AA Topic 5: calculus

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Both IB guides state that 5.1 to 5.5 are common to Analysis and Approaches and Applications and Interpretation. Those five link to the shared pages rather than being written twice. Everything from 5.6 onwards is specific to this course.

Standard Level

5.1 · Limits and the derivative Where a gradient at a point comes from, estimating a limit, and reading a derivative as a rate of change. Watch a chord collapse into a tangent while the gradient closes on 2. 5.2 · Increasing, decreasing, stationary What the sign of f′ tells you, and what a zero gradient does not guarantee. A zero gradient turns out not to mean a maximum. 5.3 · Differentiating axⁿ The power rule term by term, with integer indices. The rule checked against a real chord, so it is not a recipe from nowhere. 5.4 · Tangents and normals Both lines at a point, and the negative reciprocal. The usual slip draws a tangent that touches the curve nowhere at all. 5.5 · Integration and area Anti-differentiation, the boundary condition, and area under a curve. The region fills as you slide the limit, with the integral written above it. 5.6 · Chain, product and quotient rules The three rules, the standard derivatives, and the index that becomes a fraction. Check any derivative you produce against a measured gradient. 5.7 · The second derivative and the three graphs How f, f′ and f″ line up, and reading one from another. Hide the labels and work out which graph is which. 5.8 · Maximum, minimum, optimisation, inflexion Two tests for a stationary point, optimisation in context, and inflexions with a non-zero gradient. Find the cheapest can, and meet the bend that is not a turn. 5.9 · Kinematics Displacement, velocity, acceleration and total distance travelled. Four metres away, having travelled twelve. 5.10 · Indefinite integrals and the reverse chain rule The standard integrals, composites with ax + b, and integration by inspection. Differentiate your answer back and watch the readouts stay locked. 5.11 · Definite integrals and areas between curves Analytic definite integrals, areas below the axis, and regions trapped between two curves. Slide the strip and watch top minus bottom close to nothing at each end.

Higher Level

Analysis and Approaches HL

Eight more sub-topics. The analytic definition of a derivative arrives here, which is why 5.1 was allowed to be numerical.

5.12 · The derivative from first principles HL The limit definition done with algebra, continuity and differentiability, and higher derivatives. The same chord as 5.1, with the algebra that proves the answer. 5.13 · l’Hôpital’s rule HL Indeterminate forms, repeated application, and when the rule does not apply. Zoom in until both curves become straight lines through the same point. 5.14 · Implicit differentiation HL Differentiating a curve that is not a function, related rates, and optimisation including endpoints. A circle has a gradient everywhere, and never needs rearranging. 5.15 · Further derivatives and partial fractions HL Inverse trigonometric derivatives, their integrals, and splitting a fraction before integrating it. Verify the whole table against a measured gradient rather than trusting it. 5.16 · Substitution and integration by parts HL Choosing u, repeated parts, and the integral that comes back to itself. See the wrong choice make the integral worse, side by side with the right one. 5.17 · Volumes of revolution HL Discs about either axis, and areas measured against the y-axis. Switch the axis and the same curve makes a completely different solid. 5.18 · First order differential equations HL Separable, homogeneous, integrating factor and Euler, and how to tell which is which. Pick an equation and the method, the reason and the working appear together. 5.19 · Maclaurin series HL The six standard series, and building new ones by substitution, product, differentiation and integration. Add terms and watch a polynomial grip the curve, further out each time.

If you teach both courses

The two syllabuses diverge in ways that are easy to get wrong, and three of them cost real time:

	Analysis and Approaches	Applications and Interpretation
Index n at SL	integer at 5.3, rational from 5.6	integer only, all of SL
Kinematics	Standard Level, 5.9	Higher Level only, 5.13
∫1/x dx	Standard Level, 5.10	Higher Level only, 5.11
First principles	Higher Level, 5.12	not on the course
Phase portraits	not on the course	Higher Level, 5.17

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