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Functions, Transformations, and Calculus: A Comprehensive Study, Lecture notes of Calculus

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

  • How do you find the inverse of a function?
  • What is the modulus function and how is it used?
  • What is a function and how is it defined?
  • What is the difference between a one-to-one function and a many-to-one function?
  • What are the transformations of a graph and how do they affect the function?

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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The Not-Formula Book for C

Everything you need to know for Core 3 that wonโ€™t be in the formula book

Examination Board: AQA

Brief

This document is intended as an aid for revision. Although it includes some examples

and explanation, it is primarily not for learning content, but for becoming familiar

with the requirements of the course as regards formulae and results. It cannot

replace the use of a text book, and nothing produces competence and familiarity with

mathematical techniques like practice. This document was produced as an addition

to classroom teaching and textbook questions, to provide a summary of key points

and, in particular, any formulae or results you are expected to know and use in this

module.

Contents

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 ๐‘ฆ = ๐‘“(๐‘ฅ โˆ’ ๐‘Ž) + ๐‘ is a translation of [๐‘Ž๐‘].

Eg: The function ๐‘ฆ = (๐‘ฅ + 4)^3 + 7 is a translation of 4 to the left and 7 up from ๐‘ฆ = ๐‘ฅ^3.

The transformation ๐‘ฆ = ๐‘˜๐‘“(๐‘ฅ) is a stretch by a factor of k in the y direction. The transformation ๐‘ฆ = ๐‘“ (๐‘ฅ๐‘) is a stretch by a factor of c in the x direction.

Eg: The function y = 3sin 4๐‘ฅ is a stretch of factor 3 in the y direction and a stretch of factor ยผ in the x direction from ๐‘ฆ = sin ๐‘ฅ.

A transformation of ๐‘ฆ = โˆ’๐‘“(๐‘ฅ) is a reflection in the x-axis. A transformation of ๐‘ฆ = ๐‘“(โˆ’๐‘ฅ)^ is a reflection in the y-axis.

Eg: The function ๐‘ฆ = โˆ’๐‘ฅ^2 is a reflection of the function ๐‘ฆ = ๐‘ฅ^2 in the x-axis. The function ๐‘ฆ = (โˆ’๐‘ฅ)^3 + (โˆ’๐‘ฅ) is a reflection of the function ๐‘ฆ = ๐‘ฅ^3 + ๐‘ฅ 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‘“(๐‘ฅ)| = ๐‘“(๐‘ฅ) ๐‘คโ„Ž๐‘’๐‘› ๐‘“(๐‘ฅ) โ‰ฅ 0 |๐‘“(๐‘ฅ)| = โˆ’๐‘“(๐‘ฅ) ๐‘คโ„Ž๐‘’๐‘› ๐‘“(๐‘ฅ) < 0

Eg: |5๐‘ฅ โˆ’ 10| = 5๐‘ฅ โˆ’ 10 ๐‘คโ„Ž๐‘’๐‘› 5๐‘ฅ โˆ’ 10 โ‰ฅ 0 ๐‘–๐‘’ ๐‘ฅ โ‰ฅ 2 |5๐‘ฅ โˆ’ 10| = โˆ’(5๐‘ฅ โˆ’ 10) = 10 โˆ’ 5๐‘ฅ ๐‘คโ„Ž๐‘’๐‘› 5๐‘ฅ โˆ’ 10 < 0 ๐‘–๐‘’ ๐‘ฅ < 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

sinโˆ’1^ ๐‘ฅ has domain โˆ’1 โ‰ค ๐‘ฅ โ‰ค 1 and range โˆ’ ๐œ‹ 2 โ‰ค sinโˆ’1^ ๐‘ฅ โ‰ค ๐œ‹ 2 cosโˆ’1^ ๐‘ฅ has domain โˆ’1 โ‰ค ๐‘ฅ โ‰ค 1 and range 0 โ‰ค cosโˆ’1^ ๐‘ฅ โ‰ค ๐œ‹ tanโˆ’1^ ๐‘ฅ has domain ๐‘ฅ โˆŠ โ„ and range โˆ’ ๐œ‹ 2 < tanโˆ’1^ ๐‘ฅ < ฯ€ 2

Secant, cosecant and cotangent are defined as: sec ๐‘ฅ = (^) cos ๐‘ฅ^1 , csc ๐‘ฅ = (^) sin ๐‘ฅ^1 , cot ๐‘ฅ = (^) tan ๐‘ฅ^1.

