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

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The Not-Formula Book for C

Everything you need to know for Core 3 that won’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 – Functions

Chapter 2 – Transformations of graphs and the modulus function

Chapter 3 – Inverse trigonometric functions and secant, cosecant and cotangent

Chapter 4 – The number e and calculus

Chapter 5 – Further differentiation and the chain rule

Chapter 6 – Differentiation using the product rule and the quotient rule

Chapter 7 – Numerical solutions of equations and iterative methods

Chapter 8 – Integration by inspection and substitution

Chapter 9 – Integration by parts and standard integrals

Chapter 10 – Volume of revolution and numerical integration

Chapter 2 – Transformations of graphs and the modulus function

The transformation 𝑦 = 𝑓(𝑥 − 𝑎) + 𝑏 is a translation of [𝑎𝑏].

Eg: The function 𝑦 = (𝑥 + 4)^3 + 7 is a translation of 4 to the left and 7 up from 𝑦 = 𝑥^3.

The transformation 𝑦 = 𝑘𝑓(𝑥) is a stretch by a factor of k in the y direction. The transformation 𝑦 = 𝑓 (𝑥𝑐) is a stretch by a factor of c in the x direction.

Eg: The function y = 3sin 4𝑥 is a stretch of factor 3 in the y direction and a stretch of factor ¼ in the x direction from 𝑦 = sin 𝑥.

A transformation of 𝑦 = −𝑓(𝑥) is a reflection in the x-axis. A transformation of 𝑦 = 𝑓(−𝑥)^ is a reflection in the y-axis.

Eg: The function 𝑦 = −𝑥^2 is a reflection of the function 𝑦 = 𝑥^2 in the x-axis. The function 𝑦 = (−𝑥)^3 + (−𝑥) is a reflection of the function 𝑦 = 𝑥^3 + 𝑥 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: |𝑓(𝑥)| = 𝑓(𝑥) 𝑤ℎ𝑒𝑛 𝑓(𝑥) ≥ 0 |𝑓(𝑥)| = −𝑓(𝑥) 𝑤ℎ𝑒𝑛 𝑓(𝑥) < 0

Eg: |5𝑥 − 10| = 5𝑥 − 10 𝑤ℎ𝑒𝑛 5𝑥 − 10 ≥ 0 𝑖𝑒 𝑥 ≥ 2 |5𝑥 − 10| = −(5𝑥 − 10) = 10 − 5𝑥 𝑤ℎ𝑒𝑛 5𝑥 − 10 < 0 𝑖𝑒 𝑥 < 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 – Inverse trigonometric functions and sec, cosec and cot

sin−1^ 𝑥 has domain −1 ≤ 𝑥 ≤ 1 and range − 𝜋 2 ≤ sin−1^ 𝑥 ≤ 𝜋 2 cos−1^ 𝑥 has domain −1 ≤ 𝑥 ≤ 1 and range 0 ≤ cos−1^ 𝑥 ≤ 𝜋 tan−1^ 𝑥 has domain 𝑥 ∊ ℝ and range − 𝜋 2 < tan−1^ 𝑥 < π 2

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

Note: (^) sin 𝑥^1 ≠ sin−1^ 𝑥. One is the reciprocal of the function, the other is the inverse function.

sin^2 𝑥 + cos^2 𝑥 ≡ 1 1 + cot^2 𝑥 ≡ csc^2 𝑥 tan^2 𝑥 + 1 ≡ sec^2 𝑥

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

𝑦 = sin 𝑥 𝑦 = cos 𝑥 𝑦 = tan 𝑥

𝑦 = csc 𝑥 𝑦 = sec 𝑥 𝑦 = cot 𝑥

(All horizontal scales between 0 𝑐^ and 2𝜋𝑐)

Note: You should know how to accurately sketch the primary trigonometric functions – the other three you ought to be able to produce from those.

Chapter 5 – Further differentiation and the chain rule

𝑑𝑦 𝑑𝑥 =

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

𝑑 𝑑𝑥 𝑓𝑔(𝑥) = 𝑓

Eg: (^) 𝑑𝑥𝑑 (2𝑥 + 5)^3 = 3(2𝑥 + 5)^2 (2) = 6(2𝑥 + 5)^2

The 2𝑥 + 5 part is ‘ignored’, then the end result is multiplied by the differential of the ‘ignored’ 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.

𝑑 𝑑𝑥 sin 𝑥 = cos 𝑥 𝑑 𝑑𝑥 cos 𝑥 = − sin 𝑥

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

An increasing function is a function 𝑓(𝑥) such that 𝑓′(𝑥) ≥ 0 for all 𝑥. A decreasing function is a function 𝑓(𝑥) such that 𝑓′(𝑥) ≤ 0 for all 𝑥.

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: 𝑓(𝑥) = 𝑥^3 + 2𝑥 is an increasing function because 𝑓′(𝑥) = 3𝑥^2 + 2 > 0 for all 𝑥 (since 𝑥^2 ≥ 0).

