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Calculus, grade 10 (5 units)

The derivative: a number that says how fast a function changes at each point, and with it you find where the function rises, where it falls and where it reaches a peak.Subjects

The course builds on what you learnt in grades 7–9: the linear function, the parabola and its shifts, factoring and the quadratic formula. The topics, in order:

  • pre-analysis: the graph of a function, power functions, even and odd functions, a polynomial as a product of factors, inequalities solved from a graph and by intervals, shifts, stretches and reflections, the absolute value and composing functions;
  • the derivative: the average rate of change, the derivative at a point as a limit, the derivative function and the rules of differentiation, speed and acceleration;
  • the derivative of a polynomial and its uses: the tangent line, where a function rises and falls and its extremum points, investigating and sketching, the graph of a function and the graph of its derivative, the derivative of a product and of a composite function, and extremum problems;
  • the square root function: the inverse of x2\displaystyle x^2, its derivative, the tangent line and investigating it;
  • the rational function: the domain, vertical and horizontal asymptotes, a removable discontinuity, the derivative of a quotient, investigating it;
  • extremum problems with square root and rational functions.

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. The graph of a function: zeros, sign, rising and fallingLearningWhat a graph of a function tells: the points where it meets the axes, where it is positive and negative, where it rises and falls, and its extremum points; a recap of the linear and the quadratic function.
  2. Power functions, even and odd functionsLearningThe functions $y=x^n$: the graph for an even and for an odd power, what happens as $x$ grows or falls without end, and which power grows faster; a polynomial of odd degree and its zero; even and odd functions.
  3. A polynomial as a product of factorsLearningA qualitative look at a polynomial written as a product of linear and quadratic factors: its zeros and their multiplicity, a quadratic factor with no roots, the ends of the graph, a table of signs and a sketch.
  4. Inequalities: from a graph and by intervalsLearningSolving an inequality from a graph, the interval method for a polynomial written as a product of factors, the biquadratic equation and inequality, and a rational inequality of the form $\frac{f(x)}{g(x)}\geq0$.
  5. Shifts, stretches and reflectionsLearningThe graphs of $f(x)+k$, $f(x-p)$, $a\cdot f(x)$, $f(ax)$, $-f(x)$ and $f(-x)$; the family $a(x-p)^n+k$; what happens to the zeros and to the extremum points.
  6. The absolute value, operations on functions and compositionLearningThe function $|x|$ and the graph of $|f(x)|$; equations and inequalities with an absolute value; the sum, difference, product and quotient of functions; composing functions, the inner and the outer function.
  7. Practice: pre-analysisPracticeEquations and inequalities by the interval method, the biquadratic equation, even and odd functions, a polynomial as a product of factors, shifts and reflections, the absolute value and composing functions.
  8. Rate of change: uniform, varying and averageLearningA uniform and a varying rate of change; steps of equal width; the average rate of change – the slope of the secant; average speed; estimating a rate of change from a table.
  9. The derivative at a point: the limit of the secants' slopesLearningThe tangent – the line closest to the graph near a point; the slope of the secant as h tends to 0; the derivative at a point as a limit, the slope of the tangent and the instantaneous rate of change; instantaneous speed.
  10. The derivative function and the rules of differentiationLearningThe derivative as a function by the definition; the derivative of a power, of a constant, of a constant multiple and of a sum, and the proofs of the rules; stretching a graph and its slopes; speed and acceleration.
  11. The equation of the tangentLearningThe equation of the tangent to a graph at a point; a tangent of a given slope and a tangent parallel to a line; a horizontal tangent; where a tangent meets the axes; a tangent and a parameter.
  12. Where a function rises and falls, and extremum pointsLearningThe sign of the derivative decides where the function rises and where it falls; a point where the derivative is 0 may be an extremum; a table of signs; local and absolute extremum points, and an extremum at the end of a closed interval.
  13. Investigating a polynomialLearningThe steps of investigating a polynomial – the domain, the points on the axes, even and odd, extremum points, where it rises and falls, and a sketch; investigating in a closed interval, a parameter, and the number of solutions of the equation f(x)=k.
  14. The graph of a function and the graph of its derivativeLearningFrom the graph of a function to the graph of its derivative and back: rising and falling by the sign of the derivative, extremum points where it changes sign; an antiderivative of a polynomial, and a function from its derivative and a point.
  15. The derivative of a product and of a composite functionLearningThe rule for the derivative of a product of functions and its proof from the definition of the derivative; the composite function – the inner and the outer function, and its rule of differentiation; the derivative of (ax+b)ⁿ and of f(ax); tangents and extremum points.
  16. Extremum problems with polynomialsLearningThe steps of solving an extremum problem; a fence along a wall – also without the derivative, an open box from a sheet, numbers, a profit and the height of a ball; an open and a closed domain, and the meaning of the extremum.
  17. Practice: the derivative of a polynomial and its usesPracticeThe average rate of change, the derivative from the definition, the rules of differentiation, speed, the tangent line, investigating a polynomial, a closed interval, a parameter, the graph of the derivative, an antiderivative and an extremum problem.
  18. The square root function and the inverse functionLearningThe inverse function and the reflection in the line y=x; the square root function as the inverse of x² and the cube root function; √x²=|x|; shifts and stretches of √x; the domain of √f(x); equations with a root.
  19. The derivative of the square root functionLearningThe derivative of √x proved from the definition of the derivative and by the reflection in y=x; a vertical tangent at the origin; the derivative of √f(x) by the chain rule; the tangent to the graph of a root function.
  20. Investigating a square root functionLearningInvestigating √f(x) and a product with a root: the domain, extremum points at the ends of the domain, the derivative and a table of signs, where the function rises and falls, and a sketch.
  21. The rational function: domain, asymptotes and a removable pointLearningThe functions 1/x and 1/x², their shifts and asymptotes; the function 1/f(x) from f; a general rational function – the domain, a vertical asymptote, a removable discontinuity, a horizontal asymptote and the sign – before the derivative.
  22. The derivative of a quotient and the tangent to a rational functionLearningThe derivative of $\frac{1}{x}$ by the definition, the derivative of a quotient of functions from the product and chain rules, and the tangent to the graph of a rational function.
  23. Investigating a rational functionLearningThe steps of an investigation: the domain, asymptotes and a removable discontinuity, even or odd, the points on the axes, the derivative, a table of signs, extremum points, where it rises and falls, and a sketch.
  24. Extremum problems with square root and rational functionsLearningA rectangle of given area with the least perimeter, a box of given volume with the least area, the average cost, the shortest distance from a point to a graph, swimming and running; open and closed domains.
  25. Practice: square root and rational functionsPracticeEquations with a root, the domain, the derivatives of a root and of a quotient, tangents, asymptotes and a removable discontinuity, investigation and extremum problems.
  26. Review: calculusPracticeQuestions from the whole course: a polynomial as a product of factors and a reflection, the definition of the derivative, the tangent, investigating a polynomial, the derivative of a product, investigating a square root function and a rational function, and extremum problems.
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