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Calculus, grade 11 (4 units)

Functions that are one polynomial divided by another, and root functions: what their graphs look like, where they rise and fall, and how to work out an area they enclose.Subjects

The course goes on from the calculus of grade 10 (4 units): the derivative of a polynomial and of a square root, the product and chain rules, the equation of a tangent, extremum points, even and odd functions, shifting and stretching a graph, and extremum problems. The topics, in order:

  • before the derivative: the function 1f(x)\displaystyle \frac{1}{f(x)}, f\displaystyle f a polynomial of degree at most 2, and transformations of a graph – shifts, reflections, stretches and absolute value;
  • rational functions: the domain, asymptotes parallel to the axes, the points on the axes, where the function is positive and negative, the quotient rule, the equation of a tangent, investigating a function, the derivative’s graph, even and odd functions and a parameter;
  • root functions: the derivative of f(x)g(x)\displaystyle \sqrt{\frac{f(x)}{g(x)}} and of g(x)⋅f(x)\displaystyle g(x)\cdot\sqrt{f(x)}, and an extremum at the end of the domain;
  • extremum problems: numbers, economics, functions and graphs, plane geometry and the box;
  • integral calculus: the antiderivative, the definite integral, the area between a graph and the x\displaystyle x-axis and the area between two graphs.

There are two kinds of lessons:

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

How the page works: Using a page.

  1. One over a functionLearningThe function $g(x)=\frac{1}{f(x)}$ from the properties of $f$: the domain, no point on the $x$-axis, the same sign, opposite rising and falling, the extremum point and the asymptotes.
  2. Transformations of a graphLearningHorizontal and vertical shifts, reflections in the axes, stretches and squeezes, absolute value – of $\frac{1}{f(x)}$; the asymptotes move with the graph.
  3. Practice: one over a function and transformationsPracticeThe domain, asymptotes, extremum point and rising and falling of $\frac{1}{f(x)}$; shifts, reflections, stretches and absolute value.
  4. The rational function: domain and asymptotesLearningOne polynomial divided by another, and its sum with a polynomial; the domain, a vertical asymptote at a zero of the denominator, and the horizontal asymptote from the degrees of the numerator and the denominator.
  5. Points on the axes, positive and negativeLearningThe points where the graph of a rational function meets the axes; a sign table; solving the inequalities $\frac{f(x)}{g(x)}>0$ and $\frac{f(x)}{g(x)}\geq0$.
  6. The derivative of a quotientLearningA reminder: the rules of differentiation from grade 10; the derivative of a quotient of two functions, of $\frac{c}{x}$ and of a polynomial plus a fraction; $f'(x)$ and $f'(a)$ of a given function.
  7. The tangent to a rational function's graphLearningThe equation of the tangent at a point on the graph; a tangent parallel to the $x$-axis or to a given line – the equation $f'(x)=m$; where a tangent meets the asymptotes.
  8. Investigating a rational functionLearningThe steps of an investigation: the domain, asymptotes, points on the axes, the points where $f'(x)=0$, the sign table of $f'$ – on both sides of a point outside the domain too –, where the function rises and falls, the kind of each extremum and a sketch.
  9. The derivative's graph, parity and a parameterLearningWhat the graph of the derivative shows about the function; even and odd rational functions – algebraically and on the graph; finding a parameter from an asymptote, an extremum point and the slope of a tangent.
  10. Practice: rational functionsPracticeInvestigating rational functions in full: the domain, asymptotes, the derivative, extremum points, rising and falling; parity; tangents; a parameter; reading a function from its derivative's graph.
  11. The derivative of a root functionLearningThe functions $\sqrt{\frac{f(x)}{g(x)}}$ and $g(x)\cdot\sqrt{f(x)}$: the domain – an inequality of a fraction, the derivative by the root, product and quotient rules, and the equation of a tangent.
  12. Investigating a root functionLearningInvestigating a root function: the domain, the derivative, extremum points inside the domain and at its end, where it rises and falls, the points on the axes and a sketch.
  13. Extremum problems: numbers and economicsLearningThe way to solve an extremum problem: a variable, a function, its interval, the derivative, its sign and the answer; a number and its reciprocal, the largest fraction, the average cost of an item, on an open and on a closed interval.
  14. Extremum problems: geometry and the boxLearningExtremum problems in plane geometry – an isosceles triangle with a given perimeter, a rectangle with a vertex on a graph; the shortest distance from a point to a graph; a box with a given volume and the least surface area, and its lateral area.
  15. Practice: root functions and extremum problemsPracticeThe derivative and the domain of a root function, an investigation with an extremum at the end of the domain, a tangent; extremum problems – numbers, the average cost, a box, a rectangle in a semicircle and a closed interval.
  16. The antiderivativeLearningThe antiderivative and the indefinite integral: the inverse of differentiating, the constant $C$, the integral of a polynomial and of $\frac{c}{(ax+b)^2}$, and an antiderivative through a point.
  17. The definite integral and the area to the x-axisLearningThe definite integral $F(b)-F(a)$; the area between a graph and the $x$-axis – above the axis, below it, and a function that changes sign.
  18. The area between two graphsLearningThe area between two graphs – the integral of the upper minus the lower between the meeting points; a composite area of a graph, a tangent and an axis; an area with a parameter.
  19. Practice: integrals and areasPracticeIntegrals of a polynomial and of $\frac{c}{(ax+b)^2}$, an antiderivative through a point, the area between a graph and the $x$-axis, the area between two graphs, a graph and a tangent, and a parameter.
  20. Review: calculusPracticeQuestions from the whole course: one over a function and a shift, a tangent and an area, investigating a rational function, an inequality, a parameter, an extremum problem with a root and an antiderivative.
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