Polynomials
I like to think of a positive quadratic as a happy face (like mine) and a negative quadratic as a sad face.
When discussing polynomials, I like to recall that poly is Greek for many and nomial is Greek for terms.
When simplifying polynomials I find it helpful to visualise a square root as an index of a half (and in general an nth root as an index of 1 over n.
If I have a polynomial which has been factorised as (x – a)(x – b)(x – c), then I know it intersects the x-axis at a, b and c.
When discussing polynomials, I like to recall that poly is Greek for many and nomial is Greek for terms.
When simplifying polynomials I find it helpful to visualise a square root as an index of a half (and in general an nth root as an index of 1 over n.
If I have a polynomial which has been factorised as (x – a)(x – b)(x – c), then I know it intersects the x-axis at a, b and c.
Completing the Square
Whenever I need to demonstrate that a quadratic expression is never negative, I complete the square and show that the number on the end is positive.
I tend to halve the coefficient of x when I’m completing the square.
'Completing the Square' is a quick and easy way of locating the minimum (or maximum) point of a quadratic curve.
I tend to halve the coefficient of x when I’m completing the square.
'Completing the Square' is a quick and easy way of locating the minimum (or maximum) point of a quadratic curve.
Inequalities
If u squared > 500, then u is either very positive or very negative.
If you square a negative number, the answer is positive. So if x squared is greater than 16, then as well as x being greater than 4, x can also be less than –4.
Never write 20 > x > 7 or any other backwards inequality attempting to combine two regions into one. It’s very ugly and to be avoided at all costs. It’s not attractive to my way of thinking.
If you square a negative number, the answer is positive. So if x squared is greater than 16, then as well as x being greater than 4, x can also be less than –4.
Never write 20 > x > 7 or any other backwards inequality attempting to combine two regions into one. It’s very ugly and to be avoided at all costs. It’s not attractive to my way of thinking.
Simultaneous Equations
In English, the word ‘simultaneous’ means ‘at the same time’. That said, it’s rather difficult to solve simultaneous equations at separate times so for me, it’s the same time every time.
The solutions to simultaneous equations can be considered as intersection points of graphs.
The solutions to simultaneous equations can be considered as intersection points of graphs.
Roots
When Michael Lewis (failed Michael Jackson impersonator on X Factor 2010) returned to our screens for the live auditions in 2011 proclaiming ‘There’s so much negativity in this room, it’s unreal’, he was, of course, referring to the value of the discriminant which is positive for real roots and negative for non (or un-) real roots.
b squared – 4ac is part of the quadratic formula, where it is square-rooted. So if b squared – 4ac is evil, then the answer is money (which is the root of all evil). Something like that.
Whenever I’m stuck on a question concerning the discriminant, I go to my dentist’s for some root surgery.
When a female singer reached diva status, she is often accused of ‘forgetting her roots’. So I would just like to remind any such lady that if b squared – 4ac < 0 there are no real roots, if b squared – 4ac = 0 there is one real root and if b squared – 4ac > 0 there are two distinct real roots.
The roots of an equation are its solutions and I like to use b squared – 4ac to help me determine how many roots a quadratic equation has.
b squared – 4ac is part of the quadratic formula, where it is square-rooted. So if b squared – 4ac is evil, then the answer is money (which is the root of all evil). Something like that.
Whenever I’m stuck on a question concerning the discriminant, I go to my dentist’s for some root surgery.
When a female singer reached diva status, she is often accused of ‘forgetting her roots’. So I would just like to remind any such lady that if b squared – 4ac < 0 there are no real roots, if b squared – 4ac = 0 there is one real root and if b squared – 4ac > 0 there are two distinct real roots.
The roots of an equation are its solutions and I like to use b squared – 4ac to help me determine how many roots a quadratic equation has.
Arithmetic Progressions
I know that un stands for nth term because the UN (United Nations) has its own terms which have to be met before you can invade another country.
To remember that a represents the first term of an AP, I use the fact that a is the first letter of the alphabet (well, the one I use).
On the odd occasion I have dabbled with recurrence relations I have found it useful to think of un + 1 like the satellite TV channel equivalent of un delayed by an hour (e.g. E4 and E4+1).
To remember that a represents the first term of an AP, I use the fact that a is the first letter of the alphabet (well, the one I use).
On the odd occasion I have dabbled with recurrence relations I have found it useful to think of un + 1 like the satellite TV channel equivalent of un delayed by an hour (e.g. E4 and E4+1).
Surds
If I need to rationalise the denominator of a fraction where the denominator is of the form (a + root b), I times the fraction by another fraction with a numerator and denominator of (a – root b)
Coordinate Geometry
Perpendicular lines meet at right angles.
I don’t know how many of mkyou2tube’s readership would be aware of this, but the Paralympics are so named because they run alongside (i.e. parallel) to the Olympics (but never meet).
I don’t know how many of mkyou2tube’s readership would be aware of this, but the Paralympics are so named because they run alongside (i.e. parallel) to the Olympics (but never meet).
