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- The Binomial Expansion
Pascal's Triangle is used to expand binomial expressions like (a+b)ⁿ. It is created by summing adjacent pairs to find the number beneath this pair, starting from 1. Here are the first five rows: The (n + 1)th row of Pascal's triangle gives the coefficients in the expansion of (a+b)ⁿ Factorial Notation Parts of Pascal's triangle can be calculated quickly using factorial notation, ⁿCr (spoken "n choose r"): Expanding (a+b)ⁿ When n ∈ ℕ (when n is a positive integer) the binomial expansion is in its simplest form: The general term in an expansion of (a+b)ⁿ is given as: Expanding (1+x)ⁿ If n is a fraction or a negative number, you need to use this form of the binomial expansion: It is valid when |x| < 1 and when n ∈ ℝ The general term in this expansion is given as: x-term Coefficient When the x term has a coefficient, so the binomial is in the form (1+bx)ⁿ, treat (bx) as x, and follow the standard expansion for (1+x)ⁿ The expansion for (1+bx)ⁿ is valid for |bx| < 1, or |x| < 1/|b| Double Coefficients If the binomial is in the form (a+bx)ⁿ, you have to take a factor of aⁿ out of each term: The expansion for (a+bx)ⁿ, where n is negative or a fraction, is valid when |bx/a| < 1, or |x| < |a/b| Often, complex expressions can be simplified first by splitting them into partial fractions (see notes sheet on algebraic methods), then by using a binomial expansion.
Often when trying to find the equation of a circle, you will be given a line that intersects with the circle twice (it may be the diameter). To work out the circle from this you need to know the midpoint: The perpendicular bisector of a cord will always pass through the centre of a circle. To find this, find the midpoint of the cord and find the equation of the line perpendicular to it (the negative reciprocal of the gradient) Equation of a Circle The points on a circle are all related through the equation of a circle. For a circle with centre (0,0) and radius r, the equation is: x² + y² = r² For a circle with centre (a, b) and radius r, the equation is: (x-a)² + (y-b)² = r² Expanded Form Sometimes, the equation is given or needed in expanded form: x² + y² + 2fx + 2gy + c = 0 (-f, -g) is the centre of the circle √(f² + g² -c) is the radius To go back from expanded form to the standard factorised form for an equation of a circle, it is quickest to complete the square: Intersecting Lines & Circles A straight line can intersect a circle once, twice, or not at all: To find out how many intersections there are between a circle and a straight line without sketching it, you can solve simultaneously: Circle Theorems
- Parametric Equations
An alternative to defining curves with Cartesian equations (one equation per curve with only x and y variables), we can express curves as separate equations of x and y, each in terms of the third variable t. A curve y = f(x) can also be described using x = p(t) and y = q(t). The coordinates of each point on the curve are (p(t), q(t)) It is possible to convert between Cartesian and parametric forms with substitution to eliminate one variable: To convert x = 4t, y = 2t ² + 4 into Cartesian: x = 4t t = ¼x Rearrange to make t the subject y = 2t ² + 4 y = 2(¼x) ² + 4 Substitute into y = q(t) y = ⅛x² + 4 Simplify Domain & Range For parametric equations x = p(t) and y = q(t) where the Cartesian equation is y = f(x): The domain of f(x) is the range of p(t) The range of f(x) is the range of q(t) Trigonometry Often, parametric equations will be given as trigonometric functions. When this is the case, it is often possible to use trig identities (see notes sheet on trigonometry) to convert between parametric and Cartesian. Angles should always be taken in radians. To convert x = sin t + 2, y = cos t - 3 into Cartesian: x = sin t + 2 sin t = x - 2 Rearrange to make sin t the subject y = cos t - 3 cos t = y + 3 Rearrange to make cos t the subject (x - 2)² + (y + 3)² = 1 Use identity sin²t + cos²t ≡ 1 Note that this is the equation of a circle. Sketching Parametric Curves To sketch a curve from parametric equations, either convert into Cartesian form - for example the equation above turned out to be a simple circle, radius 1 and centre (2, -3) - or tabulate the coordinates for different values of t and then plot these.
