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An in-depth exploration of various functions, their transformations, and the concept of calculus. Topics covered include one-to-one and many-to-one functions, inverse functions, transformations of graphs, the modulus function, inverse trigonometric functions, the number e, further differentiation, and integration. The document also discusses the chain rule, product rule, quotient rule, numerical solutions of equations, and iterative methods.
What you will learn
Typology: Lecture notes
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Examination Board: AQA
Chapter 1 โ Functions
Chapter 2 โ Transformations of graphs and the modulus function
Chapter 3 โ Inverse trigonometric functions and secant, cosecant and cotangent
Chapter 4 โ The number e and calculus
Chapter 5 โ Further differentiation and the chain rule
Chapter 6 โ Differentiation using the product rule and the quotient rule
Chapter 7 โ Numerical solutions of equations and iterative methods
Chapter 8 โ Integration by inspection and substitution
Chapter 9 โ Integration by parts and standard integrals
Chapter 10 โ Volume of revolution and numerical integration
Chapter 2 โ Transformations of graphs and the modulus function
The transformation ๐ฆ = ๐(๐ฅ โ ๐) + ๐ is a translation of [๐๐].
Eg: The function ๐ฆ = (๐ฅ + 4)^3 + 7 is a translation of 4 to the left and 7 up from ๐ฆ = ๐ฅ^3.
The transformation ๐ฆ = ๐๐(๐ฅ) is a stretch by a factor of k in the y direction. The transformation ๐ฆ = ๐ (๐ฅ๐) is a stretch by a factor of c in the x direction.
Eg: The function y = 3sin 4๐ฅ is a stretch of factor 3 in the y direction and a stretch of factor ยผ in the x direction from ๐ฆ = sin ๐ฅ.
A transformation of ๐ฆ = โ๐(๐ฅ) is a reflection in the x-axis. A transformation of ๐ฆ = ๐(โ๐ฅ)^ is a reflection in the y-axis.
Eg: The function ๐ฆ = โ๐ฅ^2 is a reflection of the function ๐ฆ = ๐ฅ^2 in the x-axis. The function ๐ฆ = (โ๐ฅ)^3 + (โ๐ฅ) is a reflection of the function ๐ฆ = ๐ฅ^3 + ๐ฅ in the y-axis.
Note: the x-axis reflection changes the sign of the output values, while the y-axis reflection changes the sign of the input values.
The modulus function gives the absolute value of its input , so that: |๐(๐ฅ)| = ๐(๐ฅ) ๐คโ๐๐ ๐(๐ฅ) โฅ 0 |๐(๐ฅ)| = โ๐(๐ฅ) ๐คโ๐๐ ๐(๐ฅ) < 0
Eg: |5๐ฅ โ 10| = 5๐ฅ โ 10 ๐คโ๐๐ 5๐ฅ โ 10 โฅ 0 ๐๐ ๐ฅ โฅ 2 |5๐ฅ โ 10| = โ(5๐ฅ โ 10) = 10 โ 5๐ฅ ๐คโ๐๐ 5๐ฅ โ 10 < 0 ๐๐ ๐ฅ < 2
Note: To solve simple equations involving the modulus function, deal with the separate cases (function negative, function non-negative) individually, then discard any solutions not within the valid range. To solve more complex equations or inequalities, sketch a graph.
Chapter 3 โ Inverse trigonometric functions and sec, cosec and cot
sinโ1^ ๐ฅ has domain โ1 โค ๐ฅ โค 1 and range โ ๐ 2 โค sinโ1^ ๐ฅ โค ๐ 2 cosโ1^ ๐ฅ has domain โ1 โค ๐ฅ โค 1 and range 0 โค cosโ1^ ๐ฅ โค ๐ tanโ1^ ๐ฅ has domain ๐ฅ โ โ and range โ ๐ 2 < tanโ1^ ๐ฅ < ฯ 2
Secant, cosecant and cotangent are defined as: sec ๐ฅ = (^) cos ๐ฅ^1 , csc ๐ฅ = (^) sin ๐ฅ^1 , cot ๐ฅ = (^) tan ๐ฅ^1.
