Unit 5 · Topic 5: Calculus
The Complete IB Mathematics: Analysis and Approaches Guide · IB Mathematics: Analysis and ApproachesUnit 5 · Topic 5: Calculus
- The Complete IB Mathematics: Analysis and Approaches Guide
- IB Mathematics: Analysis and Approaches
- 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. Differentiation measures local change, while integration measures signed accumulation. Stationary points require classification, geometric area may require splitting at roots, and an optimization result must be compared with feasible endpoints.
- How IB Mathematics: Analysis and Approaches assesses this International Baccalaureate publishes no per-unit percentage for IB Mathematics: Analysis and Approaches; 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 graph with stationary points and signed regions, a derivative, bounded integral, or differential equation, a local Maclaurin series with its first omitted order
- Key skills Limits, the Derivative, and Rules of Differentiation, Tangents, Normals, Increasing and Decreasing Functions, Stationary Points, and Inflexion, Optimization and Kinematics, Antiderivatives and Integration by Inspection and Substitution, Definite Integrals and Areas Under and Between Curves, Continuity, L'Hôpital's Rule, Implicit Differentiation, and Related Rates (HL), Further Integration: Substitution, By Parts, Partial Fractions, and Volumes of Revolution (HL), Differential Equations and Maclaurin Series (HL)
- How to study for Unit 5 This page turns a function graph with stationary points and signed regions, a derivative, bounded integral, or differential equation, a local Maclaurin series with its first omitted order into one route: Write the relevant derivative, integral bounds, feasible interval, initial condition, or requested series order..
- The organizing decision Decide whether the task concerns local change, signed accumulation, optimization, a rate model, or a local series before choosing a calculus method.
What IB Mathematics: Analysis and Approaches Unit 5 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Limits, the Derivative, and Rules of Differentiation
limit; derivative; local slope02Tangents, Normals, Increasing and Decreasing Functions, Stationary Points, and Inflexion
stationary point; second derivative; inflexion03Optimization and Kinematics
feasible interval; global maximum; negative velocity04Antiderivatives and Integration by Inspection and Substitution
antiderivative; substitution; constant of integration05Definite Integrals and Areas Under and Between Curves
definite integral; signed area; integral bounds06Continuity, L'Hôpital's Rule, Implicit Differentiation, and Related Rates (HL)
continuity; implicit differentiation; related rate07Further Integration: Substitution, By Parts, Partial Fractions, and Volumes of Revolution (HL)
integration by parts; partial fractions; volume of revolution08Differential Equations and Maclaurin Series (HL)
differential equation; initial condition; Maclaurin seriesInternational Baccalaureate does not publish a per-unit weighting for this exam.
Official unit name and weighting: IB subject guide / subject brief.
Rates, accumulation, and local approximation
Connect the published share to the unit model
Differentiation measures local change, while integration measures signed accumulation. Stationary points require classification, geometric area may require splitting at roots, and an optimization result must be compared with feasible endpoints.
Kinematics links displacement, velocity, and acceleration by differentiation and integration. Integration technique is diagnosed from structure; differential equations retain equilibrium solutions and use initial conditions to select one curve.
A Maclaurin series describes local behavior near zero and must retain enough terms for the requested coefficient, approximation, or limit. Numerical results still need bounds, precision, and interpretation.
The decision that organizes this unit
Define the system and choose the route before calculating
Decide whether the task concerns local change, signed accumulation, optimization, a rate model, or a local series before choosing a calculus method.
Write the relevant derivative, integral bounds, feasible interval, initial condition, or requested series order.
Mechanism route and repair branches
Relationships to preserve
- A derivative describes local change, while a definite integral describes signed accumulation.
- A stationary point needs a sign or second-derivative test before it can be classified.
- An initial condition selects one solution from a family of differential-equation solutions.