The graphs of sin, cos and tan are periodic - this means they repeat themselves for certain intervals. Both y = sinθ and y = cosθ have a repeating period of 360° or 2π Both y = sinθ and y = cosθ have a a maximum value of 1 and a minimum of -1 y = sinθ and y = cosθ are the same shape, but offset by 90° or 1/2π. y=tanθ or 1π. repeats every 180° or 1π. There are vertical asymptotes at 90° and then every 180° after in each direction. This means tanθ is undefined at these values. Standard graph transformations apply to trigonometric graphs. See notes sheet on Functions, Graphs & Transformations Sine & Cosine Rules Often you will only know some side lengths or angles in a triangle, and will need to work out the missing measurements. When you know two side lengths and one of their corresponding (opposite) angles or vice versa, use the sine rule: Sometimes the sine rule produces two possible solutions for a missing angle, an acute angle and an obtuse angle. Make sure you get the right one. Cosine Rule If you know just the three side lengths and no angles, use the cosine rule: Area of a Triangle To calculate the area of a triangle, you need to know two side lengths and the angle between these two sides: CAST Diagrams & Quadrants As well as their standard graphs, we can represent trig functions as circles in four quadrants. These diagrams are called CAST diagrams, and work by measuring an angle anticlockwise from the positive x-axis. CAST is an acronym for which functions are positive in which quadrants (sections of the graph), starting in Q4 and going anticlockwise through Q4, Q1, Q2, Q3: In Q1, all of sinθ, cosθ and tanθ are all positive In Q2, only sinθ is positive In Q3, only tanθ is positive In Q4, only cosθ is positive CAST diagram are useful for angles greater than 360°, as you can keep going round in circles and see which quadrant you end up in. Radians A radian is an alternative measure for angles. Typically, we use radians instead of degrees for angles because we can differentiate and integrate functions in radians, but not ones in degrees. A radian is the angle subtended by a circular arc with a length equal to the radius of the circle: There are 2π in a circle, and half that in a semicircle. Therefore, to convert from degrees to radians, divide the angle in degrees by 180/π to convert from radians to degrees, multiply the angle in radians by 180/π Arc Length When using radians, calculating arc length is a lot easier than when using degrees: l = rθ arc length = radius x angle (when using radians) The diagram above shows two arcs: the minor arc is the smaller one in red, with an angle θ the major arc is the larger dotted arc, with an angle of (2π-θ) Sector Area Again, this is a lot easier to calculate using radians. A = 1/2 r²θ Sector Area = 1/2 x radius² x angle (when using radians) Just like with arc, there are minor and major sectors. Segment Area Using sector area and the area of a triangle, you can easily work out the area of a segment: Work out the area of the sector using A = 1/2 r²θ Work out the area of the triangle inside using A = 1/2 ab sinC Subtract the area of the triangle from the sector area Small Angle Approximations When θ is small and measured in radians: sinθ ≈ θ tanθ ≈ θ cosθ ≈ 1 - θ²/2 Trigonometric Ratios There are a few exact values that are helpful to know off by heart for each of sine, cosine and tangent: Sine sin 30° = sin π/6 = ½ sin 60° = sin π/3 = (√3)/2 sin 45° = sin π/4 = 1/√2 = (√2)/2 Cosine cos 30° = cos π/6 = (√3)/2 cos 60° = cos π/3 = ½ cos 45° = cos π/4 = 1/√2 = (√2)/2 Tangent tan 30° = tan π/6 = 1/√3 = (√3)/2 tan 60° = tan π/3 = √3 tan 45° = tan π/4 = 1 Sec, Cosec & Cot Each of sine, cosine and tangent have their respective reciprocal functions: secant (sec), cosecant (cosec) and cotangent (cot): Secant, sec sec x = ⅟cos x sec x is undefined for values of x where cos x = 0, so has vertical asymptotes y = sec x is symmetrical about the y-axis y = sec x has a period of 360° or 2π radians the range of y = sec x is y ≤ -1 and y ≥ 1 Cosecant, cosec cosec x = ⅟sin x cosec x is undefined for values of x where sin x = 0, so has vertical asymptotes y = sec x has a period of 360° or 2π radians the range of y = sec x is y ≤ -1 and y ≥ 1 Cotangent, cot cot x = ⅟tan x cot x is undefined for values of x where tan x = 0 y = sec x has a period of 180° or 1π radians the range of y = sec x is y ∈ ℝ Trigonometric Identities For all values of θ, whether in radians or degrees, sin² θ + cos² θ ≡ 1 tan θ ≡ sin θ / cos θ You can use these identities to solve trigonometric equations. However, it is important to note a few things: sin θ = k and cos θ = k only have solutions when -1 ≤ k ≤ 1 tan θ = k for all values of k when using a calculator to find sin‾¹, cos‾¹ or tan‾¹, the calculator will give the principle values only - this will probably not be all the values you need cot x = cos x / sin x 1 + tan² x ≡ sec² x 1 + cot² x ≡ cosec² x Again, there are a few things to note: sec x = k and cosec x = k have no solutions when -1 < k <1 cot x = k for all values of k Angle Addition Formulae sin(A ± B) ≡ sinA cosB ± cosA sinB cos(A ± B) ≡ cosA cos B ∓ sinA sinB tan(A ± B) ≡ (tanA ± tanB) / (1 ∓ tanA tanB) These are sometimes also referred to as the compound-angle formulae. They work for degrees and radians. Double Angle Formulae These are derived from the angle addition formulae above. sin2A ≡ 2 sinA cosA cos2A ≡ cos²A - sin²A ≡ 2cos²A - 1 ≡ 1 - 2sin²A tan2A ≡ (2 tanA) / (1 - tan²A) The R-Addition Formula Trigonometric expressions in the form a sinx ± b sinx or a cosx ± b sinx, where a and b are positive, can be simplified: a sinx ± b sinx can be expressed as R sin(x ± α) a cosx ± b sinx can be expressed as R cos(x ∓ α) Where R > 0 and 0° < α < 90° (or ½π) and: R cos α = a R sin α = b R = √(a² + b²) Inverse Trigonometric Functions Sine, cosine and tangent all have inverse functions, known as arcsin, arccos and arctan. Arcsin & Arccos The domain of y = arcsinx and y = arccosx is -1 ≤ x ≤ 1 The range of y = arcsinx is -½π ≤ arcsinx ≤ ½π or -90° ≤ arcsinx ≤ 90° The range of y = arccosx is 0 ≤ arccosx ≤ π or 0° ≤ arccosx ≤ 180° Arctan The domain of y = arctanx is x ∈ ℝ The range of y = arctanx is -½π ≤ arctanx ≤ ½π or -90° ≤ arctanx ≤ 90°
