CEUET Mathematics — Calculus — Limits, Derivatives & IntegralsStudy Notes
Detailed study notes for CEUET Mathematics — Calculus — Limits, Derivatives & Integrals. These are the kind of notes you would take if you were reviewing with someone who has already scored well on the CEUET: organised by what Centro Escolar University tests first, followed by the nice-to-knows, and ending with the traps to avoid.
Exam context
Centro Escolar University runs the Centro Escolar University Entrance Test on Q3–Q4 2026. Its Mathematics section sits under a "Core" weighting, and Calculus — Limits, Derivatives & Integrals is the 9th chapter in the 9-chapter CEUET Mathematics rotation. The CEUET passing mark is Competitive overall score, and the most recent 2026 paper drew about a meaningful share of questions from Mathematics.
Calculus — Limits, Derivatives & Integrals - Study notes
Calculus is one of the most important branches of mathematics that deals with continuous change. It consists of three main components: limits, derivatives, and integrals. Think of calculus as a powerful tool that helps us understand how things change over time - like the speed of a moving car, the growth rate of a population, or the area under a curve. In this chapter, we'll explore these fundamental concepts with step-by-step examples and practical applications that are essential for UPCAT and other entrance examinations.
Summary
Calculus consists of three interconnected concepts: limits (approaching values), derivatives (rates of change), and integrals (accumulated change). Master the basic rules and theorems, practice step-by-step problem solving, and always verify your answers. For UPCAT success, focus on: (1) Limit evaluation techniques including factoring for indeterminate forms, (2) Derivative rules - power, product, quotient, and chain rules, (3) Integration techniques and the Fundamental Theorem of Calculus, and (4) Real-world applications like motion problems and area calculations. Remember to work systematically, show all steps clearly, and check your work by differentiating integrals or using alternative methods.
Sections
A limit describes what happens to a function as the input approaches a particular value. Imagine you're walking towards a wall - even though you never actually touch it, you get closer and closer. That's the idea behind limits. Definition: The limit of f(x) as x approaches a is L, written as lim(x→a) f(x) = L, if f(x) gets arbitrarily close to L as x gets arbitrarily close to a. Let's work through some examples: Example 1: Find lim(x→2) (3x + 1) Step 1: Since this is a linear function, we can substitute directly Step 2: lim(x→2) (3x + 1) = 3(2) + 1 = 6 + 1 = 7 Example 2: Find lim(x→3) (x² - 9)/(x - 3) Step 1: Direct substitution gives 0/0 (indeterminate form) Step 2: Factor the numerator: x² - 9 = (x + 3)(x - 3) Step 3: Simplify: (x + 3)(x - 3)/(x - 3) = x + 3 Step 4: Now substitute: lim(x→3) (x + 3) = 3 + 3 = 6 Common Mistake: Never just say 0/0 = 0. Always try to simplify first by factoring or other algebraic methods.
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Understanding Limits
Examples
- lim(x→5) (2x - 3) = 2(5) - 3 = 7
- lim(x→0) sin(x)/x = 1 (special limit)
- lim(x→∞) 1/x = 0
- lim(x→1) (x² - 1)/(x - 1) = lim(x→1) (x + 1) = 2
Key Points
- Limits describe the behavior of functions as inputs approach specific values
- Use direct substitution when possible for continuous functions
- For indeterminate forms like 0/0, factor and simplify first
- The limit may exist even if the function is undefined at that point
- Master the limit theorems for sum, difference, product, and quotient
Derivatives measure how fast something is changing. If you're driving and your speedometer shows 60 km/h, that's the derivative of your position with respect to time. Definition: The derivative of f at point a is f'(a) = lim(x→a) [f(x) - f(a)]/(x - a) Let's master the basic derivative rules with examples: Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1) Example 1: Find the derivative of f(x) = x³ Step 1: Apply power rule with n = 3 Step 2: f'(x) = 3x^(3-1) = 3x² Example 2: Find the derivative of g(x) = 5x² + 3x - 7 Step 1: Use sum rule - differentiate each term separately Step 2: d/dx(5x²) = 5 × 2x^(2-1) = 10x Step 3: d/dx(3x) = 3 × 1x^(1-1) = 3 Step 4: d/dx(-7) = 0 (constant rule) Step 5: g'(x) = 10x + 3 Product Rule Example: Find derivative of h(x) = (2x + 1)(x² - 3) Step 1: Let f(x) = 2x + 1 and g(x) = x² - 3 Step 2: f'(x) = 2 and g'(x) = 2x Step 3: Using product rule: h'(x) = f(x)g'(x) + f'(x)g(x) Step 4: h'(x) = (2x + 1)(2x) + (2)(x² - 3) Step 5: h'(x) = 4x² + 2x + 2x² - 6 = 6x² + 2x - 6
