USTET Mathematics — Calculus — Limits, Derivatives & IntegralsSlides
The slide format strips Calculus — Limits, Derivatives & Integrals to the visuals that USTET reviewers remember best. University of Santo Tomas's 2026 Mathematics papers reward reviewers who recognise the structure of a concept before they parse the item text; these slides train that recognition.
Exam context
University of Santo Tomas runs the University of Santo Tomas Entrance Test on Early Q4 2026. Its Mathematics section sits under a "Core section" weighting, and Calculus — Limits, Derivatives & Integrals is the 9th chapter in the 9-chapter USTET Mathematics rotation. The USTET 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 - Slides
Calculus is a fundamental branch of mathematics that deals with rates of change and accumulation. This chapter covers three core concepts: limits (the foundation of calculus), derivatives (measuring rates of change), and integrals (measuring accumulation). These concepts are essential for solving real-world problems in physics, engineering, economics, and other fields. Understanding these principles will prepare you for advanced mathematical applications in your future studies and career.
Slides
Introduction to Calculus
Calculus allows us to analyze continuously changing quantities. Unlike algebra which deals with static relationships, calculus helps us understand how things change over time or space.
Notes
This overview slide introduces students to the broad scope of calculus and its practical applications.
Topic
Introduction
Slide Id
S1
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mermaid
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1
Mermaid Diagram
Code
mindmap root((Calculus)) Limits Foundation of calculus Behavior near points Continuity Derivatives Rate of change Slope of tangent Optimization Integrals Area under curves Accumulation Antiderivatives
Type
mermaid_mindmap
Description
Mind map showing the three main branches of calculus and their key concepts
Understanding Limits - Definition
A limit describes what happens to f(x) as x gets arbitrarily close to some value a. Think of it as asking: 'What value is f(x) approaching as x gets closer and closer to a?'
Notes
Students often confuse limits with function values. Emphasize that limits are about approaching behavior, not the actual value at the point.
Topic
Limits - Definition
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S2
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mermaid
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2
Mermaid Diagram
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flowchart TD A[Function f x] --> B{x approaches a} B --> C[f x approaches L] C --> D[lim x to a f x equals L] B --> E[Note: f a may not exist] E --> D
Type
mermaid_flowchart
Description
Flowchart illustrating the concept of limits and how functions behave as x approaches a value
Basic Limit Theorems - Part 1
These basic theorems are intuitive. A constant function always has the same value, and the identity function x simply equals whatever value we're approaching.
Notes
These basic limits are foundational. Students should master these before moving to more complex limit operations.
Topic
Limit Theorems - Basic
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S3
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none
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3
Mermaid Diagram
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Limit Operations - Sum, Product, Quotient
These theorems allow us to break complex limits into simpler parts. The key requirement is that the individual limits must exist and, for quotients, the denominator limit cannot be zero.
Notes
Emphasize the importance of checking that individual limits exist before applying these rules.
Topic
Limit Operations
Slide Id
S4
Visual Type
mermaid
Image Prompt
Slide Number
4
Mermaid Diagram
Code
flowchart TD A[Two functions f x and g x] --> B{Find individual limits} B --> C[lim f x equals L] B --> D[lim g x equals M] C --> E[Apply operation rules] D --> E E --> F[Sum: L plus M] E --> G[Product: L times M] E --> H[Quotient: L over M if M not zero]
Type
mermaid_flowchart
Description
Process for applying limit operations to combine simpler limits into more complex ones
Solving Limit Problems - Step by Step
Most limit problems can be solved by direct substitution. When this fails (indeterminate forms), we need special techniques to find the limit.
Notes
Common mistake: Students often panic when they see 0/0. Teach them this is just the signal to use algebraic techniques.
Topic
Limit Problem Solving
Slide Id
S5
Visual Type
mermaid
Image Prompt
Slide Number
5
Mermaid Diagram
Code
flowchart TD A[Limit Problem] --> B[Try Direct Substitution] B --> C{Result?} C -->|Real Number| D[Answer Found] C -->|0/0 or infinity/infinity| E[Indeterminate Form] E --> F[Factor or Rationalize] F --> G[Simplify] G --> H[Substitute Again] H --> D
Type
mermaid_flowchart
Description
Step-by-step flowchart for solving limit problems, showing when to use different techniques
Introduction to Derivatives
A derivative tells us how fast a function is changing at a specific point. It's like finding the exact speed of a car at a precise moment, rather than average speed over a journey.
