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Calculus, grade 11 (5 units)

The derivative says how fast a function changes; in this course you also ask how that rate itself changes, and investigate new families of functions.Subjects

The course builds on the calculus of grade 10: the rules of differentiation, the tangent line, investigating a function, the square root function and the rational function. The topics, in order:

  • how the rate of change changes: concavity, the second derivative, inflection points, the second derivative and the type of an extremum, the graphs of f\displaystyle f, f′\displaystyle f' and f′′\displaystyle f'', acceleration and real-life problems;
  • rational functions: their behaviour near the points where they are not defined and at the ends of the domain, a full investigation with concavity, families with a parameter and tangent problems;
  • functions with square roots: the domain, different asymptotes at ∞\displaystyle \infty and at −∞\displaystyle -\infty, the derivative, equations with a root, investigation, and the graph of f(x)\displaystyle \sqrt{f(x)} from the graph of f\displaystyle f;
  • extremum problems on a closed and on an open interval, with graphs, in geometry and in solids;
  • the trigonometric functions in radians: their properties and graphs, the derivatives of sin⁡x\displaystyle \sin x, cos⁡x\displaystyle \cos x and tan⁡x\displaystyle \tan x, tangents, investigation and extremum problems.

Two kinds of lessons:

  • learning lessons – an explanation, solved examples and a few questions to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. Recap: the calculus of grade 10LearningWhat you learnt in grade 10: the rules of differentiation, the tangent line, investigating a function with a table of signs, the square root function and the rational function – the domain and asymptotes.
  2. Concavity: how the rate of change changesLearningWhere the slope of the graph grows the graph is concave up, where it shrinks concave down; the tangents and chords of a concave graph; a fall that grows steeper.
  3. The second derivative and concavityLearningThe second derivative – the rate of change of the derivative; its sign gives the concavity; the intervals of concavity of a polynomial and of a rational function.
  4. Inflection pointsLearningA point where the function changes its concavity; possible points – $f''=0$ or $f''$ not defined; a horizontal, a slanted and a vertical inflection; $f''=0$ with no inflection; every cubic has one inflection point.
  5. The second derivative and the type of an extremumLearningWhere $f'=0$: $f''>0$ – a minimum, $f''<0$ – a maximum; the rule works one way, and $f''=0$ gives no conclusion; other ways to tell the type of an extremum; a family with a parameter.
  6. The graphs of f, f′ and f″LearningWhat the graph of the derivative and the graph of the second derivative tell about the function: rising, extremum points, concavity and inflection; an extremum point of $f'$ is an inflection point of $f$; which graph is the function, which the derivative and which the second derivative.
  7. The rate of change of a rateLearningDistance, speed and acceleration – $a(t)=v'(t)=s''(t)$; prices that rise more slowly; the length of the day in Tel Aviv – when it grows fastest; filling a vessel; the fastest growth at the inflection point.
  8. Practice: the second derivativePracticeSecond derivatives, intervals of concavity, inflection points, the type of an extremum from $f''$, the graph of the derivative, acceleration and a family with a parameter.
  9. Rational functions near undefined points and at the endsLearningThe sign of a rational function on each side of a vertical asymptote, a horizontal asymptote by the degrees, approaching it from above or from below, a graph that crosses its horizontal asymptote, and a removable discontinuity.
  10. A full investigation of a rational functionLearningThe steps of an investigation with the second derivative: concavity and inflection points; x/(x²+1) with three inflection points; 1/(x²−1), whose concavity changes across its asymptotes with no inflection point; the oxygen in a lake.
  11. Rational functions: parameters and tangentsLearningA family of rational functions with a parameter – the parameter from an extremum point, the values with no extremum; the tangent at a point of the graph and tangents with a given slope; the asymptotes of a shifted graph.
  12. Practice: rational functionsPracticeAsymptotes and holes, inequalities, the second derivative and concavity, inflection points, tangents, a parameter, an extremum problem and a function from its graph.
  13. Functions with square roots: the domain and asymptotesLearningThe domain of a function with a root in a sum, a product, a numerator or a denominator; √(x²) = |x|; a function with different horizontal asymptotes at ∞ and at −∞, or with one on one side only; a vertical asymptote at the end of the domain.
  14. Derivatives of functions with roots, and root equationsLearningThe derivative of a product and of a quotient with a root, a vertical tangent at the end of the domain, equations with a root – squaring and checking every solution – and the mistake of taking x into the root.
  15. Investigating functions with square rootsLearningA full investigation of a function with a root, the second derivative included: x√(4−x²) – extremum points inside the domain and at its ends, and an inflection point; x/√(x²+1) – rising throughout, two asymptotes and an inflection point.
  16. The graph of √f(x) from the graph of fLearningWhat the graph of g(x) = √f(x) takes from the graph of f: the domain where f ≥ 0, the same zeros, extremum points at the same x inside the domain, vertical tangents at the zeros, where √f is above f and where below, and the asymptotes.
  17. Practice: functions with square rootsPracticeDomains, derivatives with roots, an equation with a root, two asymptotes, investigations with extremum points at the ends and an inflection point, a tangent, √f from f, the nearest point of a graph and a graph to recognise.
  18. Extremum problems on a closed and an open intervalLearningThe greatest and least values of a function on a closed interval – the extremum points and the ends of the interval; on an open interval there may be no such value; the point of a graph nearest a point; the scouts' camps – the shortest and the longest run.
  19. Extremum problems in geometry and solidsLearningA rectangle inscribed in a semicircle, a cylinder of given volume with the least surface area, a box from a sheet, a right-angled triangle with a given hypotenuse, the cost of a journey on a closed interval.
  20. Practice: extremum problemsPracticeThe greatest and least values on a closed and on an open interval, the shortest distance from a point to a graph, a fence by a wall, an open cylinder, a rectangle in a semicircle, the shortest way via a river.
  21. The trigonometric functions in radiansLearningRadians, the graphs of sine, cosine and tangent, the period, even and odd, zeros and asymptotes; shifts and stretches: y = a·sin(bx+c)+d; a period and the smallest period.
  22. The derivatives of the trigonometric functionsLearningThe limit of sin x over x, the derivative of sine by the definition, why radians, the derivatives of cosine and tangent, the chain rule and the tangent line.
  23. Investigating trigonometric functionsLearningAn investigation on an interval of one period: the derivative, the second derivative, extremum points, concavity and inflection points; x + 2sin x; √(sin x) and extremum points at the ends of the domain; the family y = sin x + ax; x²·sin x and its envelope.
  24. Extremum problems with trigonometric functionsLearningAn angle as the variable: the area of an isosceles triangle without a derivative, a trapezoid inscribed in a semicircle, a rectangle in a semicircle, a Ferris wheel – when the car rises fastest, a gutter with the greatest cross-section, a rectangle with a given diagonal.
  25. Practice: trigonometric functionsPracticeDerivatives of trigonometric functions, the tangent line, extremum and inflection points on an interval of a period, the period and the values, the second derivative, an extremum problem, √(cos x).
  26. Course reviewPracticeQuestions from the whole course: the second derivative, concavity and inflection points, the second derivative and the type of an extremum, the graph of f′, investigating a rational function and a function with a root, a tangent, an extremum problem and trigonometric functions.
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