Note: (^) sin ๐‘ฅ^1 โ‰  sinโˆ’1^ ๐‘ฅ. One is the reciprocal of the function, the other is the inverse function.

sin^2 ๐‘ฅ + cos^2 ๐‘ฅ โ‰ก 1 1 + cot^2 ๐‘ฅ โ‰ก csc^2 ๐‘ฅ tan^2 ๐‘ฅ + 1 โ‰ก sec^2 ๐‘ฅ

Note: The second two identities can both be produced easily from the first one by dividing through by either sin^2 ๐‘ฅ or cos^2 ๐‘ฅ.

๐‘ฆ = sin ๐‘ฅ ๐‘ฆ = cos ๐‘ฅ ๐‘ฆ = tan ๐‘ฅ

๐‘ฆ = csc ๐‘ฅ ๐‘ฆ = sec ๐‘ฅ ๐‘ฆ = cot ๐‘ฅ

(All horizontal scales between 0 ๐‘^ and 2๐œ‹๐‘)

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

๐‘‘๐‘ฆ ๐‘‘๐‘ฅ =

This is the chain rule proper. However, it is often easier to use this expression of it:

๐‘‘ ๐‘‘๐‘ฅ ๐‘“๐‘”(๐‘ฅ) = ๐‘“

Eg: (^) ๐‘‘๐‘ฅ๐‘‘ (2๐‘ฅ + 5)^3 = 3(2๐‘ฅ + 5)^2 (2) = 6(2๐‘ฅ + 5)^2

The 2๐‘ฅ + 5 part is โ€˜ignoredโ€™, then the end result is multiplied by the differential of the โ€˜ignoredโ€™ 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‘‘ ๐‘‘๐‘ฅ sin ๐‘ฅ = cos ๐‘ฅ ๐‘‘ ๐‘‘๐‘ฅ cos ๐‘ฅ = โˆ’ sin ๐‘ฅ

Note: A number of more complex trigonometric (and other) results are given in the formula book.

An increasing function is a function ๐‘“(๐‘ฅ) such that ๐‘“โ€ฒ(๐‘ฅ) โ‰ฅ 0 for all ๐‘ฅ. A decreasing function is a function ๐‘“(๐‘ฅ) such that ๐‘“โ€ฒ(๐‘ฅ) โ‰ค 0 for all ๐‘ฅ.

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‘“(๐‘ฅ) = ๐‘ฅ^3 + 2๐‘ฅ is an increasing function because ๐‘“โ€ฒ(๐‘ฅ) = 3๐‘ฅ^2 + 2 > 0 for all ๐‘ฅ (since ๐‘ฅ^2 โ‰ฅ 0).

Chapter 6 โ€“ Differentiation using the product and quotient rule

Product Rule: ๐‘‘ ๐‘‘๐‘ฅ ๐‘“(๐‘ฅ)๐‘”(๐‘ฅ) = ๐‘“(๐‘ฅ)๐‘”

This is the function notation form. A more easily memorable one may be written as:

๐‘‘ ๐‘‘๐‘ฅ ๐‘ข๐‘ฃ = ๐‘ข๐‘ฃ

Eg: (^) ๐‘‘๐‘ฅ๐‘‘ (๐‘ฅ^2 sin ๐‘ฅ) = ๐‘ฅ^2 cos ๐‘ฅ + 2๐‘ฅ sin ๐‘ฅ

Quotient Rule: ๐‘‘ ๐‘‘๐‘ฅ (

๐‘ฃ๐‘ขโ€ฒ^ โˆ’ ๐‘ข๐‘ฃโ€ฒ

๐‘ฃ^2

This is given (in function notation) in the formula book, but this form may be more memorable.

Eg: (^) ๐‘‘๐‘ฅ๐‘‘ ( (^) ๐‘ฅ 2 ๐‘ฅ+1) = (๐‘ฅ

(^2) +1)โˆ’๐‘ฅ(2๐‘ฅ) (๐‘ฅ^2 +1)^2 =^

1โˆ’๐‘ฅ^2 (๐‘ฅ^2 +1)^2

Chapter 7 โ€“ Numerical solutions of equations and iterative methods

To show that๐‘“(๐‘ฅ) = 0 has a root ๐›ผ such that ๐‘Ž < ๐›ผ < ๐‘, it is sufficient to demonstrate that either ๐‘“(๐‘Ž) < 0 ๐‘Ž๐‘›๐‘‘ ๐‘“(๐‘) > 0 or ๐‘“(๐‘Ž) > 0 ๐‘Ž๐‘›๐‘‘ ๐‘“(๐‘) < 0.

Note: The converse is not always true. ๐‘“(๐‘Ž) > 0 ๐‘Ž๐‘›๐‘‘ ๐‘“(๐‘) > 0 does not imply that there isnโ€™t a root between ๐‘Ž and ๐‘.

If you are required to show a function has a root between two values, first rearrange into the form ๐‘“(๐‘ฅ) = 0, then apply the rule above.