Calculus
The purpose of differentiation is to provide you with a function for the gradient of a curve at each point.
The second differential informs you whether a turning point is a maximum or a minimum. Positive for a minimum and negative for a maximum. This is analogous to the negative attitude displayed by Maxi to the inexperienced but enthusiastic Roary the Racing Car in the eponymous animated television show.
Further to the observation about Roary earlier, I find that rain macs give off a negative vibe and mini skirts give off a positive vibe (in some people’s opinions) .
Integration is good for two things: reversing differentiation and calculating the exact area under a curve.
Dividing by m/n is the same as multiplying by n/m.
Imagine laying on the ground on the Earth's surface. You could be correctly described as a tangent to the Earth, but your behaviour would not be normal. Normal would be perpendicular to the Earth (i.e. standing up).
I have lived a long and prosperous life, never wanting for anything. And yet I have never made a single decision in my whole life. How come? Simples, I just differentiate any situation in which I find myself and put the result equal to zero. The optimal solution every time!
The word 'calculus' comes from the Latin word for 'rock' and that's why calculus rocks! Go, Newton and Leibniz! Actually, it comes from the Latin word for 'pebble' but that's close enough.
The area under a straight line will always be a shape you already know such as a trapezium, triangle or rectangle.
You don’t need to add c when integrating between limits.
Optimisation is good for making economical cardboard boxes.
If ever you have an equation in which your variable has a negative index (e.g. your equation has a term in x to the power of minus a), multiply through by x to the power of a.
You need to use the known measurement to eliminate the unwanted variable in the variable you wish to optimise.
The second differential informs you whether a turning point is a maximum or a minimum. Positive for a minimum and negative for a maximum. This is analogous to the negative attitude displayed by Maxi to the inexperienced but enthusiastic Roary the Racing Car in the eponymous animated television show.
Further to the observation about Roary earlier, I find that rain macs give off a negative vibe and mini skirts give off a positive vibe (in some people’s opinions) .
Integration is good for two things: reversing differentiation and calculating the exact area under a curve.
Dividing by m/n is the same as multiplying by n/m.
Imagine laying on the ground on the Earth's surface. You could be correctly described as a tangent to the Earth, but your behaviour would not be normal. Normal would be perpendicular to the Earth (i.e. standing up).
I have lived a long and prosperous life, never wanting for anything. And yet I have never made a single decision in my whole life. How come? Simples, I just differentiate any situation in which I find myself and put the result equal to zero. The optimal solution every time!
The word 'calculus' comes from the Latin word for 'rock' and that's why calculus rocks! Go, Newton and Leibniz! Actually, it comes from the Latin word for 'pebble' but that's close enough.
The area under a straight line will always be a shape you already know such as a trapezium, triangle or rectangle.
You don’t need to add c when integrating between limits.
Optimisation is good for making economical cardboard boxes.
If ever you have an equation in which your variable has a negative index (e.g. your equation has a term in x to the power of minus a), multiply through by x to the power of a.
You need to use the known measurement to eliminate the unwanted variable in the variable you wish to optimise.
Trigonometry
When I was a young child my nana used to make a 300 pie (blueberry and spinach) and a 60 pie (ham and gooseberry). This helped tremendously in my exams, remembering that 360 degrees was two pi(e).
In my neighbour’s house she’s the one with the degrees whilst her partner exudes a certain radiance. So we always know which is which.
The 'SATC' approach to solving trigonometric equations tells you how to label the diagram left-to-right and top-to-bottom.
Regardless of whether your trig expression is positive or negative, evaluate the inverse trig function as if it were positive before employing the SATC method.
In my neighbour’s house she’s the one with the degrees whilst her partner exudes a certain radiance. So we always know which is which.
The 'SATC' approach to solving trigonometric equations tells you how to label the diagram left-to-right and top-to-bottom.
Regardless of whether your trig expression is positive or negative, evaluate the inverse trig function as if it were positive before employing the SATC method.
Binomial Theorem
If a linear (or binomial) expression is raised to the positive integer power n, the expansion will contain (n + 1) terms.
The coefficients for a binomial expansion can be taken straight from Pascal’s triangle (although the nCr button is often quicker).
A binomial is so-called because it contains two terms (cf other words with the prefix bi such as bicycle, bilingual, binary and bisector).
Binomial expansions can be used to approximate decimals raised to powers by using the first few terms (in fact, that’s how calculators work).
When approximating, put the value required equal to the stated expression (pre-expansion) and solve.
Approximations are only permitted for (1 + ax) to the power of n if ax < 1.
One-third minus one is negative two-thirds, which in turn is negative five-thirds once one has been subtracted for a second time.
mkyou2tube's Oliver tribute to 'The Binomial Theorem' is totes amaze.
Impress the C4 examiner by refactorising their 1-less expression so that it does contain a 1 and then reconfiguring your expansion by multiplying it by the removed number (to the relevant power).
You can check a binomial expansion by choosing a permitted value for x and substituting it into, say, the first four terms and comparing with the actual value stated on a calculator - they ought to be quite similar.
The term independent of x has no x in it.