Before going too far down this notes sheet, ensure you're on top form for everything differentiation... else this will be harder than it needs to be! Integration is the opposite of differentiation, and can be used to find the initial function form its derivative. For example, to find f(x) from f'(x), you integrate. It is also commonly used to find the area between a curve and the x-axis. Integrating xⁿ When differentiating xⁿ, you multiply by the power and then reduce the power by one. The reverse of this is to add one to the power and divide by this: If f'(x) = xⁿ, then f(x) = (xⁿ⁺¹) / (n+1) If there is a coefficient of x, you do the same but: If f'(x) = kxⁿ, then f(x) = k(xⁿ⁺¹) / (n+1) Note the difference between differentiating and integrating xⁿ: Just like when differentiating polynomials (functions with multiple terms), integrate one term at a time. Definite & Indefinite Integrals Integration is often noted using ∫ () dx, where the elongated S tells you to integrate the function in the brackets while the dx tells you to do so with respect to x. It is important to get this notation right. When there are no limits with the ∫, the integral is called indefinite, because it can only give a function with an unknown y-intercept, c. If there are limits attached to the ∫, a and b, the integral is definite, as it produces a value for a defined interval [a, b]. Indefinite Integral Functions Unless you are given more information, indefinite integrals give an unspecific function. This means they produce a function but do not locate it - as in it could intersect anywhere on the y-axis. This is because when you differentiate a function, the y-intercept is not an x-term, so disappears. Therefore, when integrating the derivative, you have no idea where the y-intercept is, so we note it as an unknown, c. You always need to add ' +c ' to unspecific integral functions, else it is not complete. Finding the y-intercept, c However, when you are given more information than just what to integrate, for example a coordinate that the function passes through, you can find the value of c, the y-intercept to find the particular integral: Areas under Graphs When given a definite integral, you integrate the function, and then evaluate it by substituting in the limits and subtracting the lower from the upper: ∫[a, b] f'(x) dx = f(b) - f(a) For example, to evaluate 4x+3: While differentiation is used to find the gradient of a function, integration is used to give the area enclosed by the graph and the x-axis and the limits. Therefore, the example above is actually how to calculate the area under the line y = 4x + 3 between x=1 and x-3: Evaluating definite integrals gives the area enclosed by the function, the limits and the x-axis. Areas under the x-axis If the graph is below the x-axis, the value will be negative. Therefore, when there is a root in an interval, you need to find the root and split it up into two separate definite integrals and evaluate them separately. Find the x-coordinate of the root, r Evaluate two integrals, one on the interval [r, b] and one over [a, r] Add the positive integral to the modulus of the negative integral Area under Parametric Curves This requires the standard function for integrating 1/x - see below It is not necessary to find the Cartesian form of a curve given as parametric to integrate it. Instead, we can use the chain rule to write dx as dx/dt, and integrate with respect to the parameter, t: If x and y are given as functions of t, ∫y dx = ∫y dx/dt dt It is important to remember to change the limits into t, as you are integrating with respect to t. Area Between Graphs When trying to find the area enclosed by two or more graphs, it is easiest to split it up into the separate parts from each graph and the add or subtract accordingly. Areas between curves & lines To find the area enclosed between a curve and a straight line, you need to find the area between each and the x-axis and then subtract one from the other. Often, you will have to use equations of triangle and trapezium area: Area of a triangle = ½ ab sinC Area of a trapezium = ½ (a+b)h Areas between curves To find the area between two curves, you use integration to calculate the area between each curve and the x-axis, and then subtract one from another. Integrating Standard Functions We have covered how to integrate xⁿ, but there are nine more standard functions you need to know: ∫ xⁿ dx = (xⁿ⁺¹) / (n+1) + c ∫ e ˣ dx = e ˣ + c ∫ 1/x dx = ln|x| + c ∫ cos(x) dx = sin(x) + c ∫ sin(x) dx = -cos(x) + c ∫ tan(x) dx = ln|sec(x)| + c ∫ sec²(x) dx = tan(x) + c ∫ cosec(x) cot(x) dx = -cosec(x) + c ∫ cosec²(x) dx = -cot(x) + c ∫ sec(x) tan(x) dx = sec(x) + c ∫ f'(ax+b) dx = f(ax+b) / a + c Vital things to know The last standard function only works for functions in the form f(ax+b) and not for trigonometric functions like cos(ax+b). Only the trigonometric functions listed here can be integrated. For other functions, such as to integrate tan(x), you need to use trigonometric identities (see notes sheet on trigonometry) to get them into a form that can be integrated. Often, to get a function