November 2014

• Informal ideas of limit and convergence.
• Limit notation.
• Definition of derivative from first principles as
• Derivative interpreted as gradient function and as rate of change.
• Tangents and normals, and their equations.

Not required:

• analytic methods of calculating limits.

Example: 0.3, 0.33, 0.333, ... converges to 1/3
.Technology should be used to explore ideas of limits, numerically and graphically.

Example:

• Links to 1.1, infinite geometric series; 2.5–2.7, rational and exponential functions, and asymptotes.
• Use of this definition for derivatives of simple polynomial functions only.
• Technology could be used to illustrate other derivatives.

Link to 1.3, binomial theorem.

• Use of both forms of notation, 
• for the first derivative.
• Identifying intervals on which functions are increasing or decreasing.
• Use of both analytic approaches and technology.
• Technology can be used to explore graphs and their derivatives.

• Appl: Economics 1.5 (marginal cost, marginal revenue, marginal profit).
• Appl: Chemistry 11.3.4 (interpreting the gradient of a curve).
• Aim 8: The debate over whether Newton or Leibnitz discovered certain calculus concepts.
• TOK: What value does the knowledge of limits have? Is infinitesimal behaviour applicable to real life?
• TOK: Opportunities for discussing hypothesis formation and testing, and then the formal proof can be tackled by comparing certain cases, through an investigative approach.

• Derivative of e^x and ln x .
• Differentiation of a sum and a real multiple of these functions.
• The chain rule for composite functions.
• The product and quotient rules.
• The second derivative.
• Extension to higher derivatives.

• Link to 2.1, composition of functions.
• Technology may be used to investigate the chain rule.
• Use of both forms of notation,

• Local maximum and minimum points. Testing for maximum or minimum.
• Points of inflexion with zero and non-zero gradients.
• Graphical behaviour of functions, including the relationship between the graphs of f , f ′ and f ′′ .
• Optimization.
• Applications.

Not required:

• points of inflexion where f ′′(x) is not defined: for example, y = x¹³ at (0,0).

• Using change of sign of the first derivative and using sign of the second derivative.
• Use of the terms “concave-up” for f ′′(x) > 0 ,and “concave-down” for f ′′(x) < 0 .
• At a point of inflexion , f ′′(x) = 0 and changes sign (concavity change).
• f ′′(x) = 0 is not a sufficient condition for a point of inflexion: for example, y = x4 at (0,0) .
• Both “global” (for large |x| ) and “local” behaviour.
• Technology can display the graph of a derivative without explicitly finding an expression for the derivative.
• Use of the first or second derivative test to justify maximum and/or minimum values.
• Examples include profit, area, volume.
• Link to 2.2, graphing functions.

• Indefinite integration as anti-differentiation.
• Indefinite integral of
• The composites of any of these with the linear function ax + b .
• Integration by inspection, or substitution of the form ∫ f (g(x))g '(x)dx .

• Anti-differentiation with a boundary condition to determine the constant term.
• Definite integrals, both analytically and using technology.
• Areas under curves (between the curve and the x-axis).
• Areas between curves.
• Volumes of revolution about the x-axis.

• 
• The value of some definite integrals can only be found using technology.
• Students are expected to first write a correct expression before calculating the area.
• Technology may be used to enhance understanding of area and volume.

• Int: Successful calculation of the volume of the pyramidal frustum by ancient Egyptians (Egyptian Moscow papyrus).
• Use of infinitesimals by Greek geometers.
• Accurate calculation of the volume of a cylinder by Chinese mathematician Liu Hui
• Int: Ibn Al Haytham: first mathematician to calculate the integral of a function, in order to find the volume of a paraboloid.

• Kinematic problems involving displacement s, velocity v and acceleration a.
• Total distance travelled.

• Concepts of population, sample, random sample, discrete and continuous data.
• Presentation of data: frequency distributions (tables); frequency histograms with equal class intervals;
• box-and-whisker plots; outliers.
• Grouped data: use of mid-interval values for calculations; interval width; upper and lower interval boundaries; modal class.

Not required:

• frequency density histograms.

• Continuous and discrete data.
• Outlier is defined as more than 1.5 × IQR from the nearest quartile.
• Technology may be used to produce histograms and box-and-whisker plots.