The coefficients for a binomial expansion can be taken straight from Pascal’s triangle (although the nCr button is often quicker).
A binomial is so-called because it contains two terms (cf other words with the prefix bi such as bicycle, bilingual, binary and bisector).
Binomial expansions can be used to approximate decimals raised to powers by using the first few terms (in fact, that’s how calculators work).
When approximating, put the value required equal to the stated expression (pre-expansion) and solve.
Approximations are only permitted for (1 + ax) to the power of n if ax < 1.
One-third minus one is negative two-thirds, which in turn is negative five-thirds once one has been subtracted for a second time.
mkyou2tube's Oliver tribute to 'The Binomial Theorem' is totes amaze.
Impress the C4 examiner by refactorising their 1-less expression so that it does contain a 1 and then reconfiguring your expansion by multiplying it by the removed number (to the relevant power).
You can check a binomial expansion by choosing a permitted value for x and substituting it into, say, the first four terms and comparing with the actual value stated on a calculator - they ought to be quite similar.
The term independent of x has no x in it.
Geometric Progressions
Should you ever find yourself stuck on a question about sequences in which each term is multiplied by a common ratio to generate the next term, do not attempt to consult your GP, because in this case a GP stands for General Practitioner and not Geometric Progression.
Should you choose to ignore the aforementioned advice on consulting your GP make sure you do not abbreviate the phrase sum-to-infinity to STI when describing your problem.
If the magnitude of r is less than 1 then the series converges.
A proof relating to convergence is a con, whereas if you try to count to the end of a diverging series you will die before you get there.
Should you choose to ignore the aforementioned advice on consulting your GP make sure you do not abbreviate the phrase sum-to-infinity to STI when describing your problem.
If the magnitude of r is less than 1 then the series converges.
A proof relating to convergence is a con, whereas if you try to count to the end of a diverging series you will die before you get there.
Factor and Remainder Theorems
The Factor Theorem states that if f(a) = 0, then (x - a) is a factor of f(x).
An alternative Factor Theorem states that whenever the lineup of judges/mentors on UK X Factor changes, the new series is won by the youngest female judge. It is yet to be disproven.
An alternative Factor Theorem states that whenever the lineup of judges/mentors on UK X Factor changes, the new series is won by the youngest female judge. It is yet to be disproven.
Logarithms
Logs are the opposite of indices. When you multiply terms with indices, you add the indices. When you add logs, you multiply. Similarly when you divide terms with indices you subtract and vice versa.
If you want logs to stick in your head, there are tree things to remember. A coefficient wood turn into an index. Log366 is ½ because 6 is root 36. And they're nothing to do with twigonometry.
Log(1) is zero in any base because it’s the opposite rule of ‘Anything to the power zero equals one’.
If you want logs to stick in your head, there are tree things to remember. A coefficient wood turn into an index. Log366 is ½ because 6 is root 36. And they're nothing to do with twigonometry.
Log(1) is zero in any base because it’s the opposite rule of ‘Anything to the power zero equals one’.
Circles
If a circle’s equation is not already bracketed, you should use 'Completing the Square'.
Any tangent to a circle is perpendicular to its radius.
If you are given three points on the circumference of a circle, the centre of the circle is the point of intersection of all the perpendicular bisectors of each pair of points.
Any tangent to a circle is perpendicular to its radius.
If you are given three points on the circumference of a circle, the centre of the circle is the point of intersection of all the perpendicular bisectors of each pair of points.
Trapezium Rule
Applying the Trapezium Rule to a positive quadratic (happy face) would give an overestimate whereas applied to a negative quadratic (frowny face) would give an underestimate.
Advanced Calculus
If y = ex, then the hundredth differential of y is also ex, but I have never yet found a use for this curious fact.
When I was dumped by my then-girlfriend I won over my ex by quoting the differential of the natural logarithm of x, which coincidentally was ‘one over x’.
They say that a lack of community is a product of the times, which is strange because a product is a product of the timesing (or multiplying) of two (or more) quantities.
If u is HI and v is HO, then the Quotient Rule can be rephrased as HODEHI minus HIDEHO, all divided by HOHO.
The differential of tan x is sec2x which has to be pronounced carefully.
You can prove the differentials of tan x, sec x, cosec x and cot x by using the quotient rule.
I count up to the third letter of cot, cosec or sec in order to determine which of the original three trig functions has been reciprocated.
When you differentiate a trig function beginning with c, the answer is negative.
Where there’s a tan, there’s two secs (which also has to be pronounced carefully).
ex > 0 so ex cannot equal zero, hence when finding turning points you need not solve ex = 0.
To differentiate a chain, like sin 3x for example, where the new bit is linear, you may differentiate as usual but multiply the answer by the x-coefficient.
The product rule may be conceived thus: times one bit by the differential of the other bit, and vice versa, and add them together.
Where there is a cot, there are two cosecs.
Parallel tangents have the same gradient.
Differentiating the square of a function can be approached in two different ways: the chain rule whence y = t squared or the product rule whence u and v are both equal to the function (before squaring).
