# IBDP Maths Applications and Interpretation: Syllabus, Study Notes

### Topic 1: Number and algebra– SL content

• Topic : SL 1.1
• Operations with numbers in the form a × 10k where 1 ≤ a < 10 and k is an integer.
• Topic : SL 1.2
• Arithmetic sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence.
• Use of sigma notation for sums of arithmetic sequences.
• Applications.
• Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.
• approximate common differences.
• Topic : SL 1.3
• Geometric sequences and series Use of the formulae for the n th term and the sum of the first n terms of the sequence.
• Use of sigma notation for the sums of geometric sequences.
• Applications
• Examples include the spread of disease, salary increase and decrease and population growth
• Topic : SL 1.4
• Financial applications of geometric sequences and series:
• compound interest
• annual depreciation
• Topic : SL 1.5
• Laws of exponents with integer exponents.
• Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology.
• Topic : SL 1.6
• Approximation: decimal places, significant figures
• Upper and lower bounds of rounded numbers.
• Percentage errors.
• Estimation.
• Topic : SL 1.7
• Amortization and annuities using technology.
• Topic : SL 1.8
• Use technology to solve:
• Systems of linear equations in up to 3 variables
• Polynomial equations

### Topic 1: Number and algebra– AHL content

• Topic : AHL 1.9
• Laws of logarithms.
• logaxy = logax + logay
• loga$$\frac{x}{y}$$=logax – logay
• logaxm = mlogax for a, x, y > 0
• Topic : AHL 1.10
• Simplifying expressions, both numerically and algebraically, involving rational exponents.
• Topic : AHL 1.11
• The sum of infinite geometric sequences.
• Topic : AHL 1.12
• Complex numbers: the number i, where i2 = − 1.
• Cartesian form z = a + bi; the terms real part, imaginary part, conjugate, modulus and argument.
• The complex plane.
• Complex numbers as solutions to quadratic equations of the form ax2+bx+c=0, a≠0, with real coefficients where b2 – 4ac<0.
• Topic : AHL 1.13
• Modulus–argument (polar) form $$z = r\left( {\cos \theta + {\text{i}}\sin \theta } \right) = r{\text{cis}}\theta$$
• Exponential form:
• $$z =r{e^{{\text{i}}\theta }}$$.
• Conversion between Cartesian, polar and exponential forms, by hand and with technology.
• Calculate products, quotients and integer powers in polar or exponential forms.
• Adding sinusoidal functions with the same frequencies but different phase shift angles.
• Geometric interpretation of complex numbers.
• Topic : AHL 1.14
• Definition of a matrix: the terms element, row, column and order for m×n matrices.
• Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for m×n matrices.
• Multiplication of matrices.
• Properties of matrix multiplication: associativity, distributivity and non-commutativity.
• Identity and zero matrices
• Determinants and inverses of n×n matrices with technology, and by hand for 2×2 matrices.
• Awareness that a system of linear equations can be written in the form Ax=b.
• Solution of the systems of equations using inverse matrix.
• Topic : AHL 1.15
• Eigenvalues and eigenvectors.
• Characteristic polynomial of 2×2 matrices.
• Diagonalization of 2×2 matrices (restricted to the case where there are distinct real eigenvalues).
• Applications to powers of 2×2 matrices.

### Topic 2: Functions– SL content

• Topic: SL 2.1
• Different forms of the equation of a straight line.
• Lines with gradients m1 and m2
• Parallel lines m1 = m2.
• Perpendicular lines m1 × m2 = − 1.
• Topic: SL 2.2
• Concept of a function, domain, range and graph. Function notation, for example f(x), v(t), C(n). The concept of a function as a mathematical model.
• Informal concept that an inverse function reverses or undoes the effect of a function. Inverse function as a reflection in the line y = x, and the notation f−1(x).
• Topic: SL 2.3
• The graph of a function; its equation $$y = f\left( x \right)$$ .
• Creating a sketch from information given or a context, including transferring a graph from screen to paper. Using technology to graph functions including their sums and differences.
• Topic: SL 2.4
• Determine key features of graphs.
• Finding the point of intersection of two curves or lines using technology
• Topic: SL 2.5
• Modelling with the following functions:
• Linear models.
• f(x)=mx+c.
• f(x)=ax2+bx+c ; a≠0. Axis of symmetry, vertex, zeros and roots, intercepts on the x-axis and y -axis.
• Exponential growth and decay models.
• f(x)=kax+c
• f(x)=ka-x+c (for a>0)
• f(x)=kerx+c
• Equation of a horizontal asymptote.
• Direct/inverse variation:
• f(x)=axn, n∈ℤ
• The y-axis as a vertical asymptote when n<0.
• Cubic models:
• f(x)=ax3+bx2+cx+d.
• Sinusoidal models:
• f(x)=asin(bx)+d,
• f(x)=acos(bx)+d.
• Topic: SL 2.5
• Composite functions.
• (f ∘ g)(x) = f(g(x))
• Identity function.
• Finding the inverse function f−1(x)
• (f ∘ f−1)(x) = (f−1∘ f)(x) = x
• Topic: SL 2.6
• Modelling skills:
• Use the modelling process described in the “mathematical modelling” section to create, fit and use the theoretical models in section SL2.5 and their graphs.
• Develop and fit the model:
• Given a context recognize and choose an appropriate model and possible parameters.
• Determine a reasonable domain for a model.
• Find the parameters of a model.
• Test and reflect upon the model:
• Comment on the appropriateness and reasonableness of a model.
• Justify the choice of a particular model, based on the shape of the data, properties of the curve and/or on the context of the situation.
• Use the model:
• Reading, interpreting and making predictions based on the model
• Topic: SL 2.7
• The discriminant $$\Delta = {b^2} – 4ac$$ and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.
• Topic: SL 2.8
• The reciprocal function f(x) = 1 x , x ≠ 0: its graph and self-inverse nature.
• The rational function $$x \mapsto \frac{{ax + b}}{{cx + d}}$$ and its graph. Equations of vertical and horizontal asymptotes.