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Derivatives - The Rate of Change
Examples
- d/dx(x⁴) = 4x³
- d/dx(6x³ - 2x + 5) = 18x² - 2
- d/dx(x²)(x + 1) = x²(1) + 2x(x + 1) = x² + 2x² + 2x = 3x² + 2x
- d/dx(x²/x) = d/dx(x) = 1
Key Points
- Derivatives represent instantaneous rate of change
- Power rule: d/dx(x^n) = nx^(n-1)
- Constant multiple rule: d/dx[cf(x)] = c × f'(x)
- Sum rule: d/dx[f(x) + g(x)] = f'(x) + g'(x)
- Product rule: d/dx[f(x)g(x)] = f(x)g'(x) + f'(x)g(x)
- Quotient rule: d/dx[f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)]/[g(x)]²
Integration is the reverse of differentiation. While derivatives tell us the rate of change, integrals tell us the accumulated change. Think of it as finding the total distance traveled when you know the speed at each moment. Definition: If F'(x) = f(x), then ∫f(x)dx = F(x) + C, where C is the constant of integration. Basic Integration Rules: Power Rule for Integration: ∫x^n dx = x^(n+1)/(n+1) + C (where n ≠ -1) Example 1: Find ∫x³ dx Step 1: Apply power rule with n = 3 Step 2: ∫x³ dx = x^(3+1)/(3+1) + C = x⁴/4 + C Example 2: Find ∫(2x² - 5x + 3) dx Step 1: Integrate each term separately Step 2: ∫2x² dx = 2 × x³/3 = 2x³/3 Step 3: ∫(-5x) dx = -5 × x²/2 = -5x²/2 Step 4: ∫3 dx = 3x Step 5: Final answer: 2x³/3 - 5x²/2 + 3x + C Definite Integration using Fundamental Theorem of Calculus: ∫[a to b] f(x)dx = F(b) - F(a) Example 3: Evaluate ∫[1 to 3] x² dx Step 1: Find antiderivative: ∫x² dx = x³/3 Step 2: Apply limits: [x³/3] from 1 to 3 Step 3: F(3) - F(1) = 3³/3 - 1³/3 = 27/3 - 1/3 = 9 - 1/3 = 26/3 Verification Method: Always check your integration by differentiating your answer. If you get back to the original function, you're correct!
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Integration - Finding the Area
Examples
- ∫5x⁴ dx = 5x⁵/5 + C = x⁵ + C
- ∫(3x² + 2x - 1) dx = x³ + x² - x + C
- ∫[0 to 2] x dx = [x²/2] from 0 to 2 = 4/2 - 0 = 2
- ∫1 dx = x + C
Key Points
- Integration is the reverse process of differentiation
- Power rule for integration: ∫x^n dx = x^(n+1)/(n+1) + C
- Always add constant of integration C for indefinite integrals
- Definite integrals give numerical values (areas)
- Fundamental Theorem connects derivatives and integrals
- Check your work by differentiating your answer
Success in calculus requires recognizing problem patterns and applying the right techniques. Here's a systematic approach: Step-by-Step Problem Solving Method: 1. Read the problem carefully and identify what's being asked 2. Determine whether you need limits, derivatives, or integrals 3. Choose the appropriate rule or theorem 4. Set up the problem algebraically 5. Perform the calculation step by step 6. Verify your answer makes sense Application Example - Velocity and Acceleration: If position is s(t) = t³ - 6t² + 9t, find velocity and acceleration at t = 2. Step 1: Velocity is the derivative of position v(t) = s'(t) = 3t² - 12t + 9 Step 2: Acceleration is the derivative of velocity a(t) = v'(t) = 6t - 12 Step 3: Evaluate at t = 2 v(2) = 3(2)² - 12(2) + 9 = 12 - 24 + 9 = -3 m/s a(2) = 6(2) - 12 = 12 - 12 = 0 m/s² Area Under Curve Example: Find the area under y = x² from x = 0 to x = 3. Step 1: Set up definite integral Area = ∫[0 to 3] x² dx Step 2: Find antiderivative ∫x² dx = x³/3 Step 3: Apply limits Area = [x³/3] from 0 to 3 = 3³/3 - 0³/3 = 27/3 = 9 square units Common Mistakes to Avoid: 1. Forgetting the constant of integration in indefinite integrals 2. Making sign errors in the quotient rule 3. Not simplifying indeterminate forms in limits 4. Mixing up the order in the quotient rule formula
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Problem-Solving Strategies and Common Applications
Examples
- Finding maximum/minimum points using f'(x) = 0
- Related rates problems using chain rule
- Area between curves: ∫[a to b] |f(x) - g(x)| dx
- Average value of function: (1/(b-a))∫[a to b] f(x) dx
Key Points
- Always identify whether you need limits, derivatives, or integrals
- Work systematically through each step
- Velocity is derivative of position, acceleration is derivative of velocity
- Definite integrals calculate areas under curves
- Always verify your answers by checking units and reasonableness
- Practice recognizing standard problem types
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