On the rare occasion I have been forced to prove why the differential of a reciprocal trig function (or indeed tan x), I have carried out the demonstration by writing the trig function as a fraction and invoking the quotient rule.
The circle of trigonometric differentiation can be used both anticlockwise for integration purposes.
Implicit differentiation comes in two types: functions of y only and functions of x and y. Use the chain rule for the former and the product rule for the latter.
To differentiate y = a to the power of x, take logs of both sides and then use implicit differentiation.
The differential of the ever-popular y2 is 2yy’.
To differentiate e to the power of (x+y) split it into a product using the laws of indices (i.e. e to the x times e to the y, then go implicit.
On the frequent occasion I need to evaluate a circle's tangent, I use implicit differentiation.
To make a rate of change represent a decreasing event (eg a cooling curve or radioactive decay), put a minus sign at the front.
To answer questions on rates of change, begin by writing what is required as a differential, then set up a 'fraction multiplication' chain rule effect.
[dJ by dt] means the rate of change of J (not as is commonly thought when a chap on the decks at a party positions himself by the sandwiches).
Integration is the opposite of differentiation.
You can integrate between limits to calculate the exact area or volume under a curve or surface.
When you differentiate a number you get nothing at all. So if you integrate nothing at all you get a number. we don't know what the number is, it might even be zero, but we denote it as c.
Before integrating a polynomial, rewrite each term individually with an index.
The integral of one over cabin is log cabin by the c.
Substitute was a hit single for The Who back in 1966, who being the word I might use to describe you if you were being a mystery Mr X type figure. As a result whenever I hear the phrase 'u in terms of x' it makes me think of substituting (and pinball wizards).
I like to think of d as standing for 'don't forget'. So when I'm substituting for x and see the dx it reminds me not to forget to replace the dx as well.
When substituting, ensure you also substitute for the limits as well as the x.
I never despair if I am unable to substitute for every single occurrence of x until I have evaluated du/dx, just in case the expression for du/dx includes the occurrence I am attempting to replace.
Fool your friends by asking them to integrate ln x by parts, and when they say there’s only one part, write a 1 in front of the ln x.
When integrating by parts split the expression into one bit you can differentiate and one you can integrate.
Further to Wilma’s tip, if one bit is a polynomial take that as the bit you'll differentiate unless the other bit is ln x.
Further to the previous two tips, if one bit is ex and the other bit is sin or cos, you'll find yourself going round in circles unless you ‘plug the gap’.
‘Sin...um minus’ is my way of remembering the sign in the middle of the integral of sin2x, in which ‘sin... um’ is an anagram of ‘minus’.
I prefer to use the formula book when integrating sec x or cosec x.
When integrating tan2x or cot2x, consider using the identities to convert them into sec2x and cosec2x respectively.
To integrate tan x or cot x, rewrite them in terms of sin x and cos x and substitute for the denominator.
Trapezium Rule: Don’t confuse intervals and ordinates, there’s always one more ordinate than interval.
To solve a differential equation, get all the y on one side and all the x on the other, including the dy and dx.
The aim of solving a differential equation is to find out what y was in the first place.
The volume of revolution formula is based upon the volume of a cylinder.
If you use a substitution with limits, ensure that you replace the limits too.
Integration in C4 can largely be categorised into three disciplines: substitution, by parts and trig bits.
Parametrics should not be confused with paramedics.
To work out dy/dx, divide dy/dt by dx/dt.
When integrating y with respect to x (dx), remember to substitute for both the y and the dx. Similarly for a volume of revolution, substitute for the y2 and the dx but not the p.
The Paralympics are so called because they run parallel to the normal Olympics. Parametrics are not so called because they run parallel to the metrics, that would be the imperials.
When I was dumped by my then-girlfriend I won over my ex by quoting the differential of the natural logarithm of x, which coincidentally was ‘one over x’.
They say that a lack of community is a product of the times, which is strange because a product is a product of the timesing (or multiplying) of two (or more) quantities.
If u is HI and v is HO, then the Quotient Rule can be rephrased as HODEHI minus HIDEHO, all divided by HOHO.
The differential of tan x is sec2x which has to be pronounced carefully.
You can prove the differentials of tan x, sec x, cosec x and cot x by using the quotient rule.
I count up to the third letter of cot, cosec or sec in order to determine which of the original three trig functions has been reciprocated.
When you differentiate a trig function beginning with c, the answer is negative.
Where there’s a tan, there’s two secs (which also has to be pronounced carefully).
ex > 0 so ex cannot equal zero, hence when finding turning points you need not solve ex = 0.
To differentiate a chain, like sin 3x for example, where the new bit is linear, you may differentiate as usual but multiply the answer by the x-coefficient.
The product rule may be conceived thus: times one bit by the differential of the other bit, and vice versa, and add them together.
Where there is a cot, there are two cosecs.
Parallel tangents have the same gradient.
Differentiating the square of a function can be approached in two different ways: the chain rule whence y = t squared or the product rule whence u and v are both equal to the function (before squaring).
