Unit 5 · Topic 5: Calculus
The Complete IB Mathematics: Applications and Interpretation Guide · IB Mathematics: Applications and InterpretationUnit 5 · Topic 5: Calculus
- The Complete IB Mathematics: Applications and Interpretation Guide
- IB Mathematics: Applications and Interpretation
- 5 sections
International Baccalaureate does not publish a per-unit weighting for this exam. Exam structure (official): SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%); HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%). Be ready to carry the same unit skills and representations into a complete answer on every paper. This topic develops rates, accumulation, optimization, and differential equations. The official subject brief names the topic but does not publish a topic-by-topic examination percentage, so no percentage is claimed here on the “Calculus: choose, calculate, interpret” page.
- How IB Mathematics: Applications and Interpretation assesses this International Baccalaureate publishes no per-unit percentage for IB Mathematics: Applications and Interpretation; Paper 1 and Paper 2 each account for 40% of the SL total and 30% of the HL total. · SL: Paper 1 (90 min, 40%) · Paper 2 (90 min, 40%) · Internal Assessment (20%) | HL: Paper 1 (120 min, 30%) · Paper 2 (120 min, 30%) · Paper 3 (60 min, 2 questions, 20%) · Internal Assessment (20%) · show the model with a function and its derivative with units, a rate graph with signed area over an interval, a differential equation with an initial condition and solution
- Key skills Limits, the Derivative as a Rate of Change, and Differentiating Polynomials, Tangents and Normals, Local Maximum and Minimum Points and Optimization, Antiderivatives, Area Under a Curve, and the Trapezoidal Rule, Chain, Product, and Quotient Rules, Related Rates, and the Second Derivative (HL), Further Integration, Areas, and Volumes of Revolution (HL), Kinematics (HL), Differential Equations: Separation of Variables, Slope Fields, Euler's Method, and Phase Portraits (HL)
- How to study for Unit 5 This page turns a function and its derivative with units, a rate graph with signed area over an interval, a differential equation with an initial condition and solution into one route: Name the changing quantities and their units, then match the requested meaning to a derivative, integral, or differential equation..
- The organizing decision Decide whether the question asks for instantaneous change, a stationary point, accumulated change, or an initial-value model.
What IB Mathematics: Applications and Interpretation Unit 5 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Limits, the Derivative as a Rate of Change, and Differentiating Polynomials
stationary point; accumulation; initial condition02Tangents and Normals
stationary point; accumulation; initial condition03Local Maximum and Minimum Points and Optimization
stationary point; accumulation; initial condition04Antiderivatives, Area Under a Curve, and the Trapezoidal Rule
stationary point; accumulation; initial condition05Chain, Product, and Quotient Rules, Related Rates, and the Second Derivative (HL)
stationary point; accumulation; initial condition06Further Integration, Areas, and Volumes of Revolution (HL)
stationary point; accumulation; initial condition07Kinematics (HL)
stationary point; accumulation; initial condition08Differential Equations: Separation of Variables, Slope Fields, Euler's Method, and Phase Portraits (HL)
stationary point; accumulation; initial conditionInternational Baccalaureate does not publish a per-unit weighting for this exam.
Official unit name and weighting: IB subject guide / subject brief.
Calculus: choose, calculate, interpret
Connect the published share to the unit model
This topic develops rates, accumulation, optimization, and differential equations. The official subject brief names the topic but does not publish a topic-by-topic examination percentage, so no percentage is claimed here on the “Calculus: choose, calculate, interpret” page.
Vocabulary in use: derivative, gradient function, stationary point, local maximum, definite integral, accumulation, differential equation, initial condition. Each term is applied on these pages and collected again in the glossary with a checked page pointer on the “Calculus: choose, calculate, interpret” page.
A strong calculus solution identifies the changing quantities, carries derivative or integral units, checks the domain, and interprets sign and magnitude. The demand, profit, tank, and culture situations in this topic are independently written illustrations.
The decision that organizes this unit
Define the system and choose the route before calculating
Decide whether the question asks for instantaneous change, a stationary point, accumulated change, or an initial-value model.
Name the changing quantities and their units, then match the requested meaning to a derivative, integral, or differential equation.
Mechanism route and repair branches
Relationships to preserve
- A derivative gives instantaneous change and carries output units per input unit.
- A definite integral accumulates a rate over an interval and returns the units of the changing quantity.
- An initial condition selects the particular solution of a differential equation.
