Chapter 2: Curve-sketching using the derivative Notes

Sections

1. 2A Increasing, decreasing and stationary at a point Mastered

   Using the sign of the derivative to decide whether a curve is increasing, decreasing or stationary at a point, finding where a function increases or decreases, and what this says about x-intercepts.

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2. 2B Stationary points and turning points Mastered

   Maximum and minimum turning points and stationary points of inflection, local maxima and minima, classifying stationary points with a table of slopes, and finding unknown coefficients from a stationary point.

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3. 2C Some less familiar curves Mastered

   Using the derivative on rational functions with asymptotes and discontinuities, and recognising vertical tangents and cusps where the derivative is undefined.

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4. 2D Second and higher derivatives Mastered

   The second, third and higher derivatives of a function, their notations, and finding successive derivatives of polynomials, negative powers and composite functions.

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5. 2E Concavity and points of inflection Mastered

   Concave up and concave down, using the second derivative to find concavity and points of inflection, and the second-derivative test for classifying stationary points.

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6. 2F Systematic curve sketching with the derivative Mastered

   A step-by-step routine for sketching curves: domain, symmetry, intercepts, asymptotes, stationary points and inflections, and using the sketch to solve polynomial inequations.

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7. 2G Global maximum and minimum Mastered

   The difference between local and global maxima and minima, and finding the global maximum and minimum of a function on a closed interval by checking stationary points and endpoints.

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8. 2H Continuity and differentiability Mastered

   What it means for a function to be continuous or differentiable at a point, testing piecewise-defined functions at the join, corners, cusps and vertical tangents, and finding constants that make a join smooth.

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9. 2I Applications of maximisation and minimisation Mastered

   A step-by-step method for optimisation problems: choosing a variable, writing the quantity as a function of one variable, finding and testing stationary points, checking endpoints, and answering in context.

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10. 2J Maximisation and minimisation in geometry Mastered

    Optimisation problems in geometry: shapes inscribed under curves and inside solids, using similar triangles or Pythagoras to link the variables, and minimising distance by minimising its square.

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11. 2K Primitive functions Mastered

    Reversing differentiation: primitives of powers of x including negative and fractional powers, the arbitrary constant, finding a particular primitive from an initial condition, and working back from a second derivative.

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