Representations to read
- a function graph with stationary points and signed regions
- a derivative, bounded integral, or differential equation
- a local Maclaurin series with its first omitted order
Branches to reject
- calling every point with zero derivative a maximum or minimum
- changing a negative signed integral into geometric area without explanation
- dropping an integration constant before applying the initial condition
| Key concept | Why it's hard | What scores |
|---|---|---|
| Curve features | A zero derivative does not name the feature | Finds candidates and classifies with signs or a second derivative |
| Accumulation and models | Signed change can differ from total area | Shows bounds, technique, initial condition, and units |
| Series approximation | Cancellation can erase the first retained terms | Keeps sufficient order and states local use |
How IB Mathematics: Analysis and Approaches 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, and Rules of Differentiation | limit; derivative; local slope | calling every point with zero derivative a maximum or minimum |
| Tangents, Normals, Increasing and Decreasing Functions, Stationary Points, and Inflexion | stationary point; second derivative; inflexion | changing a negative signed integral into geometric area without explanation |
| Optimization and Kinematics | feasible interval; global maximum; negative velocity | dropping an integration constant before applying the initial condition |
| Antiderivatives and Integration by Inspection and Substitution | antiderivative; substitution; constant of integration | calling every point with zero derivative a maximum or minimum |
| Definite Integrals and Areas Under and Between Curves | definite integral; signed area; integral bounds | changing a negative signed integral into geometric area without explanation |
| Continuity, L'Hôpital's Rule, Implicit Differentiation, and Related Rates (HL) | continuity; implicit differentiation; related rate | dropping an integration constant before applying the initial condition |
| Further Integration: Substitution, By Parts, Partial Fractions, and Volumes of Revolution (HL) | integration by parts; partial fractions; volume of revolution | calling every point with zero derivative a maximum or minimum |
| Differential Equations and Maclaurin Series (HL) | differential equation; initial condition; Maclaurin series | changing a negative signed integral into geometric area without explanation |
Classify a stationary point and find signed area
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Classify a stationary point and retain signed area
Let f(x)=x³−3x.
(a) Classify the stationary point at x=1 and give its coordinates. (b) Find the signed accumulation from x=0 to x=1.
Show your working.
A. Since f′(1)=0 and f″(1)=6>0, (1,−2) is a local minimum; ∫ from 0 to 1 of f(x) dx=−5/4.
✅ Correct! 😊 Correct. The second derivative classifies the stationary point, and the bounded integral keeps its negative sign as signed accumulation.
B. Since f′(1)=0, (1,−2) is a local maximum; the signed accumulation is 5/4.
❌ Not quite 😭 A zero first derivative does not determine the classification. The positive second derivative indicates a minimum, and taking an absolute value changes signed accumulation into geometric area.
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C. Since f″(1)=6, (1,6) is a local minimum; the signed accumulation is −5/4.
❌ Not quite 😭 The second derivative value is not the function's y-coordinate. Evaluating f(1) gives −2, so the stated point is wrong.
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D. Since f(1)=−2, (1,−2) is a local minimum; ∫ from 0 to 1 of f′(x) dx=−5/4.
❌ Not quite 😭 The function value alone does not classify the point. Integrating f′ gives net change in f, not the requested accumulation of f.
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Full explanation
- Step 1Differentiate: f′(x)=3x²−3, so f′(1)=0 and x=1 is stationary.
- Step 2Since f″(x)=6x and f″(1)=6>0, the point is a local minimum; f(1)=−2.
- Step 3Evaluate ∫ from 0 to 1 of (x³−3x) dx=[x⁴/4−3x²/2] from 0 to 1=−5/4.
- Step 4The integral is negative because it is signed accumulation, not geometric area.
Practice 2: Use an initial condition in a rate model
A function y satisfies a separable differential equation.
(a) Solve dy/dx=3y given that y(0)=2. (b) Write the Maclaurin series for e³ˣ through the x² term.
Show your working.
State the first omitted order in the series.
A. Obtain y=Ae³ˣ, use y(0)=2 to get A=2, and write e³ˣ=1+3x+9x²/2 plus terms of order x³ and above.
✅ Correct! 😊 Correct. The integration constant is retained until the initial condition fixes it, and substituting 3x into the exponential series gives the stated coefficients.
B. Obtain y=e³ˣ+2 and write e³ˣ=1+3x+3x²/2 plus terms of order x³ and above.
❌ Not quite 😭 The constant multiplies the exponential solution rather than being added. The quadratic coefficient must contain (3x)², giving 9x²/2.
