Proof and Mathematical Reasoning

Direct Proof and Proof by Counter-Example

Mathematical proofs are essential for establishing the validity of statements. Direct proof involves a logical sequence of steps from known facts, while proof by counter-example disproves a statement by providing a specific example.

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

• Proof by Counter-Example: Disprove a general statement by showing a specific case where it fails.

• 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. The two integers on either side are $(n-1)$ and $(n+1)$. Their product is $(n-1)(n+1) = n^2 - 1$. Thus, $n^2 = (n-1)(n+1) + 1$.

• Example: Prove that the sum of the angles in a triangle is $180^{\circ}$ by drawing a line parallel to one side and using alternate angles.

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

Proof by Exhaustion

Proof by exhaustion involves checking all possible cases to confirm a statement.

• Example: Prove by exhaustion that $n^2 + 1$ is not divisible by $3$ for $n = 6, 7, 8, 9, 10$.

Algebra

Polynomials and Factorisation

A polynomial is an expression of the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$, where all powers of $x$ are positive integers or zero.

• Factorising: Breaking down polynomials into products of simpler polynomials.

• Standard Results:

  • Difference of squares: $x^2 - y^2 = (x-y)(x+y)$

  • Sum of squares: $(x+y)^2 = x^2 + 2xy + y^2$

  • Difference of cubes: $(x-y)^3 = x^3 - 2xy + y^2$

Long Division of Polynomials

Long division is used to divide polynomials, especially when the divisor is not a factor of the dividend.

• Example: Divide $2x^3 + 3x - 1$ by $x^2 - 5x + 9$.

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 $P(a) = 0$.

• Example: If $P(x) = 2x^3 + ax^2 + bx + 9$ is divided by $(x-2)$ and $(x-3)$, and the remainders are $-6$ and $1$ respectively, find $a$ and $b$.

Trigonometry

Solving Trigonometric Equations

Trigonometric equations can be solved using graphs, identities, and algebraic manipulation.

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

• Graphical Solution: Use the sine or cosine graph to find all solutions within the given interval.

Using Trigonometric Identities

Identities such as $\tan A = \frac{\sin A}{\cos A}$ and $\sin^2 x + \cos^2 x = 1$ are used to simplify and solve equations.

• Example: Solve $3 \sin x = 4 \cos x$.

• Divide both sides by $\cos x$ to get $3 \tan x = 4$, so $\tan x = \frac{4}{3}$.

Sequences and Series

Defining Sequences

A sequence is an ordered list of numbers following a specific rule. Common types include arithmetic and geometric sequences.

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

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

Series and Summation

The sum of the terms of a sequence is called a series. Important formulas include:

• 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

Increasing and Decreasing Functions

Differentiation is used to determine where a function is increasing or decreasing, and to find stationary points (where the derivative is zero).

• Stationary Points: Points where $\frac{dy}{dx} = 0$.

• Maximum and Minimum: Use the second derivative test to classify stationary points.

Integration

Definite Integrals and Area Under Curves

Integration is used to find the area under a curve between two points.

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

Numerical Integration: The Trapezium Rule

The trapezium rule approximates the area under a curve by dividing it into trapezoids.

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

• Application: Useful when the function cannot be integrated analytically.

Appendix: Binomial Coefficients and Points of Inflection

Binomial Coefficients

The binomial coefficient $\binom{n}{r}$ counts the number of ways to choose $r$ objects from $n$.

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

Points of Inflection

A point of inflection is where the curve changes concavity, i.e., the second derivative changes sign.

• Test: If $\frac{d^2y}{dx^2}$ changes sign at $x = a$, then $(a, f(a))$ is a point of inflection.

Additional info: These notes cover key topics in a college-level Calculus course, including proof techniques, algebraic manipulation, trigonometry, sequences and series, differentiation, and integration, as outlined in the Edexcel IAL Pure 2 syllabus.