into a format that can be integrated you need to split it into partial fractions. Then, you can use the standard result for 1/x, or the reverse chain rule (see below) Reverse Chain Rule The reverse chain rule can be used when functions can be written in the form k f'(x) / f(x). The numerator must be an exact multiple of the derivative of the denominator for this to apply. For example: The reverse chain rule can also be used when functions can be written in the form k f'(x) ( f(x) )ⁿ. For example: In both instances, adjustment may be required after trying the respective solutions. Integration by Substitution Often, you can make a complicated integral simpler by changing the variable. The difficult bit can be knowing what to use as your substitution, but in general: If there is a bracket to a high power, make this bracket the substitution In fractions, make the denominator your substitution If there is a root, make this the substitution To integrate by substitution: Write out your substitution as u Rearrange this to make x the subject Find du/dx Rearrange this to make dx the subject Manipulate the initial expression to replace dx with du and get it in terms of u only Integrate in terms of u Re-substitute u back to the initial form, and evaluate with limits Do not evaluate limits while using the substitution, as the limits will be invalid and give the incorrect answers. Only evaluate once you have substituted back to x Integration by Parts This is the integration equivalent of the product rule used in differentiation, and is used when you need to integrate two functions that are multiplied together: where u and v are functions of x This only works when the expression you are trying to integrate is in the form u dv/dx, so you need to choose which part to set as u and which as v. u will need to be differentiated v will need to be integrated It is a good idea to set dv/dx as the part of the expression that is easiest to integrate This is a fairly simple example, but they can get really very challenging. Often, you will end up with another integration by parts inside the first one - this is normal. If, however, you end up in never-ending circles of integration by parts, it means you have substituted the wrong part for v and u. The Trapezium Rule The trapezium rule is a numerical method used to approximate the area under a curve by dividing it into equally wide strips of varying heights: The are n number of strips, so the width of each strip, h, is given as h = (b - a)/n The formula for the trapezium rule is derived from the area of a trapezium, ½(a+b)h, where a and b are the height of each dotted line (the parallel sides of the trapezium). The first is called y₀, then y₁, y₂, y₃ etc. until you get to the last one, yₙ: To use this, you need to make a table of all the y-values for each interval h of x. Then, input these into the formula. The greater n, the more accurate the approximation - however also the more time-consuming it is. Differential Equations Wow, you've made it through. Now for the bit where any of this is actually useful... Separating Variables This is the simplest method for solving differential equations, and is the only one we will cover under single maths. As the name suggests, it works y separating the variables: When dy/dx = f(x) g(y), you can say ∫ 1/g(y) dy = ∫ f(x) dx Move all the y terms to the left where the dy is Move all the x terms to the right, including the dx This allows you to integrate each side with respect to the variable on that side to solve the equation. You only need to add the ' +c ' to one side. Just like when integrating an indefinite function, the initial result is a general solution and could be anywhere along the y-axis (see section on indefinite integral functions above). To fond the particular solution, you need to know a coordinate point on the curve - sometimes this is called a boundary condition. Modelling with Differential Equations A huge application of differentiation and integration is to model anything where the rate of change of something is dependent on the value of that something. For example, rabbit populations...
The gradient of a curve is constantly changing, and so finding the gradient is not as simple as reading the x-coefficient of the equation. Instead, we can draw tangents to the curve to calculate the gradient at the point where the tangent touches the curve. The gradient of a curves at a given point is given as the gradient of the tangent to the curve at that point Finding the Derivative However, drawing tangents is not very accurate when done by eye. Instead, we can use algebra to find the gradient of a curve at a given point. This works by drawing a cord connecting two points on the curve, y = f(x). The gradient of this cord gives an estimate for the average gradient of the curve between the two points. As you can see, the closer the two points are together, the more accurate the gradient of of the cord is as an estimation. This is because the cord gets closer and closer to being parallel to the tangent. This can be noted as a cord between points A and B, where the horizontal distance between the points (difference in x-values) is h. Therefore, the vertical distance (difference in y-values) is f(x₀+h) - f(x₀): Since gradient is defined as change in y-value over change in x-value (rise over run), the gradient of the cord is given as: As the value for h gets smaller, points A and B get closer together, and the gradient of the cord becomes a better estimation for the gradient of the curve at A. As h → 0, the gradient of the cord is identical to the gradient of the curve at the point. This is known as differentiation from first