On the rare occasion I have been forced to prove why the differential of a reciprocal trig function (or indeed tan x), I have carried out the demonstration by writing the trig function as a fraction and invoking the quotient rule.
The circle of trigonometric differentiation can be used both anticlockwise for integration purposes.
Implicit differentiation comes in two types: functions of y only and functions of x and y. Use the chain rule for the former and the product rule for the latter.
To differentiate y = a to the power of x, take logs of both sides and then use implicit differentiation.
The differential of the ever-popular y2 is 2yy’.
To differentiate e to the power of (x+y) split it into a product using the laws of indices (i.e. e to the x times e to the y, then go implicit.
On the frequent occasion I need to evaluate a circle's tangent, I use implicit differentiation.
To make a rate of change represent a decreasing event (eg a cooling curve or radioactive decay), put a minus sign at the front.
To answer questions on rates of change, begin by writing what is required as a differential, then set up a 'fraction multiplication' chain rule effect.
[dJ by dt] means the rate of change of J (not as is commonly thought when a chap on the decks at a party positions himself by the sandwiches).
Integration is the opposite of differentiation.
You can integrate between limits to calculate the exact area or volume under a curve or surface.
When you differentiate a number you get nothing at all. So if you integrate nothing at all you get a number. we don't know what the number is, it might even be zero, but we denote it as c.
Before integrating a polynomial, rewrite each term individually with an index.
The integral of one over cabin is log cabin by the c.
Substitute was a hit single for The Who back in 1966, who being the word I might use to describe you if you were being a mystery Mr X type figure. As a result whenever I hear the phrase 'u in terms of x' it makes me think of substituting (and pinball wizards).
I like to think of d as standing for 'don't forget'. So when I'm substituting for x and see the dx it reminds me not to forget to replace the dx as well.
When substituting, ensure you also substitute for the limits as well as the x.
I never despair if I am unable to substitute for every single occurrence of x until I have evaluated du/dx, just in case the expression for du/dx includes the occurrence I am attempting to replace.
Fool your friends by asking them to integrate ln x by parts, and when they say there’s only one part, write a 1 in front of the ln x.
When integrating by parts split the expression into one bit you can differentiate and one you can integrate.
Further to Wilma’s tip, if one bit is a polynomial take that as the bit you'll differentiate unless the other bit is ln x.
Further to the previous two tips, if one bit is ex and the other bit is sin or cos, you'll find yourself going round in circles unless you ‘plug the gap’.
‘Sin...um minus’ is my way of remembering the sign in the middle of the integral of sin2x, in which ‘sin... um’ is an anagram of ‘minus’.
I prefer to use the formula book when integrating sec x or cosec x.
When integrating tan2x or cot2x, consider using the identities to convert them into sec2x and cosec2x respectively.
To integrate tan x or cot x, rewrite them in terms of sin x and cos x and substitute for the denominator.
Trapezium Rule: Don’t confuse intervals and ordinates, there’s always one more ordinate than interval.
To solve a differential equation, get all the y on one side and all the x on the other, including the dy and dx.
The aim of solving a differential equation is to find out what y was in the first place.
The volume of revolution formula is based upon the volume of a cylinder.
If you use a substitution with limits, ensure that you replace the limits too.
Integration in C4 can largely be categorised into three disciplines: substitution, by parts and trig bits.
Parametrics should not be confused with paramedics.
To work out dy/dx, divide dy/dt by dx/dt.
When integrating y with respect to x (dx), remember to substitute for both the y and the dx. Similarly for a volume of revolution, substitute for the y2 and the dx but not the p.
The Paralympics are so called because they run parallel to the normal Olympics. Parametrics are not so called because they run parallel to the metrics, that would be the imperials.
Numerical Methods
Make sure you record your iterations neatly for easy referral
Interval bisection is all about bisecting intervals. Linear interpolation is all about interpolating linearly. Newton-Raphson is all about raphsoning your newton.
You can check your answer is accurate using the lower and upper bounds.
A nice diagram is always appreciated.
Sign change, therefore root.
If you type the iterative formula (using Ans for x) then every time you press Exe you generate a new operation. Make sure you store the first value as the initial Ans before you start.
If you are asked to verify that a certain value is a root to a given number of decimal places or significant figures, show sign change therefore root using the lower and upper bounds.
Iteration is the process of repeating a formula with a refined variable. Iteration is the process of repeating a formula with a refined variable. Iteration is the process of repeating a formula with a refined variable. Iteration is the process of repeating a formula with a refined variable. Iteration is the process of repeating a formula with a refined variable.
Interval bisection is all about bisecting intervals. Linear interpolation is all about interpolating linearly. Newton-Raphson is all about raphsoning your newton.
You can check your answer is accurate using the lower and upper bounds.
A nice diagram is always appreciated.
Sign change, therefore root.
If you type the iterative formula (using Ans for x) then every time you press Exe you generate a new operation. Make sure you store the first value as the initial Ans before you start.
If you are asked to verify that a certain value is a root to a given number of decimal places or significant figures, show sign change therefore root using the lower and upper bounds.