Representations to read
- a function and its derivative with units
- a rate graph with signed area over an interval
- a differential equation with an initial condition and solution
Branches to reject
- dropping the units or meaning of a negative derivative
- classifying a stationary point without a sign test, second derivative, or endpoint comparison
- evaluating an integral without interpreting the accumulated quantity
| Key concept | Why it's hard | What scores |
|---|---|---|
| derivative | Distinguishing derivative from gradient function requires their definitions and units. | Define the quantity measured by derivative and attach its units before calculation. |
| stationary point | A calculator screen does not show why stationary point is the suitable model feature. | Record the relation used for stationary point and one identifying intermediate value. |
| accumulation | A computed accumulation value still needs a domain and scale check. | Check the restrictions, precision, and contextual meaning of the accumulation result. |
How IB Mathematics: Applications and Interpretation assesses Topic 5: Calculus
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Limits, the Derivative as a Rate of Change, and Differentiating Polynomials | stationary point; accumulation; initial condition | dropping the units or meaning of a negative derivative |
| Tangents and Normals | stationary point; accumulation; initial condition | classifying a stationary point without a sign test, second derivative, or endpoint comparison |
| Local Maximum and Minimum Points and Optimization | stationary point; accumulation; initial condition | evaluating an integral without interpreting the accumulated quantity |
| Antiderivatives, Area Under a Curve, and the Trapezoidal Rule | stationary point; accumulation; initial condition | dropping the units or meaning of a negative derivative |
| Chain, Product, and Quotient Rules, Related Rates, and the Second Derivative (HL) | stationary point; accumulation; initial condition | classifying a stationary point without a sign test, second derivative, or endpoint comparison |
| Further Integration, Areas, and Volumes of Revolution (HL) | stationary point; accumulation; initial condition | evaluating an integral without interpreting the accumulated quantity |
| Kinematics (HL) | stationary point; accumulation; initial condition | dropping the units or meaning of a negative derivative |
| Differential Equations: Separation of Variables, Slope Fields, Euler's Method, and Phase Portraits (HL) | stationary point; accumulation; initial condition | classifying a stationary point without a sign test, second derivative, or endpoint comparison |
Differentiate a contextual rate model
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Interpret a demand derivative
The weekly demand Q for a product is modeled by Q(p) = 900 − 35p, where p is the price in dollars.
Find dQ/dp and interpret its value in this context.
Show your working.
A. dQ/dp = −35 units per week per dollar; each one-dollar price increase is associated with 35 fewer units demanded per week.
✅ Correct! 😊 Correct. The derivative, its units and the meaning of its negative sign are all stated.
B. dQ/dp = 900 units per week because 900 is the initial demand.
❌ Not quite 😭 The value 900 is the intercept, not the rate of change. This does not differentiate the model.
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C. dQ/dp = 35 units per week per dollar; demand rises by 35 units for each extra dollar.
❌ Not quite 😭 This drops the negative sign and reverses the price-demand association, so the contextual interpretation is wrong.
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D. dQ/dp = −35 units per week; demand decreases.
❌ Not quite 😭 The numerical derivative is recognized, but the units omit ‘per dollar’ and the interpretation does not quantify a one-dollar change.
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Full explanation
- Step 1Differentiate Q(p) = 900 − 35p: the constant 900 becomes 0 and the linear term gives −35.
- Step 2Because Q is measured in units per week and p in dollars, dQ/dp has units of units per week per dollar.
- Step 3The negative sign means demand decreases: a one-dollar increase is associated with 35 fewer units demanded each week.
Practice 2: Accumulate an inflow rate
Water enters a reservoir at a rate r(t) = 6 + 0.4t liters per minute for 0 ≤ t ≤ 10.
Find the volume of water that enters the reservoir during the first 10 minutes.
Show your working.
Give your answer with appropriate units.
A. Evaluate r(10) = 10 liters.
❌ Not quite 😭 Substitution finds the inflow rate at 10 minutes, not the accumulated volume over the interval.
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B. Evaluate ∫ from 0 to 10 of (6 + 0.4t) dt = 80 liters.
✅ Correct! 😊 Correct. Integrating the rate across the stated interval gives accumulated volume, and the result is measured in liters.
C. Differentiate r to obtain 0.4 liters per minute².
❌ Not quite 😭 Differentiation gives how the inflow rate changes. It does not total the water entering during ten minutes.
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D. Calculate 10(6 + 0.4 × 10) = 100 liters.
❌ Not quite 😭 This treats the final rate as constant throughout the interval and overestimates the volume because the rate increases gradually.
