Calculus Flashcards | IB - Wyatt's Notes
IB Maths — Calculus Flashcard Deck
Flashcards with spaced repetition. Press Space to flip, then rate your recall (1—4).
Key Concepts
Differentiation finds the rate of change of a function. The derivative f’(x) gives the gradient of the tangent at any point. Key rules include the power rule (d/dx[xⁿ] = nxⁿ⁻¹), the chain rule (d/dx[f(g(x))] = f’(g(x))·g’(x)), and the product rule (d/dx[uv] = u’v + uv’). The quotient rule is d/dx[u/v] = (u’v − uv’)/v². Second derivatives f”(x) tell you about concavity — positive means concave up, negative means concave down. At a stationary point, f’(x) = 0; the second derivative test determines whether it’s a maximum (f” < 0) or minimum (f” > 0).
Integration is the reverse of differentiation and gives the area under a curve. The fundamental theorem of calculus links the two: ∫ₐᵇ f(x)dx = F(b) − F(a). Key techniques include u-substitution (the reverse of the chain rule) and integration by parts (∫u dv = uv − ∫v du). The constant of integration +C must always be included in indefinite integrals. Definite integrals give numerical values representing signed area — areas above the x-axis are positive, below are negative.
Differential Equations relate a function to its derivatives. In IB, you in standard practice solve separable equations by rearranging to get all y terms on one side and all x terms on the other, then integrating both sides. The general solution contains an arbitrary constant C; an initial condition (a specific value of y at a specific x) gives the particular solution by determining C. These model real-world phenomena like population growth, radioactive decay, and cooling.
Applications of Calculus: The derivative gives the slope of a curve (used for finding maxima/minima by setting f’(x) = 0). The integral gives areas and volumes (volumes of revolution: V = π∫y²dx). Related rates problems use the chain rule to connect rates of change of different quantities (e.g., how fast the radius of a balloon changes as it’s inflated). Curve sketching uses f’(x) for turning points and f”(x) for concavity.
Intuition
Think of differentiation as a “magnifying glass” that zooms in on a curve until it looks straight — the slope of that straight line is the derivative. Integration is like stacking infinitely many thin rectangles under a curve to find the total area. The fundamental theorem of calculus tells you these two operations undo each other, just like addition and subtraction. Differential equations are the “word problems” of calculus — they describe real situations and ask you to find the function that fits.
Why it matters: Calculus is the mathematics of change. It’s essential for physics (describing motion and forces), economics (optimising profit), biology (modelling population growth), and engineering (designing structures that withstand changing loads). Without calculus, we couldn’t describe most of the natural world.
Common Pitfalls
- Forgetting the constant of integration (+C) in indefinite integrals. This is an easy mark to lose and appears in almost every exam. Always include it unless the question specifies a definite integral.
- Applying the chain rule incorrectly: you must multiply by the derivative of the inner function. A common mistake is to forget this multiplier, especially with trigonometric and exponential functions. For example, d/dx[sin(3x)] = 3cos(3x), not cos(3x).
- Confusing the conditions for a maximum vs minimum. After finding critical points (f’(x) = 0), you must test the second derivative or use a sign chart to determine which is which. f”(x) > 0 means minimum; f”(x) < 0 means maximum.
- Forgetting that integration gives signed area. Areas below the x-axis are negative. To find total area, split the integral at the x-intercepts and take absolute values.
Cross-References
- Geometry and Vectors: Vector calculus uses differentiation and integration to describe curves, surfaces, and motion in space.
- Statistics: Probability density functions use integration to find probabilities, and regression uses calculus to find best-fit lines.
- Number and Algebra: Algebra provides the foundation for manipulation of expressions before and after calculus operations.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.