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Algebra and calculus, grade 10 (4 units)

The derivative: a tool that shows where a function rises, where it falls and where it reaches a peak.Subjects

The course builds on grade 9’s algebra: factoring, algebraic fractions, the quadratic formula, the parabola and its shifts. The topics, in order:

  • algebraic technique: factoring, powers and roots, algebraic fractions;
  • equations and inequalities: of the first and second degree, systems, equations with fractions, with a square root, and the biquadratic equation;
  • pre-analysis: the properties of a function, even and odd functions, the functions y=xn\displaystyle {y}={x^n} and y=x\displaystyle {y}={\sqrt{x}}, shifts, reflections, stretches and the absolute value;
  • the derivative of a polynomial: the rate of change, the derivative at a point, the rules of differentiation, the tangent line, where a function rises and falls, extremum points, investigating and sketching, the graph of the derivative, and parameters;
  • the derivative of a square root function, and investigating a square root function;
  • extremum problems: numbers, economics, graphs and plane shapes.

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. Algebraic technique: factoring, powers and rootsLearningFactoring by a common factor, by the short multiplication formulas and by factoring a trinomial; reducing and adding algebraic fractions; the laws of powers; multiplying and dividing roots, taking a factor out of a root and putting one in.
  2. Equations and inequalities of the first and second degreeLearningFirst-degree and quadratic equations, the discriminant and the number of solutions, a system of two equations up to the second degree, first- and second-degree inequalities, and what each of them means on a graph.
  3. Equations with fractions, with a root and by substitutionLearningAn equation with fractions and its domain, an equation with one root or two – squaring and checking, the biquadratic equation and other equations by substitution, and the number of solutions as the number of meeting points of two graphs.
  4. The properties of a function: even and odd functionsLearningThe domain, the points where the graph meets the axes, where a function is positive and negative, where it increases and decreases, and the extremum points of the linear and the quadratic function; even and odd functions – by algebra and on the graph.
  5. Power functions and the square root functionLearningPower functions with a natural exponent – even and odd – and their properties; the square root function, its domain and an extremum point at the end of the domain; and an equation in which a power of x equals a number.
  6. Shifting and reflecting a graphLearningAdding a number to a function or to its x shifts its graph up, down or sideways; a minus sign before the function or before x reflects it in an axis.
  7. Stretching, shrinking and the absolute valueLearningMultiplying a function by a number stretches or shrinks its graph; the absolute value reflects the part below the x-axis above it; and several transformations together.
  8. Practice: algebra and functionsPracticeFactoring, roots, equations and inequalities of several kinds, even and odd functions, shifts, reflections, stretches and the absolute value.
  9. The rate of change and the derivative at a pointLearningA constant rate of change is the slope of a line; the average rate of change is the slope of the chord; bringing the second point closer to the first gives the derivative at the point – the slope of the tangent.
  10. The derivative function and the rules of differentiationLearningThe derivative at every point is a function of its own; the derivative of a power, of a constant, of a constant multiple and of a sum, and the value of the derivative at a point.
  11. The derivative of a product and of a composite functionLearningThe derivative of the product of two functions, and the derivative of a composite function in two steps: the outer derivative times the inner one.
  12. The equation of the tangentLearningThe tangent to a graph at a point: the slope from the derivative, the point from the function, and the equation of a line; the point where the tangent has a given slope, a horizontal tangent and a tangent parallel to a line.
  13. Where a function increases and decreases, and extremum pointsLearningThe sign of the derivative decides where the function increases and where it decreases; the points where the derivative is 0, a table of signs, local and absolute extremum points and an extremum at the end of the domain.
  14. Investigating a polynomialLearningThe steps of investigating a polynomial function – the domain, even or odd, the points on the axes, increasing and decreasing, extremum points – a sketch of the graph, and the number of solutions of the equation f(x)=k.
  15. The graph of the derivative, and parametersLearningWhat the graph of the derivative tells about the function, a sketch of the derivative's graph from the function's, finding a parameter from a tangent or an extremum point, and the extremum points after a shift and a reflection.
  16. Practice: the derivative and investigating a polynomialPracticeThe average rate of change, the derivative at a point, the rules of differentiation, the tangent line, where a function increases and decreases, extremum points, the graph of the derivative and parameters.
  17. The derivative of a square root functionLearningThe domain of the root of a function, the derivative of the square root and of the root of a function, sums and products of a root and a polynomial, and the tangent to the graph of a square root function.
  18. Investigating a square root functionLearningThe domain, the points where the graph meets the axes, where the function increases and decreases, extremum points – at an end of the domain too – and a sketch of the graph of a square root function.
  19. Extremum problemsLearningThe steps of an extremum problem on an open and on a closed domain: problems of numbers, economic problems, functions and graphs, and plane shapes.
  20. Review: algebra and calculusPracticeQuestions from the whole course: equations, transformations, the derivative and the tangent, investigating a polynomial and a square root function, the graph of the derivative, parameters and an extremum problem.
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