principles: Terminology The derivative of a function is the differentiated form of the function The derivative of f(x) can be noted as f'(x) The derivative of y = f(x) can be noted as dy/dx h → 0 means "as h tends to zero" - its value becomes negligible Differentiating xⁿ You do not have to differentiate from first principles every time - it is only used as a proof. Instead, we use general rules to make differentiation quicker. To differentiate xⁿ: multiply by the power, n reduce the power by one, n becomes n-1 If f(x) = xⁿ, then f'(x) = nxⁿ‾¹ If f(x) = axⁿ, then f'(x) = anxⁿ‾¹ This applies when n is real, and a is a constant Differentiating Quadratics For functions with more than one term, you differentiate one term at a time, using the rules above. A quadratic is a function where the highest power of x is x², and can be differentiated as such: If f(x) is ax² + bx + c, then f'(x) = 2ax + b This applies when a, b and c are constants and x is the only variable Note that the derivative is now linear, because all x-terms were reduces by one order. c has disappeared all together, because it can be seen as cx⁰, so when c is multiplied by the power of x, 0, it becomes 0. Differentiating Cubics The exact same rules apply for all functions: differentiate each term individually. If f(x) = ax³ + bx² + cx + d, then f'(x) = 3ax² + 2bx + c This applies when a, b, c and d are constants and x is the only variable Note that the cubic becomes a quadratic when differentiated. Second Order Derivatives The first order derivative is the rate of change of the function, or the gradient. The second order derivative is the rate of change of the gradient itself, or the rate of change of the rate of change of the function. It is noted as f''(x) or d²y/dx² This sounds complicated, but all it means is you differentiate a function twice: If f(x) = ax³ + bx² + cx + d: f'(x) = 3ax² + 2bx + c f''(x) = 6ax + 2b As well as second order derivatives, you can have third, fourth fifth etc order derivatives. However, these are rarely encountered at this stage. Gradients, Tangents & Normals Often, differentiation is used to find the equation of a tangent or normal to a curve at a given point. The tangent to the curve y = f(x) at the point (a, f(a)) has gradient f'(a) The tangent to the curve y = f(x) at the point (a, f(a)) has equation y - f(a) = f'(a)(x - a) The normal is the line that is perpendicular to the tangent, and intersects the curve at point A. The gradient of the normal is given as the negative reciprocal of the gradient of the tangent: The normal to the curve y = f(x) at the point (a, f(a)) has gradient 1 / f'(a) The normal to the curve y = f(x) at the point (a, f(a)) has equation y - f(a) = -(x - a) / f'(a) The negative reciprocal means that the product of the gradients of the tangents and normal equals -1 Increasing and Decreasing Functions A function can either be increasing or decreasing over a given interval, depending on whether its derivative is positive or negative. f(x) is increasing on the interval [a, b] if f'(x) ≥ 0 for all values a < x < b f(x) is decreasing on the interval [a, b] if f'(x) ≤ 0 for all values a < x < b Stationary Points A stationary point is a point on a curve where the gradient is zero. It can take one of three forms: At a local maximum, the gradient changes from positive to negative At a local minimum, the gradient changes from negative to positive At a point of inflection, the gradient does not change sign - it can either stay positive or negative throughout. You can determine which type of point it is by looking at what the second order derivative is doing, or by what the first order differential is on either side of the point. To determine the nature of a stationary point, A: If f''(a) > 0, the point is a local minimum If f''(a) < 0, the point is a local maximum If f''(a) = 0, look at the gradient on either side of A as it could be any of the three Convex & Concave Curves All curves are either convex or concave over a particular interval. It is easy to remember which is which, because concave curves look like the entrance to a cave, while convex curves look like a v. A point of inflection is where the curve changes from being concave to being convex or vice versa Sketching Gradient Functions To sketch a gradient function of a particular function, this guide applies: Maximum or minimum points on f(x) cut the x-axis on f'(x) Points of inflection on f(x) touch but do not cross the x-axis on f'(x) A positive gradient on f(x) means f'(x) is above the x-axis A negative gradient on f(x) means f'(x) is below the x-axis Vertical asymptotes on f(x) remain as vertical asymptotes on f'(x) Horizontal asymptotes of f(x) become horizontal asymptotes at the x-axis on f'(x) The Chain Rule The chain rule is used to differentiate composite functions: Where y is a function of u, and u is a function of x For example, to differentiate y = (3x⁴ + x)⁵: For functions in the form y = ( f(x) )ⁿ, finding dy/dx can be generalised as such: Multiply by n Multiply by f'(x), the derivative of the bracket Reduce the power of the bracket by one If y = ( f(x) )ⁿ, then dy/dx = n f'(x) ( (f(x) )ⁿ‾¹ A special form of the chain rule can be used when functions are not in the form y = f(x): The Product Rule The product rule is used to differentiate the product of two functions: where u and v are function of x This can be noted as: If f(x) = g(x) h(x), then f'(x) = g(x) h'(x) + h(x) g'(x) For example, to differentiate y = x² √(3x - 1): The Quotient Rule The quotient rule is used for fractions where both numerator and denominator are functions of x: where u and v are functions of x Note the minus sign in the numerator - order is