USTET Mathematics — Calculus — Limits, Derivatives & IntegralsCheat Sheet
Calculus — Limits, Derivatives & Integrals cheat sheet — the reference card you wish you had on exam day. Condensed from the full study notes, this is the high-yield core of Calculus — Limits, Derivatives & Integrals for USTET Mathematics. Download, print, revise.
Exam context
The University of Santo Tomas Entrance Test is conducted by University of Santo Tomas and is scheduled for Early Q4 2026. The Mathematics subtest is marked as "Core section" in the official pattern, and Calculus — Limits, Derivatives & Integrals appears in position 9th of 9 in the USTET Mathematics review rotation. Passing mark: Competitive overall score. Recent USTET 2026 papers have drawn roughly a meaningful share of questions from this subject.
Calculus — Limits, Derivatives & Integrals - Cheat sheet
Your last-minute revision companion for mastering limits, derivatives, and integrals before your UPCAT exam
Sections
Formulas
Formula
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
Formula
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
Formula
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
Formula
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
Formula
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
Formula
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
Section Title
Limits
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
Term
Limit
Example
lim(x→2) (x² - 1) = 3
Definition
The value that a function approaches as the input approaches a specific value
Term
Continuity
Example
f(x) = x² is continuous everywhere
Definition
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
Formulas
Formula
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
Formula
d/dx(c) = 0
Meaning
c = any constant
Watch Out
Constants always have zero derivative
When To Use
When differentiating constants
Formula
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
Formula
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
Formula
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
Formula
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
Formula
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
Section Title
Derivatives
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
Term
Derivative
Example
f'(x) represents the slope of f(x) at any point x
Definition
The instantaneous rate of change of a function at a specific point
Term
Differentiable
Example
f(x) = |x| is not differentiable at x = 0
Definition
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
Formulas
Formula
∫[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
Formula
∫ 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
Formula
∫ 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
Formula
∫[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
Formula
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
Section Title
Integrals
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
Term
Integral
Example
∫ 2x dx = x² + C
Definition
The area under a curve or the antiderivative of a function
Term
Antiderivative
Example
F(x) = x² is an antiderivative of f(x) = 2x
Definition
A function whose derivative is the given function
Term
Definite Integral
Example
∫[0 to 2] x dx = 2
Definition
Integral with specific upper and lower limits
Term
Indefinite Integral
Example
∫ x dx = x²/2 + C
Definition
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
Rows
Values
- Find rate of change
- Find area under curve
Property
Purpose
Values
- f'(x) or d/dx
- ∫ f(x) dx
Property
Symbol
Values
- nx^(n-1)
- x^(n+1)/(n+1) + C
Property
Power Rule
Values
- 0
- cx + C
Property
Constant Rule
Values
- Function
- Function + C or Number
Property
Result Type
Columns
- Aspect
- Derivatives
- Integrals
Table Title
Derivatives vs Integrals
Ready to practise for the USTET 2026?
Super Tutor's AI review plan adapts to your weak areas and builds a weekly practice schedule around your target USTET exam date.