Exam Programme VWO Mathematics A The exam. The exam programme recognizes the following domains: Domain A Domain B Domain C Domain D Domain E Mathematical skills Algebra and systematic counting Relationships between variables Change Probability and statistics The exam topics per domain. Domain A: Mathematical skills The candidate is able to think and reason according to the mathematical programme. This includes problem modeling, translating a problem to an algebraic equivalent, ordering and structuring data, problem solving, the ability to manipulate formulas, and the ability to provide a proof by using logical reasoning. Domain B: Algebra and systematic counting Subdomain B1: Algebra The candidate is able to perform calculations on numbers and variables using arithmetic and algebra. Furthermore, the candidate can use the rules for powers, roots and logarithms and is able to work with parentheses. Subdomain B2: Problems involving counting The candidate is able to structure and schematize counting problems. He or she is able to work with factorials, permutations and combinations. Domain C: Relationships between functions Subdomain C1: Standard functions The candidate is able to recognize and work with linear functions, quadratic function, (higherorder) power functions, sine functions, exponential functions and logarithmic functions represented by a graph, a table or a functional description.
Subdomain C2: Functions, graphs, equalities and inequalities The candidate is able to create and manipulate functional descriptions of standard functions (see C1) in relation to a given context. He or she is also able to solve equations and inequalities either by applying algebraic techniques or where stated differently with a Graphing Calculator to get to an answer. He or she is also able to interpret results in relation to the given context. Domain D: Change Subdomain D1: Sequences The candidate is able to recognize the behaviour of a sequence and describe this using the notation uu nn. He or she is able to perform calculations with sequences, such as sums of terms in arithmetic and geometric sequences. Subdomain D2: Slopes The candidate is able to connect the change of a function to a differential quotient or the slope of a graph in a given context. Subdomain D3: Derivatives The candidate is able to find the derivative of power functions (linear functions, quadratic functions, root functions and higher order power functions), exponential functions and logarithmic functions. He or she is able to apply the rules of differentiation (product rule, quotient rule and chain rule.) Derivatives of exponential and logarithmic functions also include functions of the type ff(xx) = ee aaaabb or ff(xx) = ln(aaxx bb ). The candidate is able to describe the change of a function with its derivative function. The candidate is able to find the local extremes. Domain E: Probability and Statistics Subdomain E1: Research Context and Problem Statement The candidate is able to choose appropriate variables in order to obtain a statistical result of a problem statement within a certain research context. Subdomain E2: Representation of data The candidate is able to interpret data from a table or graph and evaluate these data on their merit.
Subdomain E3: Quantification of data The candidate is able to summarize data using measures of central tendency (mean, median and mode) and measures of spread (standard deviation) and interpret these numbers within the context. Subdomain E4: Probability The candidate is able to determine the probability of a certain random event from a diagram or by using combinatorics. The candidate is able to use the sum rule and the product rule for probabilities. Subdomain E5: Probability distributions The candidate is able to recognize when random variables behave according to a probability distribution. If so, the candidate is able to perform calculations on these distributions in cases where it involves a binomial or a normal distribution. The candidate is also able to compute the expected value and the standard deviation for these distributions and is able to use the square root N-law. Subdomain E6: Interferential statistics Within a context, the candidate is able to perform a statistical test. The candidate is able to o Determine whether the statistical test concerns a proportion (binomial) or a mean (normal) o Formulate a null and an alternative hypothesis. o Determine whether the testing procedure is one-sided or two-sided. o Use the result of the experiment or sample to calculate the pp-value. o Interpret the pp-value and draw a conclusion within the given context. Subdomain E7: Statistics and the use of the Graphing Calculator The candidate is familiar with the use of the Graphing Calculator with respect to subdomains E1-E6 in order to analyze small datasets.
Differentiation Exam VWO Math A Formula sheet rule function derivative sum rule ss(xx) = ff(xx) + (xx) ss (xx) = ff (xx) + (xx) difference rule ss(xx) = ff(xx) (xx) ss (xx) = ff (xx) (xx) product rule pp(xx) = ff(xx) (xx) pp (xx) = ff (xx) (xx) + ff(xx) (xx) quotiënt rule qq(xx) = ff(xx) (xx) qq (xx) = ff (xx) (xx) ff(xx) (xx) (xx) 2 chain rule kk(xx) = ff((xx)) kk (xx) = ff (xx) (xx) or dddd dddd = dddd dddd dddd dddd Rules for powers aa pp aa qq = aa pp+qq pp qq aa qq = aa pp (aa bb) pp = aa pp bb pp aa pp aa qq = aapp qq aa pp = 1 pp aa pp aa bb = aapp bb pp (aa pp ) qq = aa pppp Rules for logarithms rule + log (bb) = log (aaaa) condition > 0, 1, aa > 0, bb > 0 log (bb) = log aa bb > 0, 1, aa > 0, bb > 0 kk = log (aa kk ) > 0, 1, aa > 0 = pp pp > 0, 1, aa > 0, pp > 0, pp 1 log ()
Sequences The sum of an arithmetic sequence is given by: in which NN is the number of terms. SS = 1 2 NN(uu ffffffffff + uu llllllll ) The sum of a geometric sequence is given by: SS = uu llllllll+1 uu ffffffffff rr 1 with rr 1 Rules for random variables For the sum of two random variables XX and YY the expected value is given by: EE(XX + YY) = EE(XX) + EE(YY) For the sum two independent random variables XX and YY the standard deviation is given by: σσ(xx + YY) = σσ 2 (XX) + σσ 2 (YY) The nn-law: for the sum SS and the sample mean XX of nn independent and identically distributed random variables, it holds that: EE(SS) = nn EE(XX) EE(XX ) = EE(XX) σσ(ss) = nn σσ(xx) σσ(xx ) = σσ(xx) nn Binomial distribution The probability of kk successes in a binomial distribution XX, where nn is the number of trials and pp is the probability of a success in a single trial, is given by: PP(XX = kk) = nn kk ppkk (1 pp) nn kk Furthermore: EE(XX) = nn pp and σσ(xx) = nn pp (1 pp) Normal distribution If XX is normally distributed with mean μμ and standard deviation σσ, it holds that: ZZ = XX μμ σσ follows a standard normal distribution with: PP(XX ) = PP(ZZ μμ σσ )