Chapter 6 – Differentiation using the product and quotient rule

Product Rule: 𝑑 𝑑𝑥 𝑓(𝑥)𝑔(𝑥) = 𝑓(𝑥)𝑔

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

𝑑 𝑑𝑥 𝑢𝑣 = 𝑢𝑣

Eg: (^) 𝑑𝑥𝑑 (𝑥^2 sin 𝑥) = 𝑥^2 cos 𝑥 + 2𝑥 sin 𝑥

Quotient Rule: 𝑑 𝑑𝑥 (

𝑣𝑢′^ − 𝑢𝑣′

𝑣^2

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

Eg: (^) 𝑑𝑥𝑑 ( (^) 𝑥 2 𝑥+1) = (𝑥

(^2) +1)−𝑥(2𝑥) (𝑥^2 +1)^2 =^

1−𝑥^2 (𝑥^2 +1)^2

Chapter 7 – Numerical solutions of equations and iterative methods

To show that𝑓(𝑥) = 0 has a root 𝛼 such that 𝑎 < 𝛼 < 𝑏, it is sufficient to demonstrate that either 𝑓(𝑎) < 0 𝑎𝑛𝑑 𝑓(𝑏) > 0 or 𝑓(𝑎) > 0 𝑎𝑛𝑑 𝑓(𝑏) < 0.

Note: The converse is not always true. 𝑓(𝑎) > 0 𝑎𝑛𝑑 𝑓(𝑏) > 0 does not imply that there isn’t a root between 𝑎 and 𝑏.

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

Iteration requires a formula linking each approximation of 𝑥 to the next: 𝑥𝑛+1 = 𝑓(𝑥𝑛).

An iteration 𝑥𝑛+1 = 𝑓(𝑥𝑛) converges if |𝑓′(𝑥)| < 1 for certain values of 𝑥.

If the iteration converges to a limit, that limit can be found by setting 𝑥𝑛+1 = 𝑥𝑛 = 𝐿.

Eg: The equation 𝑥^3 − 4𝑥 = 7 produces the iteration formula 𝑥𝑛+1 = √7 + 4𝑥^3 𝑛. Starting with 5: 𝑥 1 = 5 𝑥 2 = √7 + 4𝑥^3 1 = √7 + 4(5)^3 = 3 𝑥 3 = √7 + 4𝑥^3 2 = √7 + 4(3)^3 = 2.6684 …

Chapter 9 – Integration by parts and standard integrals

Note: This result is given in the formula book – it is given here for reference only.

You will be most commonly expected to apply Integration by Parts to the following forms of integral:

∫ 𝑥𝑛^ sin 𝑚𝑥 𝑑𝑥

∫ 𝑥𝑛^ cos 𝑚𝑥 𝑑𝑥

∫ 𝑥𝑛𝑒𝑚𝑥^ 𝑑𝑥

∫ 𝑥𝑛^ ln 𝑚𝑥 𝑑𝑥

Note: While for most functions the simplest part (usually 𝑥𝑛) becomes 𝑢 and the other part becomes (^) 𝑑𝑥𝑑𝑣, when ln 𝑚𝑥 is involved, this should be made 𝑢, since it is not readily integrable.

Eg:

∫ 𝑥^2 ln 𝑥 𝑑𝑥

𝑢 = ln 𝑥

2 𝑑𝑢 𝑑𝑥 =

𝑥^3

∫ 𝑥^2 ln 𝑥 𝑑𝑥 =

𝑥^3

3 ln 𝑥 − ∫

𝑥^3

𝑥^3

3 ln 𝑥 − ∫

𝑥^2

𝑥^3

3 ln 𝑥 −

𝑥^3

𝑥^3

(3 ln 𝑥 − 1) + 𝐶

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 – Volume of revolution and numerical integration

The volume of a function 𝑦 = 𝑓(𝑥) rotated fully around the x-axis is given by:

𝑉 = ∫ 𝜋𝑦^2 𝑑𝑥

𝑏 𝑎

Note: To find a volume rotated around the y-axis, rearrange to give 𝑥 = 𝑔(𝑦) and use the rule:

𝑉 = ∫ 𝜋𝑥^2 𝑑𝑦

𝑏 𝑎

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.

Simpson’s Rule approximates the curve as a series of quadratics, and since this is usually a much closer fit to the actual function, Simpson’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 ∫ √1 + 𝑒 02 𝑥^ 𝑑𝑥using Simpson’s Rule with five ordinates (four strips):

𝑥 0 = 0 𝑦 0 = √1 + 𝑒^0 = 1.4142 …

𝑥 1 = 0.5 𝑦 1 = √1 + 𝑒0.5^ = 1.6274 …

𝑥 2 = 1 𝑦 2 = √1 + 𝑒^1 = 1.9282 …

𝑥 3 = 1.5 𝑦 3 = √1 + 𝑒1.5^ = 2.3413 …

𝑥 4 = 2 𝑦 4 = √1 + 𝑒^2 = 2.8963 …

∫ √1 + 𝑒𝑥^ 𝑑𝑥

2 0

Note: Simpson’s Rule with 𝑛 ordinates usually gives a result accurate to 𝑛 decimal places.

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