Where the derivative comes from, what its sign tells you, and the two lines at a point. Each page has something that moves.
Sub-topics 5.1 to 5.5 are shared between Applications and Interpretation and Analysis and Approaches, and taken at both levels, so one set of pages serves four course variants. Where the two courses differ, the page says so.
All eight Standard Level sub-topics are here, in teaching order, not guide order. Everything to do with differentiating runs together, then integration and area arrive as a block of their own.
5.1 · Limits and the derivative Where a gradient at a single point comes from, estimating a limit from a table, and reading a derivative as a rate of change. Watch a chord collapse into a tangent while the gradient closes on 2. 5.2 · Reading the derivative Increasing, decreasing and stationary, sign tables, and local against global maximum. f and f′ move together, and a zero gradient turns out not to mean a maximum. 5.3 · Differentiating powers of x The power rule term by term, negative powers, constants, and the index limit that separates the two courses. 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, their equations, and the negative reciprocal. The usual slip draws a tangent that touches the curve nowhere at all. 5.6 · Stationary points Solving f′(x) = 0, giving the coordinates, and the difference between a local maximum and the greatest value on an interval. Drag the end of the interval and watch the highest point jump to the edge. 5.7 · Optimisation in context Building the function before differentiating it, rejecting the impossible root, and answering with units. Cut the corners off a sheet and watch the box fight itself for volume. 5.5 · Integration and area Anti-differentiation, the constant from a boundary condition, and the area under a curve. The region fills as you slide the limit, with the integral written above it. 5.8 · The trapezoidal rule Estimating an area from a table or a function, and saying in advance whether the estimate is too big or too small. Add strips and watch which side of the curve every chord sits on.Ten more sub-topics that only Higher Level students take. They sit on top of the eight above, so do those first. The last four build on each other in order: a slope field is a picture of a differential equation, Euler is that picture done numerically, a phase portrait is the same idea for two coupled quantities, and a second order equation becomes exactly that with one substitution.
5.9 · The chain, product and quotient rules HL The three rules, the standard derivatives, and related rates. A ripple whose radius grows steadily while its area does not. 5.10 · The second derivative and concavity HL Concavity, the second derivative test, and points of inflexion. f, f′ and f″ on one x, with the curve coloured by which way it bends. 5.11 · Integration by inspection and substitution HL The reverse chain rule, and the logarithm case that Standard Level excludes. Differentiate your answer back and watch the two readouts stay locked. 5.12 · Area about either axis, and volumes of revolution HL Discs of radius y, rotation about either axis, and signed against unsigned area. Spin the curve and the solid builds, one disc at a time. 5.13 · Kinematics HL Displacement, velocity, acceleration, and total distance travelled. It ends 1.33 m away having travelled 4 m. Same curve, two answers. 5.14 · Differential equations HL Turning a sentence into an equation, and separating the variables. A whole family of solutions, until one measurement picks one out. 5.15 · Slope fields HL Reading a field, sketching a solution on it, and long-term behaviour. Tap anywhere and a solution curve is traced through your point. 5.16 · Euler’s method HL Stepping along an equation you cannot solve, and how wrong it is. A staircase of straight lines closing in on the true curve. 5.17 · Phase portraits and eigenvalues HL Coupled systems, the five shapes, and reading them off the eigenvalues. Release a particle anywhere and watch where the system takes it. 5.18 · Second order equations HL One substitution turns them into a coupled system you already know. Sweep the damping and watch a closed loop become a spiral, then a slide.Displacement, velocity and acceleration questions are not set at Standard Level on this course. If you are revising at SL from a general calculus book, that is a chapter you can skip; at HL it is sub-topic 5.13 above.
AA has its own Topic 5, with nineteen sub-topics. Five of them, 5.1 to 5.5, are the shared pages above; the rest are different enough to need their own. Go to AA Topic 5.
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