Calculus — Limits, Derivatives & Integrals - Cheat sheet
Your last-minute revision companion for mastering limits, derivatives, and integrals before your UPCAT exam
Limits
Formulas
lim(x→a) f(x) = L
Meaning
As x approaches a, f(x) approaches L
Watch out
The function doesn't need to be defined at x = a
When to use
When finding the value a function approaches at a specific point
lim(x→a) (mx + b) = ma + b
Meaning
m = slope, b = y-intercept, a = approaching value
Watch out
Always substitute x = a directly for linear functions
When to use
For any linear function limit
lim(x→a) c = c
Meaning
c = constant value
Watch out
Constants never change regardless of x value
When to use
When the function is just a constant
lim(x→a) [f(x) ± g(x)] = L ± M
Meaning
L = limit of f(x), M = limit of g(x)
Watch out
Both individual limits must exist first
When to use
When finding limits of sums or differences
lim(x→a) [f(x) · g(x)] = L · M
Meaning
L = limit of f(x), M = limit of g(x)
Watch out
Both individual limits must exist first
When to use
When finding limits of products
lim(x→a) [f(x)/g(x)] = L/M
Meaning
L = limit of f(x), M = limit of g(x), M ≠ 0
Watch out
Denominator limit cannot be zero
When to use
When finding limits of quotients
Important facts
- Limits can exist even if the function is undefined at that point
- For polynomial functions, substitute the value directly
- Use factoring for 0/0 indeterminate forms
- Rational functions: substitute if denominator ≠ 0
Key definitions
Limit
Example
lim(x→2) (x² - 1) = 3
The value that a function approaches as the input approaches a specific value
Continuity
Example
f(x) = x² is continuous everywhere
A function is continuous at x = a if lim(x→a) f(x) = f(a)
Diagrams to know
- Graph showing function approaching a limit
- Discontinuous function with existing limit
Derivatives
Formulas
f'(a) = lim(x→a) [f(x) - f(a)]/(x - a)
Meaning
Definition of derivative at point a
Watch out
This is the limit definition, not for quick calculations
When to use
When using the formal definition of derivative
d/dx(c) = 0
Meaning
c = any constant
Watch out
Constants always have zero derivative
When to use
When differentiating constants
d/dx(x^n) = nx^(n-1)
Meaning
n = any real number power
Watch out
Reduce the power by 1 and multiply by original power
When to use
Power rule for any polynomial term
d/dx[cf(x)] = c·f'(x)
Meaning
c = constant, f(x) = function
Watch out
Constants factor out of derivatives
When to use
When a constant multiplies a function
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
Meaning
Sum/difference rule for derivatives
Watch out
Derivative of sum equals sum of derivatives
When to use
When differentiating sums or differences
d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Meaning
Product rule: first times derivative of second plus second times derivative of first
Watch out
Don't forget both terms in the product rule
When to use
When differentiating products of two functions
d/dx[f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)]/[g(x)]²
Meaning
Quotient rule: bottom times derivative of top minus top times derivative of bottom, all over bottom squared
Watch out
Order matters: numerator derivative first, then subtract
When to use
When differentiating quotients of two functions
Common values
Value
1
Symbol
d/dx(x)
Quantity
Derivative of x
Value
2x
Symbol
d/dx(x²)
Quantity
Derivative of x²
Value
-1/x²
Symbol
d/dx(x⁻¹)
Quantity
Derivative of 1/x
Important facts
- Derivative represents slope of tangent line
- If f'(x) > 0, function is increasing
- If f'(x) < 0, function is decreasing
- Critical points occur where f'(x) = 0 or undefined
Key definitions
Derivative
Example
f'(x) represents the slope of f(x) at any point x
The instantaneous rate of change of a function at a specific point
Differentiable
Example
f(x) = |x| is not differentiable at x = 0
A function is differentiable at a point if its derivative exists at that point
Diagrams to know
- Tangent line to a curve at a point
- Graph showing increasing/decreasing intervals
Integrals
Formulas
∫[a to b] f(x) dx
Meaning
Definite integral from a to b of function f(x)
Watch out
Include limits of integration and dx
When to use
When finding area under curve between specific bounds
∫ c dx = cx + C
Meaning
c = constant, C = constant of integration
Watch out
Don't forget the constant of integration C
When to use
When integrating constants
∫ x^n dx = x^(n+1)/(n+1) + C
Meaning
n ≠ -1, C = constant of integration
Watch out
Add 1 to power and divide by new power
When to use
Power rule for integration
∫[a to b] f(x) dx = F(b) - F(a)
Meaning
F(x) = antiderivative of f(x)
Watch out
Evaluate antiderivative at upper limit minus lower limit
When to use
Fundamental Theorem of Calculus for definite integrals
F'(x) = f(x) where F(x) = ∫[a to x] f(t) dt
Meaning
Derivative of integral equals original function
Watch out
Integration and differentiation are inverse operations
When to use
First Fundamental Theorem of Calculus
Common values
Value
x + C
Symbol
∫ 1 dx
Quantity
Integral of 1
Value
x²/2 + C
Symbol
∫ x dx
Quantity
Integral of x
Value
1/2
Symbol
∫[0 to 1] x dx
Quantity
Integral from 0 to 1 of x
Important facts
- Integration is the reverse of differentiation
- Always add constant C for indefinite integrals
- Definite integrals give numerical values
- Area below x-axis counts as negative
Key definitions
Integral
Example
∫ 2x dx = x² + C
The area under a curve or the antiderivative of a function
Antiderivative
Example
F(x) = x² is an antiderivative of f(x) = 2x
A function whose derivative is the given function
Definite Integral
Example
∫[0 to 2] x dx = 2
Integral with specific upper and lower limits
Indefinite Integral
Example
∫ x dx = x²/2 + C
Integral without specific limits, includes constant C
Diagrams to know
- Area under curve representation
- Fundamental Theorem visualization
Must remember
- lim(x→a) f(x) can exist even if f(a) is undefined
- Power Rule: d/dx(x^n) = nx^(n-1)
- Product Rule: (fg)' = f'g + fg'
- Quotient Rule: (f/g)' = (gf' - fg')/g²
- ∫ x^n dx = x^(n+1)/(n+1) + C (n ≠ -1)
- Fundamental Theorem: ∫[a to b] f(x) dx = F(b) - F(a)
- Always include +C for indefinite integrals
- Derivative of constant is zero
- Integration and differentiation are inverse operations
- Check continuity before applying limit theorems
Last minute tips
- For limits: try direct substitution first, then factor if you get 0/0
- For derivatives: identify the rule needed (power, product, quotient) before calculating
- For definite integrals: find antiderivative first, then evaluate at bounds
- Common mistake: forgetting +C in indefinite integrals loses points
- Draw quick sketches to visualize limit and integral problems
Comparison tables
Derivatives vs Integrals
| Aspect | Derivatives | Integrals |
|---|---|---|
| Purpose | Find rate of change | Find area under curve |
| Symbol | f'(x) or d/dx | ∫ f(x) dx |
| Power Rule | nx^(n-1) | x^(n+1)/(n+1) + C |
| Constant Rule | 0 | cx + C |
| Result Type | Function | Function + C or Number |
Diagrams
Calculus Concepts Overview
Complete overview of all calculus concepts showing the three main branches and their key components
Problem-Solving Decision Tree
Step-by-step decision tree for solving different types of calculus problems systematically