THE MATHEMATICS CONTINUUM · getcontinuum.co.za

First Year — Formula Reference Sheet

First Year University · 13 formulas

First Year University

First Year

Limits, differentiation rules, integration techniques, linear algebra, probability distributions.

13 formulas · opens your print dialog

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Limits & Continuity

Formal Limit Definition

lim [x \to a] f (x) = L means : for every ε > 0 there exists δ > 0 such that ∣ f (x) - L ∣ < ε whenever 0 < ∣ x - a ∣ < δ

The rigorous epsilon-delta definition of a limit.

When to use

Formal proofs of limits in analysis courses.

Example

Prove lim[x→2] (2x+1) = 5: given ε, choose δ=ε/2

Limits & Continuity

L'Hôpital's Rule

lim [x \to a] \frac{f ( x )}{g ( x )} = lim [x \to a] \frac{f ^{'} ( x )}{g ^{'} ( x )} [\frac{0}{0} or \frac{\infty}{\infty} f or m]

Evaluates indeterminate limits by differentiating numerator and denominator.

When to use

When substitution gives 0/0 or ∞/∞.

Example

lim[x→0] sin(x)/x = lim[x→0] cos(x)/1 = 1

Differentiation

Trig Derivatives

\frac{d}{d x} (sin x) = cos x ∣ \frac{d}{d x} (cos x) = - sin x ∣ \frac{d}{d x} (tan x) = sec^{2} x

Standard derivatives of the three main trigonometric functions.

When to use

Differentiating any expression containing sin, cos, or tan.

Example

f(x) = 3sin x − 2cos x → f'(x) = 3cos x + 2sin x

Differentiation

Exponential & Logarithm Derivatives

\frac{d}{d x} (e^{x}) = e^{x} ∣ \frac{d}{d x} (ln x) = \frac{1}{x} ∣ \frac{d}{d x} (a^{x}) = a^{x} ln a

Derivatives of exponential and natural log functions.

When to use

Differentiating growth/decay models, compound interest models.

Example

f(x) = e^(2x) → f'(x) = 2e^(2x) (chain rule applied)

Differentiation

Implicit Differentiation

D i f f er e n t ia t e b o t h s i d es with r es p ec tt o x; t r e a t y a s a f u n c t i o n o f x (a ppl y c hain r u l e t oy t er m s)

Finds dy/dx when y cannot be expressed explicitly as a function of x.

When to use

Equations like x² + y² = r² (circle), x³ + y³ = 6xy.

Example

x² + y² = 25 → 2x + 2y(dy/dx) = 0 → dy/dx = −x/y

Integration

Trig Integrals

\int sin x d x = - cos x + C ∣ \int cos x d x = sin x + C ∣ \int sec^{2} x d x = tan x + C

Standard antiderivatives of trig functions.

When to use

Integrating any expression containing trigonometric functions.

Example

∫(sin x + 2cos x) dx = −cos x + 2sin x + C

Integration

Integration by Parts

\int u d v = uv - \int v d u

Integrates the product of two functions.

When to use

Integrating products like x·eˣ, x·sin x, x·ln x.

Example

∫x·eˣ dx: let u=x, dv=eˣ dx → xeˣ − ∫eˣ dx = xeˣ − eˣ + C

Integration

Integration by Substitution

\int f (g (x)) g^{'} (x) d x = \int f (u) d u where u = g (x)

Simplifies integrals by substituting a composite expression.

When to use

When you can identify an inner function whose derivative is also present.

Example

∫2x(x²+1)⁵ dx: let u=x²+1, du=2x dx → ∫u⁵ du = u⁶/6 + C

Linear Algebra

Determinant (2×2)

det ([ab; c d]) = a d - b c

Scalar value encoding whether a 2×2 matrix is invertible.

When to use

Solving systems, checking invertibility, finding areas of parallelograms.

Example

det([3 2; 1 4]) = 12 − 2 = 10

Linear Algebra

Inverse (2×2 Matrix)

A^{- 1} = (\frac{1}{det A}) \times [d - b; - c a]

Finds the inverse of a 2×2 matrix.

When to use

Solving matrix equations Ax = b, or reversing a linear transformation.

Example

A=[2 1; 5 3] → det=1 → A⁻¹=[3 −1; −5 2]

Linear Algebra

Eigenvalue Equation

A v = λ v \to det (A - λ I) = 0

Finds the scalars λ (eigenvalues) for which A has a special direction v.

When to use

Principal component analysis, differential equations, quantum mechanics.

Example

A=[3 1; 0 2]: det([3−λ 1; 0 2−λ]) = (3−λ)(2−λ) = 0 → λ=3 or λ=2

Probability

Bayes' Theorem

P (A ∣ B) = P (B ∣ A) \cdot \frac{P ( A )}{P ( B )}

Updates a probability estimate after receiving new evidence.

When to use

Medical testing, spam filters, any situation with conditional information.

Example

P(disease|positive test) given test accuracy and disease prevalence

Probability

Standard Normal (Z-score)

Z = \frac{X - μ}{σ}

Converts a raw score to standard deviations from the mean.

When to use

Any normal distribution problem — finding probabilities using Z-tables.

Example

X=75, μ=60, σ=10: Z = (75−60)/10 = 1.5

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