Iteration requires a formula linking each approximation of ๐‘ฅ to the next: ๐‘ฅ๐‘›+1 = ๐‘“(๐‘ฅ๐‘›).

An iteration ๐‘ฅ๐‘›+1 = ๐‘“(๐‘ฅ๐‘›) converges if |๐‘“โ€ฒ(๐‘ฅ)| < 1 for certain values of ๐‘ฅ.

If the iteration converges to a limit, that limit can be found by setting ๐‘ฅ๐‘›+1 = ๐‘ฅ๐‘› = ๐ฟ.

Eg: The equation ๐‘ฅ^3 โˆ’ 4๐‘ฅ = 7 produces the iteration formula ๐‘ฅ๐‘›+1 = โˆš7 + 4๐‘ฅ^3 ๐‘›. Starting with 5: ๐‘ฅ 1 = 5 ๐‘ฅ 2 = โˆš7 + 4๐‘ฅ^3 1 = โˆš7 + 4(5)^3 = 3 ๐‘ฅ 3 = โˆš7 + 4๐‘ฅ^3 2 = โˆš7 + 4(3)^3 = 2.6684 โ€ฆ

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:

โˆซ ๐‘ฅ๐‘›^ sin ๐‘š๐‘ฅ ๐‘‘๐‘ฅ

โˆซ ๐‘ฅ๐‘›^ cos ๐‘š๐‘ฅ ๐‘‘๐‘ฅ

โˆซ ๐‘ฅ๐‘›๐‘’๐‘š๐‘ฅ^ ๐‘‘๐‘ฅ

โˆซ ๐‘ฅ๐‘›^ ln ๐‘š๐‘ฅ ๐‘‘๐‘ฅ

Note: While for most functions the simplest part (usually ๐‘ฅ๐‘›) becomes ๐‘ข and the other part becomes (^) ๐‘‘๐‘ฅ๐‘‘๐‘ฃ, when ln ๐‘š๐‘ฅ is involved, this should be made ๐‘ข, since it is not readily integrable.

Eg:

โˆซ ๐‘ฅ^2 ln ๐‘ฅ ๐‘‘๐‘ฅ

๐‘ข = ln ๐‘ฅ

2 ๐‘‘๐‘ข ๐‘‘๐‘ฅ =

๐‘ฅ^3

โˆซ ๐‘ฅ^2 ln ๐‘ฅ ๐‘‘๐‘ฅ =

๐‘ฅ^3

3 ln ๐‘ฅ โˆ’ โˆซ

๐‘ฅ^3

๐‘ฅ^3

3 ln ๐‘ฅ โˆ’ โˆซ

๐‘ฅ^2

๐‘ฅ^3

3 ln ๐‘ฅ โˆ’

๐‘ฅ^3

๐‘ฅ^3

(3 ln ๐‘ฅ โˆ’ 1) + ๐ถ

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 ๐‘ฆ = ๐‘“(๐‘ฅ) rotated fully around the x-axis is given by:

๐‘‰ = โˆซ ๐œ‹๐‘ฆ^2 ๐‘‘๐‘ฅ

๐‘ ๐‘Ž

Note: To find a volume rotated around the y-axis, rearrange to give ๐‘ฅ = ๐‘”(๐‘ฆ) and use the rule:

๐‘‰ = โˆซ ๐œ‹๐‘ฅ^2 ๐‘‘๐‘ฆ

๐‘ ๐‘Ž

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.

Simpsonโ€™s Rule approximates the curve as a series of quadratics, and since this is usually a much closer fit to the actual function, Simpsonโ€™s 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 โˆซ โˆš1 + ๐‘’ 02 ๐‘ฅ^ ๐‘‘๐‘ฅusing Simpsonโ€™s Rule with five ordinates (four strips):

๐‘ฅ 0 = 0 ๐‘ฆ 0 = โˆš1 + ๐‘’^0 = 1.4142 โ€ฆ

๐‘ฅ 1 = 0.5 ๐‘ฆ 1 = โˆš1 + ๐‘’0.5^ = 1.6274 โ€ฆ

๐‘ฅ 2 = 1 ๐‘ฆ 2 = โˆš1 + ๐‘’^1 = 1.9282 โ€ฆ

๐‘ฅ 3 = 1.5 ๐‘ฆ 3 = โˆš1 + ๐‘’1.5^ = 2.3413 โ€ฆ

๐‘ฅ 4 = 2 ๐‘ฆ 4 = โˆš1 + ๐‘’^2 = 2.8963 โ€ฆ

โˆซ โˆš1 + ๐‘’๐‘ฅ^ ๐‘‘๐‘ฅ

2 0

Note: Simpsonโ€™s Rule with ๐‘› ordinates usually gives a result accurate to ๐‘› decimal places.

Produced by A. Clohesy; TheChalkface.net 04/06/