non-Homogeneous Differential Equations These have the form d y dy a + b + cy = f ( x) dx dx where f ( x ) will be a simple function such as a polynomial, exponential or trigonometric function. The particular integral y p is calculated by making an educated guess: if f ( x ) is a constant, try y = a if f ( x ) is linear, try y = ax + b if f ( x ) is a quadratic, try y = ax + bx + c if f ( x)= e rx, try y = ke rx p p p p if f ( x ) has the form sinnx or cos nx, try y = psinnx+ qcos nx. dy The particular integral is then found by substituting y p p, dx and d y p dx into the differential equation and solving by equating coefficients. The general solution is given by y = y + y c where y c is the complementary solution, i.e. the solution of the corresponding homogeneous equation: d y dy a + b + cy = 0. dx dx p p Page 3 HSN000