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C. Obtain y=2eˣ and write e³ˣ=1+3x+9x² plus terms of order x³ and above.
❌ Not quite 😭 The differential equation requires exponential rate 3, not 1. The factorial denominator 2! is also missing from the quadratic series term.
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D. Obtain y=Ae³ˣ and stop, because the initial condition is already represented by A; write only 1+3x.
❌ Not quite 😭 An undetermined A is a family, not the requested particular solution. Stopping before the x² term also fails the requested truncation order.
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Full explanation
- Step 1Separate variables to obtain dy/y=3 dx, then integrate to get ln|y|=3x+C.
- Step 2Write the solution as y=Ae³ˣ and apply y(0)=2, giving A=2 and y=2e³ˣ.
- Step 3Substitute u=3x into eᵘ=1+u+u²/2!+higher-order terms.
- Step 4Thus e³ˣ=1+3x+9x²/2 plus terms of order x³ and above.
Key terms for Unit 5: Topic 5: Calculus
Models, uses, and boundaries
- Curve derivative
- The derivative of x cubed minus 3x is 3x squared minus 3. Use it to locate stationary points before classifying them.
- Box volume
- The box volume is x times 20 minus 2x times 12 minus 2x. Use it on the feasible interval from zero to 6 centimeters.
- Integration by parts
- Integration by parts equals u times v minus the integral of v times du. Use it when the integrand has a product whose factors simplify after differentiation and integration.
- Separable rate equation
- Separating the rate equation gives dy divided by y equals 2 dx. Use it before applying the initial condition to select the particular solution.
- Local series
- The Maclaurin series for e to the power 2x begins 1 plus 2x plus 2x squared plus four-thirds x cubed. Use it near zero through the requested cubic term.
IB Mathematics: Analysis and Approaches Unit 5 FAQ
How much of IB Mathematics: Analysis and Approaches 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?
Write the relevant derivative, integral bounds, feasible interval, initial condition, or requested series order.
Which relationships should I preserve?
A derivative describes local change, while a definite integral describes signed accumulation. A stationary point needs a sign or second-derivative test before it can be classified. An initial condition selects one solution from a family of differential-equation solutions.
Which representations should I practice?
Practice moving among a function graph with stationary points and signed regions, a derivative, bounded integral, or differential equation, a local Maclaurin series with its first omitted order.
What error should I check before submitting an answer?
Check for calling every point with zero derivative a maximum or minimum; changing a negative signed integral into geometric area without explanation; dropping an integration constant before applying the initial condition.
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.
- Classify a stationary point and find signed area
- Optimize a box and interpret velocity
- Integrate by parts with visible choices
- Solve a rate equation and build a local series
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related IB Mathematics: Analysis and Approaches unit guides
IB Mathematics: Analysis and Approaches 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: Analysis and Approaches Unit 5
Start with the organizing decision
Before solving, restate the decision in operational terms: Decide whether the task concerns local change, signed accumulation, optimization, a rate model, or a local series before choosing a calculus method.. Your first written move should be to Write the relevant derivative, integral bounds, feasible interval, initial condition, or requested series order..
Practice the same idea in several representations
Rotate through a function graph with stationary points and signed regions, a derivative, bounded integral, or differential equation, a local Maclaurin series with its first omitted order. Use each representation to practice Limits, the Derivative, and Rules of Differentiation, Tangents, Normals, Increasing and Decreasing Functions, Stationary Points, and Inflexion, Optimization and Kinematics, Antiderivatives and Integration by Inspection and Substitution, Definite Integrals and Areas Under and Between Curves, Continuity, L'Hôpital's Rule, Implicit Differentiation, and Related Rates (HL), Further Integration: Substitution, By Parts, Partial Fractions, and Volumes of Revolution (HL), Differential Equations and Maclaurin Series (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 calling every point with zero derivative a maximum or minimum; changing a negative signed integral into geometric area without explanation; dropping an integration constant before applying the initial condition. 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: Analysis and Approaches to see every unit and plan your revision.