• Appl: Psychology: descriptive statistics, random sample (various places in the guide).
• Aim 8: Misleading statistics.
• Int: The St Petersburg paradox, Chebychev, Pavlovsky.

• Statistical measures and their interpretations.
• Central tendency: mean, median, mode.
• Quartiles, percentiles.
• Dispersion: range, interquartile range, variance, standard deviation.
• Effect of constant changes to the original data.
• Applications.

• On examination papers, data will be treated as the population.
• Calculation of mean using formula and technology. Students should use mid-interval values to estimate the mean of grouped data.
• Calculation of standard deviation/variance using only technology.

Link to 2.3, transformations.
Examples:

• If 5 is subtracted from all the data items, then the mean is decreased by 5, but the standard deviation is unchanged.
• If all the data items are doubled, the median is doubled, but the variance is increased by a factor of 4.

• Appl: Psychology: descriptive statistics (various places in the guide).
• Appl: Statistical calculations to show patterns and changes; geographic skills; statistical graphs.
• Appl: Biology 1.1.2 (calculating mean and standard deviation ); Biology 1.1.4 (comparing means and spreads between two or more samples).
• Int: Discussion of the different formulae for variance.
• TOK: Do different measures of central tendency express different properties of the data? Are these measures invented or discovered? Could mathematics make alternative, equally true, formulae? What does this tell us about mathematical truths?
• TOK: How easy is it to lie with statistics?

• Linear correlation of bivariate data.
• Pearson’s product–moment correlation coefficient r.
• Scatter diagrams; lines of best fit.
• Equation of the regression line of y on x.
• Use of the equation for prediction purposes.
• Mathematical and contextual interpretation.

Not required:

• the coefficient of determination R².

• Independent variable x, dependent variable y.
• Technology should be used to calculate r. However, hand calculations of r may enhance understanding.
• Positive, zero, negative; strong, weak, no correlation.
• The line of best fit passes through the mean point.
• Technology should be used find the equation.
• Interpolation, extrapolation.

• Appl: Chemistry 11.3.3 (curves of best fit).
• Appl: Geography (geographic skills). Measures of correlation; geographic skills.
• Appl: Biology 1.1.6 (correlation does not imply causation).
• TOK: Can we predict the value of x from y, using this equation?
• TOK: Can all data be modelled by a (known) mathematical function? Consider the reliability and validity of mathematical models in describing real-life phenomena.

• Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
• The probability of an event
• The complementary events A and A′ (not A).
• Use of Venn diagrams, tree diagrams and tables of outcomes.

• The sample space can be represented diagrammatically in many ways.
• Experiments using coins, dice, cards and so on, can enhance understanding of the distinction between (experimental) relative frequency and (theoretical) probability.
• Simulations may be used to enhance this topic.
• Links to 5.1, frequency; 5.3, cumulative frequency.

• Combined events, P(A∪ B).
• Mutually exclusive events: P(A∩ B) = 0 .Conditional probability; the definition
• Independent events; the definition P( A| B) = P(A) = P( A| B′)  .
• Probabilities with and without replacement.

• The non-exclusivity of “or”.
• Problems are often best solved with the aid of a Venn diagram or tree diagram, without explicit use of formulae.

• Aim 8: The gambling issue: use of probability in casinos. Could or should mathematics help increase incomes in gambling?
• TOK: Is mathematics useful to measure risks?
• TOK: Can gambling be considered as an application of mathematics? (This is a good opportunity to generate a debate on the nature, role and ethics of mathematics regarding its applications.)

• Concept of discrete random variables and their probability distributions.
• Expected value (mean), E(X ) for discrete data.
• Applications.

• Simple examples only, such as:
• E(X ) = 0 indicates a fair game where X represents the gain of one of the players.
• Examples include games of chance.

• Binomial distribution.
• Mean and variance of the binomial distribution.

Not required:

• formal proof of mean and variance.

• Conditions under which random variables have this distribution.
• Technology is usually the best way of calculating binomial probabilities.

• Normal distributions and curves.
• Standardization of normal variables (z-values, z-scores).
• Properties of the normal distribution.

• Probabilities and values of the variable must be found using technology.
• Link to 2.3, transformations.
• The standardized value ( z ) gives the number of standard deviations from the mean.

• Appl: Biology 1.1.3 (links to normal distribution).
• Appl: Psychology: descriptive statistics (various places in the guide).