Iteration is the process of repeating a formula with a refined variable. Iteration is the process of repeating a formula with a refined variable. Iteration is the process of repeating a formula with a refined variable. Iteration is the process of repeating a formula with a refined variable. Iteration is the process of repeating a formula with a refined variable.
Matrices
Multiply a row by a column (but not a column by a row).
Simultaneous equations can be used to solve matrix problems.
You can only multiply matrix M by matrix N if M has the same number of columns as N has rows. If you multiply an (a ´ b) matrix by a (b ´ c) matrix, the result will be an (a ´ c) matrix.
The determinant of a matrix is the area factor comparing the areas of the original shape and transformed shape.
The product of any matrix and its inverse (assuming it has an inverse) is the identity matrix.
A singular matrix is a matrix with no inverse (and for which the determinant is zero).
If applying matrix A twice brings a point back to its original position (e.g. reflection in y = x) then A squared = I.
A matrix is a rectangular array of numbers betwixt two curly brackets and not, as is commonly perceived, a simulated reality of the world set in 1999.
Rows times columns, or as my RS teacher used to say: ‘Roman Catholic’.
A square matrix has the same number of elements in each row and column.
A mistaken identity is when your matrix does not solely consist of 1s along the leading diagonal and 0s everywhere else.
Eigen is German for own and hence matrices own special vectors and values we call eigenvectors and eigenvalues.
If we imagine the UK as stopping at the end of Wales (going upwards) then a leading diagonal leads you from Anglesey to Bexleyheath (approx).
Never ever ever diagonalise a matrix.
Simultaneous equations can be used to solve matrix problems.
You can only multiply matrix M by matrix N if M has the same number of columns as N has rows. If you multiply an (a ´ b) matrix by a (b ´ c) matrix, the result will be an (a ´ c) matrix.
The determinant of a matrix is the area factor comparing the areas of the original shape and transformed shape.
The product of any matrix and its inverse (assuming it has an inverse) is the identity matrix.
A singular matrix is a matrix with no inverse (and for which the determinant is zero).
If applying matrix A twice brings a point back to its original position (e.g. reflection in y = x) then A squared = I.
A matrix is a rectangular array of numbers betwixt two curly brackets and not, as is commonly perceived, a simulated reality of the world set in 1999.
Rows times columns, or as my RS teacher used to say: ‘Roman Catholic’.
A square matrix has the same number of elements in each row and column.
A mistaken identity is when your matrix does not solely consist of 1s along the leading diagonal and 0s everywhere else.
Eigen is German for own and hence matrices own special vectors and values we call eigenvectors and eigenvalues.
If we imagine the UK as stopping at the end of Wales (going upwards) then a leading diagonal leads you from Anglesey to Bexleyheath (approx).
Never ever ever diagonalise a matrix.
Exponentials and Logarithms
e is 2.718281828 to 9 decimal places, which coincidentally contains a repeating pattern.
A logarithm is not a tune played with a stick but the opposite of an index.
When I solve equations like ex = 8, I always take logs of both sides (which in this case gives you x = ln 8).
Any curve of the form y = a to the power of x will pass through the point (0, 1) because anything to the power of zero is one.
When you write the natural log of x (ln) it looks like IN. I use this literary peculiarity to remind me that ex is the opposite (INverse!) of ln x in the same way that interior is the opposite of exterior and interpolate is the opposite of extrapolate and introvert is the opposite of extrovert.
When I need to understand when to use a natural log or a different log (e.g. base 10), I think of dying trees undergoing exponential decay because trees are themselves natural logs.
Since a negative index implies a fraction (e.g. 2 to the power of –1 = ½), I like to think of e to the power of minus infinity as a fraction with a really big denominator and hence close (if not equal) to zero.
A logarithm is not a tune played with a stick but the opposite of an index.
When I solve equations like ex = 8, I always take logs of both sides (which in this case gives you x = ln 8).
Any curve of the form y = a to the power of x will pass through the point (0, 1) because anything to the power of zero is one.
When you write the natural log of x (ln) it looks like IN. I use this literary peculiarity to remind me that ex is the opposite (INverse!) of ln x in the same way that interior is the opposite of exterior and interpolate is the opposite of extrapolate and introvert is the opposite of extrovert.
When I need to understand when to use a natural log or a different log (e.g. base 10), I think of dying trees undergoing exponential decay because trees are themselves natural logs.
Since a negative index implies a fraction (e.g. 2 to the power of –1 = ½), I like to think of e to the power of minus infinity as a fraction with a really big denominator and hence close (if not equal) to zero.
Partial Fractions
Interestingly, partial fractions are actually fully formed fractions in their own right.
Always always multiply by the denominator of the fraction you’re trying to split up.
Always always multiply by the denominator of the fraction you’re trying to split up.
Vectors
You can use Pythagoras’ Theorem in three dimensions to calculate the magnitude of a 3D vector.
In the equation r = a + lb, b is the direction. l is lambda in the wrong font.
Impress your colleagues by finding the angle between two vectors by using the scalar product.