important! For example, to differentiate y = x / (2x+5): Trigonometric Differentiation Trigonometric functions can be differentiated using small angle approximations (see notes sheet on trigonometry). These approximations are only valid when angles are measured in radians, therefore To differentiate a trigonometric function, all angles must be in radians Differentiating sin and cos From first principles, we know that: If f(x) = sin(kx), then f'(x) = k cos(kx) If f(x) = cos(kx), then f'(x) = -k sin(kx) It is important to remember the negative sign when deriving cos x. Differentiating tan We can use the quotient rule and the trigonometric identity tanx = sinx / cosx to differentiate tanx: If f(x) = tan(kx), then f'(x) = k sec²(kx) Differentiating cosec, sec & cot Similarly, the quotient rule can be used to prove the results for cosec, sec and cot. The results are given as: If f(x) = cosec(kx), then f'(x) = -k cosec(kx) cot(kx) If f(x) = sec(kx) then f'(x) = k sec(kx) tan(kx) If f(x) = cot(kx), then f'(x) = -k cosec²(kx) Parametric Differentiation When functions are given as parametric (see notes sheet on parametric equations), there is no need to convert them into Cartesian form to differentiate them. Instead, the derivative of the Cartesian equation can be found from the derivatives of the parametric equations: This is from a version of the chain rule. For example, to differentiate x = t³ + t, y = t² +1: Implicit Differentiation Sometimes, functions can be difficult or near impossible to rearrange so that a linear variable is the subject. these equations can often be differentiated implicitly without having to rearrange them at all. In general, from the chain rule: Specific results that you will come across a lot are: In practice... In practice, what these results mean is: Differentiate all terms with only x in them normally When differentiating a term with only y in it, differentiate it with respect to y and multiply by dy/dx For terms with both x and y in them, use the product rule and multiply only the term with the y-derivative in it by dy/dx At the end, rearrange to make dy/dx the subject For example, to differentiate x³ + x + y³ + 3y = 6:
A vector is a quantity with both magnitude and direction, and is typically represented visually by a line segment between two points. There are many ways of representing vectors in notation: The two points that the vector connects with an arrow above them As a bold typeface lower case letter As an underlined lower case letter As a column vector, showing displacement in the x-direction above that in the y-direction As a multiple of unit vectors, i (one unit in the positive x-direction) and j (one unit in the positive y-direction) These notes will predominantly use bold typeface of lower case letters, underlined letters and the unit vectors i and j. Vectors can be multiplied by a scalar, and added and subtracted: Magnitude & Direction The magnitude of a vector is given by Pythagoras' Theorem. Magnitude is noted using straight lines on either side of the letter, like modulus. For the vector a = xi + yj, |a| = √(x²+y²) A unit direction vector, â, can be found as a / |a| A vector can also be defined by giving its magnitude and the angle it makes with one of the coordinate axis. This is called magnitude-direction form. Position Vectors Position vectors are used to give the location of a point relative to a fixed origin. A point (p, q) has a position vector pi + qj Vector Geometry Position vectors can be used to solve geometric problems: Vectors in 3D In three-dimensional geometry, there are x, y, and z-axes. This means each coordinate has three values, (x, y, z). The position vector of this is xi + yj + zk Pythagoras' Theorem still applies, but adjusted for three points: Distance from point (x, y, z) to origin is given as √(x²+y²+z²) The distance between two points, (x₁, y₁, z₁) and (x₂, y₂, z₂) is given as: √( (x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)² ) Vector Geometry in 3D
- Exponentials and Logarithms
An exponential function is one a constant is raised to the power of a variable: The larger the coefficient, the steeper the graph All exponential functions in the form y = a^x pass through (0, 1) The value of the function decreases as x tends to 0 Functions in the form y = a^x where 0 < a < 1 are the other way around: The gradient graphs of an exponential function are always very similar, but when a = 2.72 (e), the gradient graph is exactly the same: For all real values of x: if f(x) = e^x, f'(x) = e^x if f(x) = e^kx, f'(x) = ke^kx Functions in e can be used to model growth or decline where the rate of increase in number is proportional to the number. Logarithms Logarithms are the inverse of exponential functions. Just as indices laws apply to exponentials, there are a series of laws that apply to logarithms - these are known as the log rules or laws of logarithms: When working with difficult equations with exponents, it can be helpful to apply logarithms to both sides: The Natural Logarithm, ln The natural logarithm, ln, is the logarithm in base e. The graph of ln(x) is reflection of e^x in the line y=x: The y=axis is an asymptote in this graph, meaning that ln(x) is only defined when x > 0 Logarithms and Non-Linear Data Often, it is helpful to model non-linear data as linear. This can be achieved by taking logs of one or both sides of the equation: If the equation is in the form y = ax^n, a graph of log y against log x will give a straight line where log a is the y intercept and n the gradient. If the equation is in the form y = ab^x, a graph of log y against x will give a straight line where log a is the y intercept and log b the gradient.