Note: (^) sin ๐ฅ^1 โ sinโ1^ ๐ฅ. One is the reciprocal of the function, the other is the inverse function.
sin^2 ๐ฅ + cos^2 ๐ฅ โก 1 1 + cot^2 ๐ฅ โก csc^2 ๐ฅ tan^2 ๐ฅ + 1 โก sec^2 ๐ฅ
Note: The second two identities can both be produced easily from the first one by dividing through by either sin^2 ๐ฅ or cos^2 ๐ฅ.
๐ฆ = sin ๐ฅ ๐ฆ = cos ๐ฅ ๐ฆ = tan ๐ฅ
๐ฆ = csc ๐ฅ ๐ฆ = sec ๐ฅ ๐ฆ = cot ๐ฅ
(All horizontal scales between 0 ๐^ and 2๐๐)
Note: You should know how to accurately sketch the primary trigonometric functions โ the other three you ought to be able to produce from those.
Chapter 5 โ Further differentiation and the chain rule
๐๐ฆ ๐๐ฅ =
This is the chain rule proper. However, it is often easier to use this expression of it:
๐ ๐๐ฅ ๐๐(๐ฅ) = ๐
Eg: (^) ๐๐ฅ๐ (2๐ฅ + 5)^3 = 3(2๐ฅ + 5)^2 (2) = 6(2๐ฅ + 5)^2
The 2๐ฅ + 5 part is โignoredโ, then the end result is multiplied by the differential of the โignoredโ part.
Note: The chain rule can be extended to any number of functions within functions, but you will rarely be expected to deal with more than two or three.
๐ ๐๐ฅ sin ๐ฅ = cos ๐ฅ ๐ ๐๐ฅ cos ๐ฅ = โ sin ๐ฅ
Note: A number of more complex trigonometric (and other) results are given in the formula book.
An increasing function is a function ๐(๐ฅ) such that ๐โฒ(๐ฅ) โฅ 0 for all ๐ฅ. A decreasing function is a function ๐(๐ฅ) such that ๐โฒ(๐ฅ) โค 0 for all ๐ฅ.
Note: This means that a function is only increasing if its gradient is never negative , and a function is only decreasing if its gradient is never positive.
Eg: ๐(๐ฅ) = ๐ฅ^3 + 2๐ฅ is an increasing function because ๐โฒ(๐ฅ) = 3๐ฅ^2 + 2 > 0 for all ๐ฅ (since ๐ฅ^2 โฅ 0).
Chapter 6 โ Differentiation using the product and quotient rule
Product Rule: ๐ ๐๐ฅ ๐(๐ฅ)๐(๐ฅ) = ๐(๐ฅ)๐
This is the function notation form. A more easily memorable one may be written as:
๐ ๐๐ฅ ๐ข๐ฃ = ๐ข๐ฃ
Eg: (^) ๐๐ฅ๐ (๐ฅ^2 sin ๐ฅ) = ๐ฅ^2 cos ๐ฅ + 2๐ฅ sin ๐ฅ
Quotient Rule: ๐ ๐๐ฅ (
This is given (in function notation) in the formula book, but this form may be more memorable.
Eg: (^) ๐๐ฅ๐ ( (^) ๐ฅ 2 ๐ฅ+1) = (๐ฅ
(^2) +1)โ๐ฅ(2๐ฅ) (๐ฅ^2 +1)^2 =^
1โ๐ฅ^2 (๐ฅ^2 +1)^2
Chapter 7 โ Numerical solutions of equations and iterative methods
To show that๐(๐ฅ) = 0 has a root ๐ผ such that ๐ < ๐ผ < ๐, it is sufficient to demonstrate that either ๐(๐) < 0 ๐๐๐ ๐(๐) > 0 or ๐(๐) > 0 ๐๐๐ ๐(๐) < 0.