### Topic 2: Functions– AHL content

• Topic: AHL 2.7
• Composite functions in context.
• The notation (f∘g)(x)=f(g(x)).
• Inverse function f-1, including domain restriction.
• Finding an inverse function.
• Topic : AHL 2.8
• Transformations of graphs.
• Translations: y = f(x) + b; y = f(x − a).
• Reflections (in both axes): y = − f(x); y = f( − x).
• Vertical stretch with scale factor p: y= p f(x).
• Horizontal stretch with scale factor $$\frac{1}{q}$$: y = f(qx).
• Composite transformations.
• Topic : AHL 2.9
• In addition to the models covered in the SL content the AHL content extends this to include modelling with the following functions:
• Exponential models to calculate half-life.
• Natural logarithmic models:
• f(x)=a+blnx
• Sinusoidal models:
• f(x)=asin(b(x-c))+d
• Logistic models:
• $$f(x)=\frac{L}{1+Ce^{-kx}};L,C,k>0$$
• Piecewise models.
• Topic: AHL 2.10
• Scaling very large or small numbers using logarithms.
• Linearizing data using logarithms to determine if the data has an exponential or a power relationship using best-fit straight lines to determine parameters
• Interpretation of log-log and semi-log graphs.

### Topic 3: Geometry and trigonometry-SL content

• Topic : SL 3.1
• The distance between two points in three dimensional space, and their midpoint.
• Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids.
• The size of an angle between two intersecting lines or between a line and a plane.
• Topic SL 3.2
• Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
• The sine rule including the ambiguous case.
• $$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$$
• The cosine rule.
• $$c^2 = a^2 +b^2-2abcosC;$$
• $$cosC =\frac{a^2+ b^2-c^2}{2ab}$$
• Area of a triangle as $$\frac{1}{2}ab\sin C$$ .
• Topic SL 3.3
• Applications of right and non-right angled trigonometry, including Pythagoras’s theorem.
• Angles of elevation and depression.
• Construction of labelled diagrams from written statements.
• Topic SL 3.4
• length of an arc; area of a sector.
• Topic SL 3.5
• Equations of perpendicular bisectors.
• Topic SL 3.6
• Voronoi diagrams: sites, vertices, edges, cells.
• Addition of a site to an existing Voronoi diagram.
• Nearest neighbour interpolation.
• Applications of the “toxic waste dump” problem.