Notes
Help students visualize derivatives by connecting to familiar concepts like speed and slope.
Topic
Derivatives - Introduction
Slide Id
S6
Visual Type
mermaid
Image Prompt
Slide Number
6
Mermaid Diagram
Code
flowchart TD A[Function f x] --> B[Choose point a] B --> C[Find slope of secant line] C --> D[Let x approach a] D --> E[Secant becomes tangent] E --> F[Slope of tangent is derivative] F --> G[f prime a equals rate of change]
Type
mermaid_flowchart
Description
Visual explanation of how derivatives are formed from the limit of secant line slopes
Basic Derivative Rules
These fundamental rules allow us to find derivatives without using the limit definition every time. The power rule is the most important - it works for any real number n.
Notes
Students should memorize these rules and practice applying them to polynomial functions before moving to more complex rules.
Topic
Derivative Rules - Basic
Slide Id
S7
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none
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Slide Number
7
Mermaid Diagram
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none
Product and Quotient Rules
These rules are essential when functions are multiplied or divided. The product rule shows that the derivative of a product is NOT the product of derivatives.
Notes
These rules are commonly confused. Give students memory aids and plenty of practice problems.
Topic
Product and Quotient Rules
Slide Id
S8
Visual Type
mermaid
Image Prompt
Slide Number
8
Mermaid Diagram
Code
flowchart TD A[Two functions f x and g x] --> B{Multiplication or Division?} B -->|Multiplication| C[Product Rule] B -->|Division| D[Quotient Rule] C --> E[f x times g prime x plus f prime x times g x] D --> F[g x times f prime x minus f x times g prime x] F --> G[All over g x squared]
Type
mermaid_flowchart
Description
Decision tree for when to use product rule vs quotient rule when finding derivatives
Derivative Problem Solving Example
This example demonstrates two approaches to the same problem. Using multiple methods helps verify answers and builds problem-solving flexibility.
Notes
Teaching multiple solution methods helps students develop mathematical flexibility and confidence.
Topic
Derivative Examples
Slide Id
S9
Visual Type
mermaid
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Slide Number
9
Mermaid Diagram
Code
flowchart TD A[f x equals 2x squared plus 3x times x minus 1] --> B{Choose Method} B -->|Method 1| C[Apply Product Rule] B -->|Method 2| D[Expand First] C --> E[2x squared plus 3x times 1 plus 4x plus 3 times x minus 1] D --> F[2x cubed plus x squared minus 3x] E --> G[6x squared plus 2x minus 3] F --> G G --> H[Verify: Both methods match]
Type
mermaid_flowchart
Description
Comparison of two methods for solving the same derivative problem
Introduction to Integrals
Integration is the reverse process of differentiation. While derivatives measure rates of change, integrals measure total accumulation over an interval.
Notes
Students often struggle with the conceptual difference between indefinite and definite integrals. Use concrete examples.
Topic
Integrals - Introduction
Slide Id
S10
Visual Type
mermaid
Image Prompt
Slide Number
10
Mermaid Diagram
Code
mindmap root((Integrals)) Indefinite Integrals Antiderivatives Plus constant C General form Definite Integrals Area under curve From a to b Numerical result Applications Distance from velocity Area calculations Volume problems
Type
mermaid_mindmap
Description
Mind map showing the two types of integrals and their key characteristics
Fundamental Theorem of Calculus
This theorem is the bridge between derivatives and integrals. It tells us that integration 'undoes' differentiation, and provides a practical way to evaluate definite integrals.
Notes
This is one of the most important theorems in calculus. Make sure students understand both parts.
Topic
Fundamental Theorem
Slide Id
S11
Visual Type
mermaid
Image Prompt
Slide Number
11
Mermaid Diagram
Code
flowchart TD A[Function f x] --> B[Find Antiderivative F x] B --> C[F prime x equals f x] A --> D[Want definite integral from a to b] D --> E[Use F b minus F a] C --> E E --> F[Final Answer]
Type
mermaid_flowchart
Description
Flowchart showing how the Fundamental Theorem connects antiderivatives to definite integrals
Basic Integration Rules
These rules are the reverse of differentiation rules. The power rule for integration increases the exponent by 1 and divides by the new exponent.