If cos x is negative then the angle is obtuse, but should you require an acute angle you could just subtract it from 180 degrees.
A unit vector is a vector with a length of one unit.
If two vectors are perpendicular then a dot b equals zero.
Two parallel vectors are multiples of each other.
The foot of the perpendicular is so called because the diagram looks like a leg and a foot.
The scalar/dot product is so called because the answer is a scalar and a dot is used to represent it. Likewise, the vector/cross product is so called because the answer is a vector and a cross is used to represent it.
Whenever I wish to designate a vector equation of a line and am in need of a variable with which to indicate the possibility of moving different amounts along the line, I always select a Greek letter which is in some way related to animals. For example, my first choice would be LAMBDA, featuring the young of a sheep. My second choice would be MEW, the purring sound made by a cat. In addition, a plane is labelled PI, which of course comes in many animal varieties such as chicken pie, steak pie and fish pie.
My favourite equation for a plane is r · n = p, where r is the position vector and n is the vector perpendicular to the plane and the distance between a plane in this form and the origin is given by p divided by the length of n.
n stands for normal, like at right angles to the plane.
The mirror image of a point in a plane is the same distance (or same number of normal vectors) on the other side.
A triangle is half of a parallelogram and a tetrahedron is a sixth of a parallelepiped.
Two planes intersect in a straight line.
We use the letter r to represent a position vector because r is the last letter of ‘position vector’.
In the equation r = a + lb, b is the direction. l is lambda in the wrong font.
Impress your colleagues by finding the angle between two vectors by using the scalar product.
If cos x is negative then the angle is obtuse, but should you require an acute angle you could just subtract it from 180 degrees.
A unit vector is a vector with a length of one unit.
If two vectors are perpendicular then a dot b equals zero.
Two parallel vectors are multiples of each other.
The foot of the perpendicular is so called because the diagram looks like a leg and a foot.
The scalar/dot product is so called because the answer is a scalar and a dot is used to represent it. Likewise, the vector/cross product is so called because the answer is a vector and a cross is used to represent it.
Whenever I wish to designate a vector equation of a line and am in need of a variable with which to indicate the possibility of moving different amounts along the line, I always select a Greek letter which is in some way related to animals. For example, my first choice would be LAMBDA, featuring the young of a sheep. My second choice would be MEW, the purring sound made by a cat. In addition, a plane is labelled PI, which of course comes in many animal varieties such as chicken pie, steak pie and fish pie.
My favourite equation for a plane is r · n = p, where r is the position vector and n is the vector perpendicular to the plane and the distance between a plane in this form and the origin is given by p divided by the length of n.
n stands for normal, like at right angles to the plane.
The mirror image of a point in a plane is the same distance (or same number of normal vectors) on the other side.
A triangle is half of a parallelogram and a tetrahedron is a sixth of a parallelepiped.
Two planes intersect in a straight line.
We use the letter r to represent a position vector because r is the last letter of ‘position vector’.
Functions
When I need to distinguish between domain and range I remember that Jermain Defoe is a striker and crosses the ball so Domain is a cross. I haven’t thought of anything for range.
Odd functions aren’t really odd, they just look like polynomials with odd indices.
All trig functions are many-to-one unless you restrict their domains.
I always use the letter f to represent a function unless I have some more, in which case I start using g, h and so on.
When I say the words fg and gf, they sound different. In the same way, I like to recall that fg(x) and gf(x) are different.
In response to the previous Top Tip, I think that if f and g are inverses of each other then fg(x) would equal gf(x), and in fact they would both equal x.
Inverse functions are mirror images of each other in the line y = x and should I ever wish to find the inverse of a function I exchange the x and y and then make y the subject again.
Reciprocal curves have two asymptotes, one horizontal and one vertical.
Whenever Lisa Snowdon needs to state the domain of f(x) = x squared, she simply thinks of her ex, George Clooney because he was ‘der main’ actor when ER (the medical drama) began and x E R. I call this technique the ‘ex-factor’.
To find the inverse of a function, interchange y and x , then make y the subject again.
Odd functions aren’t really odd, they just look like polynomials with odd indices.
All trig functions are many-to-one unless you restrict their domains.
I always use the letter f to represent a function unless I have some more, in which case I start using g, h and so on.
When I say the words fg and gf, they sound different. In the same way, I like to recall that fg(x) and gf(x) are different.
In response to the previous Top Tip, I think that if f and g are inverses of each other then fg(x) would equal gf(x), and in fact they would both equal x.
Inverse functions are mirror images of each other in the line y = x and should I ever wish to find the inverse of a function I exchange the x and y and then make y the subject again.
Reciprocal curves have two asymptotes, one horizontal and one vertical.
Whenever Lisa Snowdon needs to state the domain of f(x) = x squared, she simply thinks of her ex, George Clooney because he was ‘der main’ actor when ER (the medical drama) began and x E R. I call this technique the ‘ex-factor’.
To find the inverse of a function, interchange y and x , then make y the subject again.
McLaurin's and Taylor's Expansions
A MacLaurin series is just a Taylor series designed by someone who likes zero.