### Topic 3: Geometry and trigonometry-AHL content

• Topic : AHL 3.7
• The definition of a radian and conversion between degrees and radians.
• Using radians to calculate area of sector, length of arc.
• Topic : AHL 3.8
• The definitions of cosθ and sinθ in terms of the unit circle.
• The Pythagorean identity:
• cos2θ+sin2θ=1
• Definition of tanθ as $$\frac{sin\theta }{cos\theta }$$
• Extension of the sine rule to the ambiguous case.
• Graphical methods of solving trigonometric equations in a finite interval.
• Topic : AHL 3.9
• Geometric transformations of points in two dimensions using matrices: reflections, horizontal and vertical stretches, enlargements, translations and rotations.
• Compositions of the above transformations.
• Geometric interpretation of the determinant of a transformation matrix..
• Topic : AHL 3.10
• Concept of a vector; position vectors; displacement vectors.
• Representation of vectors using directed line segments.
• Unit vectors ; Base vectors i, j, k.
• Components of a vector: $$v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k$$ .
• The zero vector 0, the vector -v.
• Position vectors $$\vec{{OA}}=a$$
• Rescaling and normalizing vectors.
• Topic : AHL 3.11
• Vector equation of a line in two and three dimensions: $$r = a + \lambda b$$ .
• Topic : AHL 3.12
• Vector applications to kinematics.
• Modelling linear motion with constant velocity in two and three dimensions.
• Motion with variable velocity in two dimensions.
• Topic : AHL 3.13
• Definition and calculation of the scalar product of two vectors.
• The angle between two vectors; the acute angle between two lines.
• Definition and calculation of the vector product of two vectors.
• Geometric interpretation of |v×w|.
• Components of vectors.
• Topic : AHL 3.14
• Graph theory: Graphs, vertices, edges, adjacent vertices, adjacent edges. Degree of a vertex.
• Simple graphs; complete graphs; weighted graphs
• Directed graphs; in degree and out degree of a directed graph.
• Subgraphs; trees.
• Topic : AHL 3.15
• Walks.
• Number of k -length walks (or less than k -length walks) between two vertices.
• Construction of the transition matrix for a strongly-connected, undirected or directed graph.
• Topic : AHL 3.16
• Tree and cycle algorithms with undirected graphs.Walks, trails, paths, circuits, cycles.
• Eulerian trails and circuits.
• Hamiltonian paths and cycles.
• Minimum spanning tree (MST) graph algorithms:
• Kruskal’s and Prim’s algorithms for finding minimum spanning trees.
• Chinese postman problem and algorithm for solution, to determine the shortest route around a weighted graph with up to four odd vertices, going along each edge at least once.
• Travelling salesman problem to determine the Hamiltonian cycle of least weight in a weighted complete graph.
• Nearest neighbour algorithm for determining an upper bound for the travelling salesman problem.
• Deleted vertex algorithm for determining a lower bound for the travelling salesman problem.
•

### Topic 4 : Statistics and probability-SL content

• Topic: SL  4.1
• Concepts of population, sample, random sample and frequency distribution of discrete and continuous data.
• Reliability of data sources and bias in sampling.
• Interpretation of outliers.
• Sampling techniques and their effectiveness
• Topic: SL  4.2
• Presentation of data (discrete and continuous): frequency distributions (tables).
• Histograms. Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR).
• Production and understanding of box and whisker diagrams.
• Topic: SL  4.3
• Measures of central tendency (mean, median and mode).
• Estimation of mean from grouped data.
• Modal class.
• Measures of dispersion (interquartile range, standard deviation and variance).
• Effect of constant changes on the original data.
• Quartiles of discrete data.
• Topic: SL  4.4
• Linear correlation of bivariate data. Pearson’s product-moment correlation coefficient, r.
• Scatter diagrams; lines of best fit, by eye, passing through the mean point.
• Equation of the regression line of y on x.
• Use of the equation of the regression line for prediction purposes.
• Interpret the meaning of the parameters, a and b, in a linear regression y = ax + b.
• Topic: SL  4.5
• Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.
• The probability of an event $$A$$ is $$P\left( A \right) = \frac{{n\left( A \right)}}{{n\left( U \right)}}$$
• The complementary events $$A$$ and $${A’}$$ (not $$A$$).
• Expected number of occurrences.
• Topic: SL  4.6
• Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.
• Combined events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
• Mutually exclusive events: P(A ∩ B) = 0.
• Conditional probability; the definition $$P\left( {\left. A \right|P} \right) = \frac{{P\left( {A\mathop \cap \nolimits B} \right)}}{{P\left( B \right)}}$$.
• Independent events; the definition $$P\left( {\left. A \right|B} \right) = P\left( A \right) = P\left( {\left. A \right|B’} \right)$$ .
• Topic: SL  4.7
• Concept of discrete random variables and their probability distributions.
• Expected value (mean), for discrete data. Applications.
• Topic: SL  4.8
• Binomial distribution. Mean and variance of the binomial distribution.
• Topic: SL  4.9
• The normal distribution and curve.
• Properties of the normal distribution.
• Diagrammatic representation.
• Normal probability calculations.
• Inverse normal calculations
• Topic: SL  4.10
• Spearman’s rank correlation coefficient, rs.
• Awareness of the appropriateness and limitations of Pearson’s product moment correlation coefficient and Spearman’s rank correlation coefficient, and the effect of outliers on each.
• Topic: SL  4.11
• Formulation of null and alternative hypotheses, H0 and  H1.
• Significance levels.
• p -values.
• Expected and observed frequencies.
• The χ2 test for independence: contingency tables, degrees of freedom, critical value.
• The χ2 goodness of fit test.
• The t -test.
• Use of the p -value to compare the means of two populations.
• Using one-tailed and two-tailed tests.