- Numerical Methods
A root of a function is when the function = 0. On a graph, this is represented by an x-axis intersection, where y-value changes from positive to negative or vice versa. Therefore, we can locate roots by showing that the sign has changed. However, this only applies if the function is continuous throughout the given interval, notated using square brackets [a, b]. This means that the graph does not jump from one value to another, but changes smoothly. If there is a vertical asymptote in the interval, it is not continuous. If f(x) is continuous on the interval [a, b], and f(a) and f(b) have opposite signs, then there is a root between the two Iteration It is possible to estimate roots using an iterative method. Generally, the equation needs to be manipulated slightly before it works, as the linear variable (x) has to be the subject. To solve an equation in the form f(x) = 0 using iteration, rearrange it into the form x = g(x) and apply iteration Applying iteration means you use put the value you get out of g(x) back into the function, and repeat this multiple times. There are multiple rearrangements for each function f(x), giving different variations of x = g(x). These different iterations will either converge to a root, or diverge: If the iteration converges closer and closer to the root, always in the same direction, it forms a staircase diagram. If the iteration converges in an alternating pattern above and below the root, a cobweb diagram is formed. If the iteration diverges, it moves away from the root very quickly. The Newton-Raphson Method The Newton-Raphson method is the method used by many calculators to solve equations in the form f(x) = 0. The formula for the Newton-Raphson method is: It works using tangent lines with increasing accuracy to approximate a root: As you can see, it relies on differentiation, and so cannot work when f'(x) equals or is close to equalling zero. If the starting value x₀ is close to a turning point, the method does not work, because the gradient of the tangent is very small, so the tangent crosses the x-axis a long way away from x₀ If any value of x is on a turning point, the method does not work because the gradient of the tangent is 0, and so never crosses the x-axis.
- Work, Energy & Power
Work done is the amount of energy transferred from one form to another when a force causes movement. The units of work are Joules, J W = Fx Work Done = Force × distance moved (in the same direction as the force) When the force is not in the same direction as the movement of the object, you need to work out the correct component force using trigonometry. You must remember to do this! Energy One of the fundamental laws of physics is the idea that energy cannot be created. This is known as the Principle of Conservation of Energy and states: The total energy in a closed system always remains constant - energy can never be created or destroyed, it can only ever be transferred from one form to another. Some of these forms include: Kinetic Gravitational Potential Elastic Potential Electrical Potential Nuclear Internal (heat/thermal) Kinetic Energy All moving objects have kinetic energy: KE = 1/2 m v^2 Kinetic Energy = 1/2 x mass x velocity squared This equation can be derived easily from work done and SUVAT: Gravitational Potential Energy Gravitational Potential Energy is the energy of an object due to its positioning in a gravitational field. The greater the height, the greater the GPE. GPE gain is work done in moving an object: GPE = mgh This can be derived as such: W = Fh → F = ma = mg → GPE = mgh (g is the acceleration due to free fall, 9.81 m/s^2) Because all energy is conserved, falling objects increase in kinetic energy as they decrease in GPE. Power Power is the rate at which work is done: P = W/t power = work done / time taken This can also be written as: P = Fv power = force × velocity A derivation of this is: P = W/t → P = Fx/t → P = Fv (Because v = x/t) The units of power are Watts, W, or Joules per second, J/s Efficiency Efficiency is the proportion of input energy that is transferred into useful output energy - for example in a motor, it is the portion of input energy that comes out as kinetic energy, rather than thermal or sound energy. Efficiency can be calculated as a percentage: 100 x useful output energy / total input energy
- Energy, Power & Resistance
Potential Difference Potential Difference (p.d.) is the work done per unit charge – the amount of energy converted from electrical to another form per coulomb of charge passing through a component. W = VQ work done = p.d. × charge Electromotive Force Electromotive Force is the amount of energy converted into electrical energy per coulomb of charge through the source: W= εQ work done = e.m.f x charge Essentially, p.d. is voltage used up in a circuit, whereas e.m.f. is the voltage provided to the circuit by a cell or power supply. Both p.d. and e.m.f. are often referred to as voltage, because they both shave the Volt as their unit. The Volt is 1J of energy per 1C of charge. Accelerating Electrons A cathode-ray tube (also called an electron gun) is used to accelerate electrons. A hot metal filament releases electrons by thermionic emission, and then these are accelerated in a vacuum between the filament and a metal plate with a small hole in it. The two are connected and a very high p.d. is applied, making the filament a cathode and the plate an anode. The electrons released from the filament are attracted to the positively charged anode plate, and pass through the hole. This creates a concentrated beam of high velocity electrons. When an electron is accelerated in this way, the energy transferred equals the wok done on the electron, which equals the kinetic energy: eV = 1/2 m v^2 work done on electron = gain in kinetic energy of electron Resistance All conductors have resistance that obstructs the flow of charge through them. The higher the resistance of a conductor, the more energy is required to push the charge through, as more energy is lost in the process. Ohm's Law Ohm's Law states that the current flowing through a conductor is directly proportional to the p.d. across it, provided that the physical conditions (such as temperature) remain constant. V = IR p.d. = current x resistance The units of resistance are Ohms, Ω, defined as the resistance of a conductor when 1 volt produces a current of 1 ampere through that conductor. I-V Characteristics To investigate the I-V relationships of components, a test circuit as shown can be set up and the current recorded for a variety of potential differences. Taking averages increases reliability, measuring to 3 sig-figs increases accuracy. A graph can be plotted