Notes
Emphasize that integration is the reverse of differentiation. Students should check their answers by differentiating.
Topic
Integration Rules
Slide Id
S12
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none
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12
Mermaid Diagram
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none
Solving Integration Problems - Step by Step
Systematic approach prevents common errors like forgetting the constant of integration or making arithmetic mistakes in definite integrals.
Notes
Common errors include forgetting +C for indefinite integrals and arithmetic mistakes when evaluating definite integrals.
Topic
Integration Problem Solving
Slide Id
S13
Visual Type
mermaid
Image Prompt
Slide Number
13
Mermaid Diagram
Code
flowchart TD A[Integration Problem] --> B{Definite or Indefinite?} B -->|Indefinite| C[Find antiderivative] B -->|Definite| D[Find antiderivative] C --> E[Add constant C] D --> F[Evaluate at bounds] E --> G[Final Answer] F --> H[F b minus F a] H --> G
Type
mermaid_flowchart
Description
Step-by-step process for solving integration problems, showing different paths for definite vs indefinite integrals
Real-World Applications
Calculus provides tools to model and solve problems involving continuous change, making it essential in science, engineering, and economics.
Notes
Help students see the practical relevance of calculus by connecting it to their potential career interests.
Topic
Applications
Slide Id
S14
Visual Type
mermaid
Image Prompt
Slide Number
14
Mermaid Diagram
Code
mindmap root((Calculus Applications)) Physics Motion problems Velocity and acceleration Work and energy Economics Marginal analysis Optimization Cost functions Biology Population dynamics Growth rates Epidemiology Engineering Design optimization Control systems Signal processing
Type
mermaid_mindmap
Description
Mind map showing various real-world applications of calculus across different fields
Common Mistakes and How to Avoid Them
These are the most frequent errors students make. Being aware of them helps prevent mistakes on exams and homework.
Notes
Spend time on common mistakes - prevention is better than correction after the fact.
Topic
Common Mistakes
Slide Id
S15
Visual Type
mermaid
Image Prompt
Slide Number
15
Mermaid Diagram
Code
flowchart TD A[Calculus Problem] --> B[fa:fa-warning Check Common Mistakes] B --> C[Limits: Is this the function value?] B --> D[Derivatives: Did I use product rule correctly?] B --> E[Integrals: Did I include plus C?] C --> F[fa:fa-check Verify Answer] D --> F E --> F F --> G[fa:fa-star Final Answer]
Type
mermaid_flowchart
Description
Checklist flowchart for avoiding common calculus mistakes
Chapter Summary and Key Takeaways
These three concepts work together to give us powerful tools for analyzing continuous change. Mastering the basic rules and understanding the connections between concepts is key to success in calculus.
Notes
End with encouragement - calculus builds on itself, so strong foundations in these basics will make advanced topics much easier.
Topic
Chapter Summary
Slide Id
S16
Visual Type
mermaid
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Slide Number
16
Mermaid Diagram
Code
flowchart TD A[fa:fa-calculator Limits] --> B[fa:fa-star Foundation of Calculus] B --> C[fa:fa-arrow-up Derivatives] B --> D[fa:fa-area-chart Integrals] C --> E[Rate of Change] D --> F[Accumulation] C --> G[fa:fa-link Fundamental Theorem] D --> G G --> H[fa:fa-lightbulb Complete Understanding]
Type
mermaid_flowchart
Description
Summary flowchart showing how limits, derivatives, and integrals connect to form a complete understanding of calculus
References
- CET 2026 Comprehensive Lecture Notes — Mathematics
- Calculus concepts and limit theorems from course materials
- Derivative rules and integration techniques from lecture notes
- Fundamental Theorem of Calculus applications and examples
In summary
This chapter has introduced you to the fundamental concepts of calculus: limits, derivatives, and integrals. These concepts are interconnected and form the basis for understanding continuous change in mathematics and science. Remember that limits provide the foundation, derivatives measure rates of change, and integrals measure accumulation. The Fundamental Theorem of Calculus beautifully connects differentiation and integration, showing they are inverse operations. Practice applying these concepts to both mathematical problems and real-world situations to deepen your understanding. As you continue your mathematical journey, these tools will prove invaluable in advanced mathematics, physics, engineering, and many other fields.
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