When differentiating, ensure you use the Chain Rule or Product Rule as appropriate.
5! means 5 times 4 times 3 times 2 times 1, not ‘5’ shouted at the top of your voice.
When x is really small (and measured in radians) then sin x and tan x are almost the same as x itself (but cos x isn’t).
The differential of cos x is minus sin x whereas the differential of cosh x is sinh x.
A quick way of writing the fourth differential is f’’’’(x) (and other ordinal differentials can be denoted in an equivalent fashion).
MacLaurin series can be used for finding approximations for things.
MacLaurin series are unsuitable for functions which (or functions whose derivatives) are undefined.
When differentiating, ensure you use the Chain Rule or Product Rule as appropriate.
5! means 5 times 4 times 3 times 2 times 1, not ‘5’ shouted at the top of your voice.
When x is really small (and measured in radians) then sin x and tan x are almost the same as x itself (but cos x isn’t).
The differential of cos x is minus sin x whereas the differential of cosh x is sinh x.
A quick way of writing the fourth differential is f’’’’(x) (and other ordinal differentials can be denoted in an equivalent fashion).
MacLaurin series can be used for finding approximations for things.
MacLaurin series are unsuitable for functions which (or functions whose derivatives) are undefined.
First Order Differential Equations
Solving a FODE is a bit like using the product rule backwards.
The F in FODE stands for First, so to distinguish it from a SODE (for which the S stands for Second).
If y = vx you'll need the product rule to differentiate it.
FODEs can be solved in two ways and if the variables won't be separated then you’ll need an integrating factor.
This is how I remember the IF approach: You need an IF IF your equation isn’t exact but can be written as y’ + Py = Q (where P and Q are functions of x).
e and ln annihilate themselves when they occur simultaneously in the evaluation of an integrating factor.
Don't forget ‘+ c’, even when you're doing something as sophisticated as a FODE.
The F in FODE stands for First, so to distinguish it from a SODE (for which the S stands for Second).
If y = vx you'll need the product rule to differentiate it.
FODEs can be solved in two ways and if the variables won't be separated then you’ll need an integrating factor.
This is how I remember the IF approach: You need an IF IF your equation isn’t exact but can be written as y’ + Py = Q (where P and Q are functions of x).
e and ln annihilate themselves when they occur simultaneously in the evaluation of an integrating factor.
Don't forget ‘+ c’, even when you're doing something as sophisticated as a FODE.
Second Order Differential Equations
All quadratic equations have two roots, even if they happen to be the same.
Learn the mantra ‘booyakasha exponential real, trigonometric imaginary’. The booyakasha is unnecessary but means that the initials spell BERTI, which sounds better than ERTI.
The S in SODE stands for Second. The T in TODE stands for Third. The F in FODE could theoretically stand for First, Fourth or Fifth, but as we don't need to worry about the Fourth or Fifth, we can just pretend it stands for First.
CF + PI = GS.
Learn the mantra ‘booyakasha exponential real, trigonometric imaginary’. The booyakasha is unnecessary but means that the initials spell BERTI, which sounds better than ERTI.
The S in SODE stands for Second. The T in TODE stands for Third. The F in FODE could theoretically stand for First, Fourth or Fifth, but as we don't need to worry about the Fourth or Fifth, we can just pretend it stands for First.
CF + PI = GS.
Complex Numbers
'Imaginary Numbers' was almost a hit for John Lennon.
If you think complex numbers are difficult, try renaming them as easy numbers.
The square root of a complex number is also a complex number.
de Moivre’s theorem relates the power of a complex number to its form in terms of sines and cosines of multiples of angles.
de Moivre’s theorem requires you to remember things, so it’s been put in a handy little instructional video, appropriately entitled de Moivre’s theorem.
There are three cube roots of unity (1), but only one of them is real. Likewise there are four fourth roots, five fifth roots, etc.
Actually, any complex number has n nth roots and these are equally spaced on an Argand Diagram (2p/n apart).
Complex roots always come in complex conjugate pairs.
Complex loci include circles, half-lines, perpendicular bisectors and sunsets, or as I like to remember them, CHAPS.
mkyou2tube's video on 'de Moivre's theorem' saves it from ever having to remember what to do.
If you think complex numbers are difficult, try renaming them as easy numbers.
The square root of a complex number is also a complex number.
de Moivre’s theorem relates the power of a complex number to its form in terms of sines and cosines of multiples of angles.
de Moivre’s theorem requires you to remember things, so it’s been put in a handy little instructional video, appropriately entitled de Moivre’s theorem.
There are three cube roots of unity (1), but only one of them is real. Likewise there are four fourth roots, five fifth roots, etc.
Actually, any complex number has n nth roots and these are equally spaced on an Argand Diagram (2p/n apart).
Complex roots always come in complex conjugate pairs.
Complex loci include circles, half-lines, perpendicular bisectors and sunsets, or as I like to remember them, CHAPS.
mkyou2tube's video on 'de Moivre's theorem' saves it from ever having to remember what to do.