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Full explanation
- Step 1The question asks for accumulated water from a rate, so use a definite integral over 0 ≤ t ≤ 10.
- Step 2An antiderivative of 6 + 0.4t is 6t + 0.2t².
- Step 3Evaluate [6t + 0.2t²] from 0 to 10 to obtain 60 + 20 = 80.
- Step 4Liters per minute multiplied through integration by minutes gives 80 liters entering the reservoir.
Key terms for Unit 5: Topic 5: Calculus
Models, uses, and boundaries
- Interpret a Constant Derivative
- The derivative of a linear function is its constant gradient; interpret that gradient with both variables' units. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Differentiate a Demand Model
- Differentiate the linear model to obtain dQ/dp = −35 units per week per dollar; use the negative sign to verify that the model decreases by 35 weekly units per dollar. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Classify a Stationary Point
- Solve P′(x) = −4x + 120 = 0 to obtain x = 30 and P(30) = 900; confirm a positive-to-negative derivative change or P″(x) = −4 and compare endpoints. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Accumulate a Rate
- Evaluate ∫ from 0 to 8 of (4 + 0.5t) dt as 32 + 16 = 48 L; check that integrating liters per minute over minutes produces liters. Use this relation when the named quantities and conditions in the question match the calculation described here.
- Solve an Initial-Value Model
- Use N(t) = Ce^(0.25t), apply C = 80, and calculate N(6) = 358.54; check that positive 0.25 gives growth and that the initial value returns 80. Use this relation when the named quantities and conditions in the question match the calculation described here.
IB Mathematics: Applications and Interpretation Unit 5 FAQ
How much of IB Mathematics: Applications and Interpretation does Unit 5 carry?
International Baccalaureate does not publish a per-unit weighting for this exam.
What is the first move on a Topic 5: Calculus problem?
Name the changing quantities and their units, then match the requested meaning to a derivative, integral, or differential equation.
Which relationships should I preserve?
A derivative gives instantaneous change and carries output units per input unit. A definite integral accumulates a rate over an interval and returns the units of the changing quantity. An initial condition selects the particular solution of a differential equation.
Which representations should I practice?
Practice moving among a function and its derivative with units, a rate graph with signed area over an interval, a differential equation with an initial condition and solution.
What error should I check before submitting an answer?
Check for dropping the units or meaning of a negative derivative; classifying a stationary point without a sign test, second derivative, or endpoint comparison; evaluating an integral without interpreting the accumulated quantity.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Differentiate a contextual rate model
- Classify a stationary point
- Interpret a definite integral
- Check and answer: Differentiate a contextual rate model · Interpret a definite integral
- Solve an initial-value change model
- Check the stationary point and change model
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related IB Mathematics: Applications and Interpretation unit guides
IB Mathematics: Applications and Interpretation Exam Guide & Review
The whole exam and its official unit sequence.01Topic 1: Number and Algebra
International Baccalaureate does not publish a per-unit weighting for this exam.02Topic 2: Functions
International Baccalaureate does not publish a per-unit weighting for this exam.03Topic 3: Geometry and Trigonometry
International Baccalaureate does not publish a per-unit weighting for this exam.04Topic 4: Statistics and Probability
International Baccalaureate does not publish a per-unit weighting for this exam.How to study IB Mathematics: Applications and Interpretation Unit 5
Start with the organizing decision
Before solving, restate the decision in operational terms: Decide whether the question asks for instantaneous change, a stationary point, accumulated change, or an initial-value model.. Your first written move should be to Name the changing quantities and their units, then match the requested meaning to a derivative, integral, or differential equation..
Practice the same idea in several representations
Rotate through a function and its derivative with units, a rate graph with signed area over an interval, a differential equation with an initial condition and solution. Use each representation to practice Limits, the Derivative as a Rate of Change, and Differentiating Polynomials, Tangents and Normals, Local Maximum and Minimum Points and Optimization, Antiderivatives, Area Under a Curve, and the Trapezoidal Rule, Chain, Product, and Quotient Rules, Related Rates, and the Second Derivative (HL), Further Integration, Areas, and Volumes of Revolution (HL), Kinematics (HL), Differential Equations: Separation of Variables, Slope Fields, Euler's Method, and Phase Portraits (HL), and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for dropping the units or meaning of a negative derivative; classifying a stationary point without a sign test, second derivative, or endpoint comparison; evaluating an integral without interpreting the accumulated quantity. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official IB subject guide / subject brief for the 2027 administration.
Where to go next
Return to the IB Mathematics: Applications and Interpretation to see every unit and plan your revision.