Calculus Foundations

MYP Unit Framework

Key Concept: CHANGE Related Concepts: Approximation. Patterns. Quantity. Global Context: Scientific and Technical Innovation (How does calculus provide the mathematical tools to analyse rates of change and accumulation in science and engineering?) Statement of Inquiry: Calculus provides tools to analyse rates of change and accumulation, forming the mathematical foundation of modern science.

Inquiry Questions

1. Limits — The Foundation of Calculus

The Problem of Instantaneous Change

'How fast is an object moving at EXACTLY t = 3 seconds? You cannot measure speed at an INSTANT — because speed is distance divided by TIME, and at an instant, BOTH distance and time are ZERO. This is the PARADOX that calculus resolves.'

The Idea of a Limit

'A limit is the value a function APPROACHES as the input APPROACHES a specific value. We write: lim(x→a) f(x) = L. This means: as x gets ARBITRARILY CLOSE to a (but NOT necessarily equal), f(x) gets ARBITRARILY CLOSE to L.'

Example: 'Consider f(x) = (x² − 1)/(x − 1). At x = 1, this function is UNDEFINED (division by zero). But as x APPROACHES 1, f(x) approaches 2. The limit EXISTS even though the function is not defined at that point.'

One-Sided Limits

• lim(x→a⁻) f(x): limit as x approaches a FROM THE LEFT.
• lim(x→a⁺) f(x): limit as x approaches a FROM THE RIGHT. 'The limit EXISTS only if BOTH one-sided limits exist and are EQUAL.'

Continuity

'A function is CONTINUOUS at x = a if: (1) f(a) is defined. (2) lim(x→a) f(x) exists. (3) lim(x→a) f(x) = f(a). INFORMALLY: you can draw the graph WITHOUT lifting your pen.'

'Most functions we work with (polynomials, rational, trigonometric) are CONTINUOUS everywhere in their domains — but they can have DISCONTINUITIES at points where they are undefined (like division by zero).'

2. Differentiation — Measuring Instantaneous Rate of Change

The Derivative — What It Is

'The DERIVATIVE of a function f at a point x is the INSTANTANEOUS RATE OF CHANGE of f at that point. It is the SLOPE of the TANGENT LINE to the curve at that point.

Notation: f'(x) (Lagrange), dy/dx (Leibniz), d/dx [f(x)].'

Differentiation from First Principles

f'(x) = lim(h→0) [f(x + h) − f(x)] / h

'This is the DEFINITION of the derivative. It calculates the slope of the SECANT line between two points (x, f(x)) and (x + h, f(x + h)), and then takes the limit as the distance between the points SHRINKS to zero — giving the slope of the TANGENT line at a single point.'

The Power Rule — The First Differentiation Rule

If f(x) = xⁿ, then f'(x) = n × xⁿ⁻¹.

Examples:

• f(x) = x⁵ → f'(x) = 5x⁴
• f(x) = x² → f'(x) = 2x
• f(x) = x → f'(x) = 1
• f(x) = x⁰ = 1 → f'(x) = 0

Rules of Differentiation

3. Applications of Differentiation

Tangents and Normals

'The derivative at a point gives the slope of the TANGENT line. The NORMAL line is PERPENDICULAR to the tangent — its slope is the NEGATIVE RECIPROCAL: m_normal = −1/m_tangent (provided m_tangent ≠ 0).'

Example: 'Find the equation of the tangent to f(x) = x² at x = 3. f(3) = 9. f'(x) = 2x, so f'(3) = 6. The tangent line is: y − 9 = 6(x − 3), or y = 6x − 9.'

Rates of Change in Real-World Contexts

'The derivative is the mathematical tool for analysing ANY quantity that changes.'

• Velocity: v(t) = s'(t) — the derivative of displacement (position) with respect to time.
• Acceleration: a(t) = v'(t) = s''(t) — the derivative of velocity (second derivative of position).
• Population growth rate: P'(t) — the rate at which a population is changing at time t.
• Marginal cost: C'(x) — the cost of producing ONE MORE unit.

Optimisation — Finding Maxima and Minima

'Differentiation allows us to find WHERE a function reaches its MAXIMUM or MINIMUM value — and WHAT that value is.'

The process:

1. Find f'(x) and set it EQUAL to zero: f'(x) = 0. The solutions are CRITICAL POINTS.
2. Determine whether each critical point is a MAXIMUM, MINIMUM, or POINT OF INFLECTION using the SECOND DERIVATIVE TEST:
   • If f''(x) < 0: LOCAL MAXIMUM (the curve is CONCAVE DOWN).
   • If f''(x) > 0: LOCAL MINIMUM (the curve is CONCAVE UP).
   • If f''(x) = 0: TEST FAILS — may be a point of inflection.

Real-world example: 'A farmer wants to fence a rectangular field using 100 m of fencing. What dimensions MAXIMISE the area? A = x(50 − x) = 50x − x². dA/dx = 50 − 2x = 0 → x = 25 m. The field is 25 m × 25 m — a SQUARE. This is a MAXIMUM because d²A/dx² = −2 < 0.'

