UTS ENGINEERING
33130
Mathematics 1
Calculus, Algebra & Differential Equations
Bachelor of Engineering (Honours) | Year 1
University of Technology Sydney
, Topic 1: Functions & Limits
1.1 Types of Functions
A function f maps every input x in the domain to exactly one output f(x) in the range.
Domain: Set of all valid input values x
Range: Set of all output values f(x)
Composite function: f(g(x)) — apply g first, then f
Inverse function: f⁻¹(x) — swaps domain and range, only exists if f is one-to-one
1.2 Limits
The limit of f(x) as x approaches a is L if f(x) gets arbitrarily close to L.
lim(x→a) f(x) = L
• Left-hand limit: lim(x→a⁻) f(x)
• Right-hand limit: lim(x→a⁺) f(x)
• Limit exists only if left = right limit
Limit Laws
• Sum: lim[f(x) + g(x)] = lim f(x) + lim g(x)
• Product: lim[f(x)·g(x)] = lim f(x) · lim g(x)
• L'Hôpital's Rule: if 0/0 or ∞/∞, then lim f/g = lim f'/g'
⚡ Exam Tip: L'Hôpital's rule is very common in exams — always check you have 0/0 or ∞/∞ form first.
Topic 2: Differentiation
2.1 Definition & Basic Rules
The derivative f'(x) represents the instantaneous rate of change of f at x.
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Key Differentiation Rules
• Power rule: d/dx [xⁿ] = nxⁿ⁻¹
• Product rule: (uv)' = u'v + uv'
• Quotient rule: (u/v)' = (u'v - uv') / v²
• Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
Common Derivatives
• d/dx[sin x] = cos x
• d/dx[cos x] = -sin x
• d/dx[eˣ] = eˣ
33130
Mathematics 1
Calculus, Algebra & Differential Equations
Bachelor of Engineering (Honours) | Year 1
University of Technology Sydney
, Topic 1: Functions & Limits
1.1 Types of Functions
A function f maps every input x in the domain to exactly one output f(x) in the range.
Domain: Set of all valid input values x
Range: Set of all output values f(x)
Composite function: f(g(x)) — apply g first, then f
Inverse function: f⁻¹(x) — swaps domain and range, only exists if f is one-to-one
1.2 Limits
The limit of f(x) as x approaches a is L if f(x) gets arbitrarily close to L.
lim(x→a) f(x) = L
• Left-hand limit: lim(x→a⁻) f(x)
• Right-hand limit: lim(x→a⁺) f(x)
• Limit exists only if left = right limit
Limit Laws
• Sum: lim[f(x) + g(x)] = lim f(x) + lim g(x)
• Product: lim[f(x)·g(x)] = lim f(x) · lim g(x)
• L'Hôpital's Rule: if 0/0 or ∞/∞, then lim f/g = lim f'/g'
⚡ Exam Tip: L'Hôpital's rule is very common in exams — always check you have 0/0 or ∞/∞ form first.
Topic 2: Differentiation
2.1 Definition & Basic Rules
The derivative f'(x) represents the instantaneous rate of change of f at x.
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Key Differentiation Rules
• Power rule: d/dx [xⁿ] = nxⁿ⁻¹
• Product rule: (uv)' = u'v + uv'
• Quotient rule: (u/v)' = (u'v - uv') / v²
• Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
Common Derivatives
• d/dx[sin x] = cos x
• d/dx[cos x] = -sin x
• d/dx[eˣ] = eˣ
