Edexcel IAL Pure Mathematics 2: Revision Notes (Calculus and Related Topics)

Proof and Mathematical Reasoning

Direct Proof and Proof by Counter-Example

Mathematical proofs are essential for establishing the truth of statements in calculus and algebra. Direct proof involves a logical sequence of steps from known facts, while proof by counter-example disproves a statement by providing a single example where it fails.

• Direct Proof: Start with a known statement and use logical steps to reach the conclusion.

• Counter-Example: Disprove a general statement by showing a specific case where it does not hold.

• Example: Prove that the square of any integer is one more than the product of the two integers on either side of it.

  • Let $n$ be any integer. $(n-1)$ and $(n+1)$ are the two integers on either side.

  • Their product: $(n-1)(n+1) = n^2 - 1$

  • So, $n^2 = (n-1)(n+1) + 1$

Useful Results: Any even number can be written as $2n$, any odd number as $2n+1$, where $n$ is an integer.

Algebra

Polynomials and Factorisation

Polynomials are algebraic expressions involving powers of $x$. Factorisation is the process of expressing a polynomial as a product of its factors.

• General Form: $a_nx^n + a_{n-1}x^{n-1} + ext{...} + a_1x + a_0$

• Factorising: Use methods such as grouping, difference of squares, and the factor theorem.

• Standard Results:

  • $(x^2 - y^2) = (x-y)(x+y)$

  • $(x+y)^2 = x^2 + 2xy + y^2$

  • $(x-y)^2 = x^2 - 2xy + y^2$

• Long Division: Used to divide polynomials when factorising is not straightforward.

• Example: Factorise $2x^3 + x^2 - 13x + 6$.

  • Possible linear factors: $(x+1)$, $(x+2)$, $(x+3)$, $(x+6)$

  • Test each by substitution to find a root.

  • Divide by the found factor to get the quadratic factor.

Remainder and Factor Theorems

The remainder theorem states that the remainder of $P(x)$ divided by $(x-a)$ is $P(a)$. The factor theorem states that $(x-a)$ is a factor of $P(x)$ if $P(a)=0$.

• Remainder Theorem: $P(x) = (x-a)Q(x) + r$, where $r = P(a)$

• Factor Theorem: If $P(a) = 0$, then $(x-a)$ is a factor of $P(x)$

• Example: Find the remainder when $P(x) = 2x^2 + ax + b$ is divided by $(x-2)$ and $(x+2)$, given $P(2) = 6$ and $P(-2) = 1$.

  • Set up equations: $2(2)^2 + a(2) + b = 6$ and $2(-2)^2 + a(-2) + b = 1$

  • Solve for $a$ and $b$.

Trigonometry

Solving Trigonometric Equations

Trigonometric equations involve functions such as sine, cosine, and tangent. Solutions often require using graphs, identities, or algebraic manipulation.

• Example: Solve $\sin x = 0.453$ for $0 \leq x \leq 360^\circ$.

  • $x = \arcsin(0.453) = 27^\circ$

  • Other solution: $x = 180^\circ - 27^\circ = 153^\circ$

• Using Graphs: Graphical solutions help visualize all possible solutions within a given interval.

• Using Identities: Trigonometric identities simplify equations and help solve for unknowns.

  • Example: $\tan A = \frac{\sin A}{\cos A}$

  • To solve $3 \sin x = 4 \cos x$, divide both sides by $\cos x$ to get $3 \tan x = 4$.

  • $\tan x = \frac{4}{3}$, so $x = \arctan(\frac{4}{3})$

Sequences and Series

Defining Sequences

A sequence is an ordered list of numbers following a specific rule. Series are the sum of terms in a sequence.

• Arithmetic Sequence: Each term differs from the previous by a constant difference $d$.

• Geometric Sequence: Each term is multiplied by a constant ratio $r$.

• General Term:

  • Arithmetic: $a_n = a_1 + (n-1)d$

  • Geometric: $a_n = a_1 r^{n-1}$

• Sum of Arithmetic Series:

  • $S_n = \frac{n}{2}(a_1 + a_n)$

• Sum of Geometric Series:

  • $S_n = a_1 \frac{1 - r^n}{1 - r}$, for $r \neq 1$

Differentiation

Stationary Points and Local Maxima/Minima

Differentiation is used to find the rate of change of a function. Stationary points occur where the derivative is zero, indicating possible maxima, minima, or points of inflection.

• Derivative: $f'(x)$ gives the gradient of the curve at $x$.

• Stationary Point: $f'(x) = 0$

• Second Derivative Test:

  • If $f''(x) > 0$, the point is a minimum.

  • If $f''(x) < 0$, the point is a maximum.

• Point of Inflection: Where the curve changes concavity, $f''(x) = 0$ and $f''(x)$ changes sign.

Integration

Definite Integrals and Area Under Curves

Integration is the reverse process of differentiation and is used to find areas under curves and accumulate quantities.

• Definite Integral: $\int_a^b f(x) dx$ gives the area under $f(x)$ from $x=a$ to $x=b$.

• Trapezium Rule: A numerical method for estimating the area under a curve.

  • $\int_a^b f(x) dx \approx \frac{h}{2} [y_0 + 2y_1 + 2y_2 + ... + 2y_{n-1} + y_n]$

  • Where $h$ is the width of each subinterval, and $y_i$ are the function values at the endpoints.

Appendix: Binomial Coefficients and Useful Results

Binomial Coefficients

Binomial coefficients are used in the expansion of $(a+b)^n$ and are denoted as $\binom{n}{r}$.

• Formula: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$

• Application: Used in binomial expansions and probability.

Summary Table: Key Formulas

Topic	Formula	Description
Arithmetic Sequence	$a_n = a_1 + (n-1)d$	General term
Geometric Sequence	$a_n = a_1 r^{n-1}$	General term
Sum of Arithmetic Series	$S_n = \frac{n}{2}(a_1 + a_n)$	Sum of first $n$ terms
Sum of Geometric Series	$S_n = a_1 \frac{1 - r^n}{1 - r}$	Sum of first $n$ terms
Derivative	$f'(x)$	Gradient of curve
Definite Integral	$\int_a^b f(x) dx$	Area under curve
Trapezium Rule	$\int_a^b f(x) dx \approx \frac{h}{2} [y_0 + 2y_1 + ... + y_n]$	Numerical integration
Binomial Coefficient	$\binom{n}{r} = \frac{n!}{r!(n-r)!}$	Number of ways to choose $r$ from $n$

Additional info: These notes cover core topics in college-level calculus and algebra, including proof techniques, polynomial factorisation, trigonometric equations, sequences and series, differentiation, and integration. The content is structured to support exam preparation for Edexcel IAL Pure Mathematics 2.