4. Integration — The Reverse of Differentiation

The Indefinite Integral

'Integration is the INVERSE operation of differentiation. If differentiation finds the RATE of change, integration finds the ORIGINAL function from its rate of change.'

Notation: ∫f(x) dx = F(x) + C, where F'(x) = f(x) and C is the CONSTANT OF INTEGRATION.

'Why is C necessary? Because the derivative of a CONSTANT is ZERO. If F'(x) = f(x), then (F(x) + 5)' = f(x) too. The constant represents the FAMILY of functions that differ only by a vertical shift.'

The Reverse Power Rule

∫xⁿ dx = xⁿ⁺¹/(n + 1) + C, for n ≠ −1.

Examples:

• ∫x² dx = x³/3 + C
• ∫x dx = x²/2 + C
• ∫1 dx = x + C

The Definite Integral — Area Under a Curve

'The DEFINITE integral ∫ₐᵇ f(x) dx represents the NET AREA between the curve y = f(x) and the x-axis, from x = a to x = b.'

The Fundamental Theorem of Calculus

∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x).

'This theorem CONNECTS differentiation and integration. It says: to find the area under a curve from a to b, find the ANTIDERIVATIVE, evaluate it at b and a, and SUBTRACT. This is one of the MOST BEAUTIFUL and POWERFUL results in all of mathematics.'

5. Applications of Integration

Area Between Curves

'To find the area BETWEEN two curves, integrate the DIFFERENCE between the upper function and the lower function: Area = ∫ₐᵇ [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a, b].'

Real-World Applications

• Distance from velocity: If you know the VELOCITY function of an object, integrating gives the TOTAL DISTANCE travelled.
• Total accumulation: Total growth of a population from its growth rate. Total profit from marginal profit.
• Average value of a function: f_avg = (1/(b − a)) ∫ₐᵇ f(x) dx. 'The average height of a function over an interval — useful in physics, economics, and engineering.'
• Volume of revolution: Rotating a curve around an axis creates a 3D solid. Integration calculates its VOLUME.

6. Why Calculus Matters

The Language of Modern Science

'Calculus is the MATHEMATICAL LANGUAGE of change. Without calculus, we could not:

• Design bridges that can bear their loads.
• Model the spread of a pandemic.
• Calculate orbits for space probes.
• Optimise supply chains.
• Understand the physics of black holes.
• Model climate change.

Calculus is NOT just an abstract mathematical exercise. It is the FOUNDATION of modern engineering, physics, economics, biology, and data science.'

Your Summative Assessment — The Calculus Investigation

Task: Apply the tools of differentiation and/or integration to a REAL-WORLD PROBLEM. Choose ONE of the following options:

Option A — Optimisation: Identify a real-world scenario where a quantity can be OPTIMISED (maximised or minimised). Formulate the problem mathematically (define variables, write the function). Use differentiation to find the optimum. Verify that it is a maximum or minimum. Interpret your result in the CONTEXT of the problem.

Option B — Area/Accumulation: Find a real-world scenario involving ACCUMULATION (distance from velocity, total growth, volume of a solid). Formulate the problem. Use integration (definite integral) to find the total accumulation. Interpret your result.

Write a 1000–1200 word report including: problem statement, mathematical formulation, calculations with clear steps, interpretation of results, and discussion of LIMITATIONS and ASSUMPTIONS.

'This investigation PREPARES you for the IB DP Mathematics Analysis and Approaches (AA) course and the Internal Assessment. Understanding calculus now gives you a SIGNIFICANT head start.'

ATL Skills

Formative Assessments

Interdisciplinary Connections

• Physics: Kinematics (velocity, acceleration from position), work, centre of mass.
• Economics: Marginal analysis, profit optimisation, consumer/producer surplus.
• Biology: Population growth models, drug concentration over time.
• TOK: Is calculus DISCOVERED (a feature of the universe) or INVENTED (a tool we created)? What does it mean for mathematics to be the 'language of science'?

Service as Action

• Tutoring: Tutor younger students in algebra and precalculus concepts that build toward calculus.
• STEM mentorship: Mentor students interested in STEM careers where calculus is essential.
• Applied project: Help a local business or organisation use optimisation to improve efficiency.

IB Learner Profile Attributes

Self-Test Questions

1. Explain the concept of a LIMIT in your own words. Give an example of a function that has a limit at a point where it is not defined.

2. Differentiate from FIRST PRINCIPLES: f(x) = x².

3. Use the power rule to differentiate: f(x) = 3x⁴ − 5x² + 2x − 7.

4. Find the equation of the tangent line to f(x) = x² − 3x at x = 2.

5. A rectangular garden is to be fenced on three sides with 60 m of fencing. Find the dimensions that MAXIMISE the area.

6. Integrate: ∫(3x² − 4x + 1) dx. Evaluate: ∫₀² (x² + 1) dx.

7. State the FUNDAMENTAL THEOREM OF CALCULUS and explain why it is important.

8. Describe THREE real-world applications of calculus and explain why calculus is necessary for each.