to see its ohmic or non-ohmic properties. The ammeter must be connected in series, and the voltmeter in parallel. A Resistor/Metallic Conductor obeys Ohm's Law - the I-V characteristic is a straight line through the origin, where the inverse of the gradient is the resistance (R = V/I). A Filament Lamp is an example of temperature affecting resistance. As the p.d. increases, so does the brightness and temperature of the bulb and with it, its resistance. This is because at higher temperatures, the positive ions in the filament have greater internal energy, meaning they vibrate more. Therefore, the probability of the charge carriers colliding with them is greater. In this instance, Ohm's law does not apply. A Diode only works with current flowing thorough it in one direction - this is known as the forward bias. Typically, there is a threshold of around 0.6V before they conduct at all. Light Dependent Resistors and NTC Thermistors (Negative Temperature Coefficient) lose resistance as the energy on them increases. This is because they are semiconductors, that release more electrons for conducting when they have more energy. This means the electron density increases, so the current increases and resistance decreases. Resistivity Resistivity is a property of a material, as different structures make for different conducting properties. It is defined as the resistance from one side to another of a 1m cube. R = ρL/A resistance = resistivity × length / cross-sectional area This relationship means that resistance is directly proportional to length, while it is inversely proportional to the cross-sectional area: R ∝ L R ∝ 1/A Resistivity is measured in ohm meters, Ωm Investigating Resistivity To determine resistivity, you need to work out the cross-sectional area using a digital caliper. Then, clamp the wire in place with a voltmeter in parallel and an ammeter in series with a fixed power supply. Vary the length of wire (use a ruler) and measure voltage and current to establish the resistance. Repeat at several different lengths. The resistivity of a metal increases with temperature. This is because charge is carried by moving electrons, but heat makes this motion harder as the ions in the structure vibrate more (see I-V characteristics of a filament lamp above). The resistivity of a semiconductor decreases with temperature, as more electrons are released for conducting (See LDR/Thermistors above). Energy & Power Power is the rate of transfer of energy, where one Watt is one joule of work done per second. P = IV power = current x voltage Using the equation V = IR, we can rewrite this power equation to other, often more useful forms: If power is energy per second, then we can use it to work out the electrical energy (work done): W = Pt work done = power x time Again, this can be rewritten in multiple ways: A joule is a very small unit of measuring energy. In fact, it is so small that it is rarely used commercially, because we typically use thousands or millions of joules at a time. Therefore, the Kilowatt-hour is used domestically: 1 kWh = 3.6 million joules This can quickly be calculated: 1 kW = 1000W 1 h = 60x60s = 3600s 1 kWh = 1000W x 3600s = 3,600,000 Ws = 3,600,00 J Energy Transferred (kWh) = Power (kW) x Time (h)
Materials can have many different properties, and experience a number of external and internal forces. The two main such forces are the tensile and compressive forces, relating to extension (stretching) and compression (squishing) respectively. Hooke's Law The most common example of these forces being applied are in a helical spring: these have a natural length, and are either stretched or compressed depending on direction of force applied. The distance of this deformation is called the extension (regardless of direction) and according to Hooke's law: Extension is directly proportional to the force applied, provided the elastic limit is not passed As you can see in the graph, the relationship is linear, where the gradient is the force constant up to the elastic limit - if this is exceeded, the object will no longer return to its initial shape, - plastic deformation has occurred. F = kx Force = force constant x extension Multiple String Systems Springs (and indeed any elastic materials) can be combined either in series or in parallel, with vastly different effects: Elastic Potential Energy All springs, just like any object under tensile or compressive forces, store Elastic Potential Energy. This is equivalent to the work done in extending/compressing them, and since W = Fx, this is the area under a force-extension graph. W = 1/2 Fx E = 1/2 Fx E = 1/2 k x^2 Stress, Strain & Young Modulus Tensile Stress Stress is the force per unit cross-sectional area. Its units are Pascals, Pa. σ = F/A stress = force / cross-sectional area Tensile Strain Strain is the fractional change in length of the material. It is a ratio, so has no units. ε = x/L strain = extension / original length Young Modulus Provided the limit of proportionality is not exceeded, stress is directly proportional to strain. The constant is known as the Young Modulus, and this is a fixed property of any given material. The units are also Pa. E = σ/ε Young Modulus = tensile stress / tensile strain This graph shows the elastic profile of a typical metal wire, such as mild steel. Up until the limit of proportionality, the wire obeys Hooke's Law. Between the limit of proportionality and the elastic limit, the wire deforms elastically, and plastically hereon-after. Between the two yield points, the material extends dramatically. The stress increases until it reaches the UTS - the Ultimate Tensile Stress. This is the maximum stress the wire can take. Finally, the stress decreases until the material just breaks. Material Properties Ductile Materials curve on a stress strain graph, and even when deformed they keep their strength. See the steel graph above. Brittle materials do not curve. To start with, the graphs are identical when they obey Hooke’s law. Then, when the stress reaches a certain point, the material snaps without deforming. Stronger materials can withstand more stress before they break than weaker ones. Stiffer materials are harder to stretch or compress as they have a high young modulus. Stiffness is not a sign of strength. Polymeric materials have their molecules arranged in long chains, which gives a vast range of properties. Polythene behaves plastically, meaning applying enough stress will deform it into a new shape permanently, and it is also ductile. Rubber behaves elastically, but its loading and unloading graphs are different. This means less energy is released on the unloading graph, so less work is done, and some is lost as heat.