Note: The converse is not always true. ๐(๐) > 0 ๐๐๐ ๐(๐) > 0 does not imply that there isnโt a root between ๐ and ๐.
If you are required to show a function has a root between two values, first rearrange into the form ๐(๐ฅ) = 0, then apply the rule above.
Iteration requires a formula linking each approximation of ๐ฅ to the next: ๐ฅ๐+1 = ๐(๐ฅ๐).
An iteration ๐ฅ๐+1 = ๐(๐ฅ๐) converges if |๐โฒ(๐ฅ)| < 1 for certain values of ๐ฅ.
If the iteration converges to a limit, that limit can be found by setting ๐ฅ๐+1 = ๐ฅ๐ = ๐ฟ.
Eg: The equation ๐ฅ^3 โ 4๐ฅ = 7 produces the iteration formula ๐ฅ๐+1 = โ7 + 4๐ฅ^3 ๐. Starting with 5: ๐ฅ 1 = 5 ๐ฅ 2 = โ7 + 4๐ฅ^3 1 = โ7 + 4(5)^3 = 3 ๐ฅ 3 = โ7 + 4๐ฅ^3 2 = โ7 + 4(3)^3 = 2.6684 โฆ
Chapter 9 โ Integration by parts and standard integrals
Note: This result is given in the formula book โ it is given here for reference only.
You will be most commonly expected to apply Integration by Parts to the following forms of integral:
โซ ๐ฅ๐^ sin ๐๐ฅ ๐๐ฅ
โซ ๐ฅ๐^ cos ๐๐ฅ ๐๐ฅ
โซ ๐ฅ๐^ ln ๐๐ฅ ๐๐ฅ
Note: While for most functions the simplest part (usually ๐ฅ๐) becomes ๐ข and the other part becomes (^) ๐๐ฅ๐๐ฃ, when ln ๐๐ฅ is involved, this should be made ๐ข, since it is not readily integrable.
Eg:
โซ ๐ฅ^2 ln ๐ฅ ๐๐ฅ
๐ข = ln ๐ฅ
2 ๐๐ข ๐๐ฅ =
โซ ๐ฅ^2 ln ๐ฅ ๐๐ฅ =
3 ln ๐ฅ โ โซ
3 ln ๐ฅ โ โซ
3 ln ๐ฅ โ
(3 ln ๐ฅ โ 1) + ๐ถ
Note: The standard integrals in this chapter are all given in the formula book, but it is important to be familiar with using and applying them.
Chapter 10 โ Volume of revolution and numerical integration
The volume of a function ๐ฆ = ๐(๐ฅ) rotated fully around the x-axis is given by:
๐ = โซ ๐๐ฆ^2 ๐๐ฅ
๐ ๐
Note: To find a volume rotated around the y-axis, rearrange to give ๐ฅ = ๐(๐ฆ) and use the rule:
๐ = โซ ๐๐ฅ^2 ๐๐ฆ
๐ ๐
Approximations to integrals can be found using a numerical integration method. Three of the most commonly used methods are:
The Trapezium Rule approximates the curve to straight lines and calculates the area of the resultant trapezia.
The Mid-Ordinate Rule takes the midpoint between two ordinates as the height and approximates the area through a series of rectangles.
Simpsonโs Rule approximates the curve as a series of quadratics, and since this is usually a much closer fit to the actual function, Simpsonโs Rule usually gives a much better approximation than either of the other two. Valid only when used with an even number of strips.
Note: All of these rules are quoted in the formula book, but you will need to be familiar with them and be able to apply any of them to a function.
Eg: Approximation to โซ โ1 + ๐ 02 ๐ฅ^ ๐๐ฅusing Simpsonโs Rule with five ordinates (four strips):
2 0
Note: Simpsonโs Rule with ๐ ordinates usually gives a result accurate to ๐ decimal places.
Produced by A. Clohesy; TheChalkface.net 04/06/