### Topic 4 : Statistics and probability-AHL content

• Topic: AHL  4.12
• Design of valid data collection methods, such as surveys and questionnaires.
• Selecting relevant variables from many variables.
• Choosing relevant and appropriate data to analyse.
• Categorizing numerical data in a χ2 table and justifying the choice of categorisation.
• Choosing an appropriate number of degrees of freedom when estimating parameters from data when carrying out the χ2 goodness of fit test.
• Definition of reliability and validity.
• Reliability tests.
• Validity tests.
• Topic: AHL  4.13
• Non-linear regression.
• Evaluation of least squares regression curves using technology.
• Sum of square residuals (SSres) as a measure of fit for a model.
• The coefficient of determination (R2).
• Evaluation of R2 using technology.
• Topic: AHL  4.14
• Linear transformation of a single random variable.
• Expected value of linear combinations of n random variables.
• Variance of linear combinations of n independent random variables.
• $$\bar{x}) as an unbiased estimate of μ. • s2n-1 as an unbiased estimate of σ2. • Topic: AHL 4.15 • A linear combination of n independent normal random variables is normally distributed. In particular, • X~N(μ, σ2)⇒\(\bar{X}$$ ~N(μ,$$\frac{\sigma ^2}{n}$$)
• Central limit theorem.
• Topic: AHL  4.16
• Confidence intervals for the mean of a normal population.
• Topic: AHL  4.17
• Poisson distribution, its mean and variance.
• Sum of two independent Poisson distributions has a Poisson distribution.
• Topic: AHL  4.18
• Critical values and critical regions. Test for population mean for normal distribution.
• Test for proportion using binomial distribution.
• Test for population mean using Poisson distribution.
• Use of technology to test the hypothesis that the population product moment correlation coefficient (ρ) is 0 for bivariate normal distributions.
• Type I and II errors including calculations of their probabilities.
• Topic: AHL  4.19
• Transition matrices. Powers of transition matrices.
• Regular Markov chains.
• Initial state probability matrices.
• Calculation of steady state and long-term probabilities by repeated multiplication of the transition matrix or by solving a system of linear equations.

### Topic 5: Calculus-SL content

• Topic SL 5.1
• Introduction to the concept of a limit.
• Derivative interpreted as gradient function and as rate of change.
• Topic SL 5.2
• Increasing and decreasing functions.
• Graphical interpretation of f ′(x) > 0, f ′(x) = 0, f ′(x) < 0.
• Topic SL 5.3
• Derivative of f(x) = axn is f ′(x) = anxn−1 , n ∈ ℤ
• The derivative of functions of the form f(x) = axn + bxn−1 . . . . where all exponents are integers.
• Topic SL 5.4
• Tangents and normals at a given point, and their equations.
• Topic: SL 5.5
• Introduction to integration as anti-differentiation of functions of the form f(x) = axn + bxn−1 + …., where n ∈ ℤ, n ≠ − 1.
• Anti-differentiation with a boundary condition to determine the constant term.
• Definite integrals using technology.
• Area of a region enclosed by a curve y = f(x) and the x -axis, where f(x) > 0.
• Topic: SL 5.6
• Values of x where the gradient of a curve is zero.
• Solution of f′(x)=0. Local maximum and minimum points.
• Topic: SL 5.7
• Optimisation problems in context.
• Topic: SL 5.8
• Approximating areas using the trapezoidal rule.

### Topic 5: Calculus-AHL content

• Topic: AHL 5.9
• The derivatives of sin x, cos x, tan x, ex, lnx, xn where n∈ℚ.
• The chain rule, product rule and quotient rules.
• Related rates of change.
• Topic: AHL 5.10
• The second derivative.
• Use of second derivative test to distinguish between a maximum and a minimum point.
• Topic: AHL 5.11
• Definite and indefinite integration of xn where n∈ℚ, including n=-1 , sin x, cos x, $$\frac{1}{cos^2x}$$ and ex.
• Integration by inspection, or substitution of the form ∫f(g(x))g′(x)dx.
• Topic: AHL 5.12
• Area of the region enclosed by a curve and the x or y-axes in a given interval.
• Volumes of revolution about the x- axis or y- axis.
• Topic: AHL 5.13
• Kinematic problems involving displacement s, velocity v and acceleration a.
• Topic: AHL 5.14
• Setting up a model/differential equation from a context.
• Solving by separation of variables.
• Topic: AHL 5.15
• Slope fields and their diagrams.
• Topic: AHL 5.16
• Euler’s method for finding the approximate solution to first order differential equations.
• Numerical solution of $$\frac{dy}{dx}$$=f(x,y).
• Numerical solution of the coupled system$$\frac{dx}{dt}$$ =f1(x,y,t) and $$\frac{dy}{dt}$$ =f2(x,y,t).
• Topic: AHL 5.17
• Phase portrait for the solutions of coupled differential equations of the form:
• $$\frac{dx}{dt}$$=ax+by
• $$\frac{dy}{dt}$$=cx+dy.
• Qualitative analysis of future paths for distinct, real, complex and imaginary eigenvalues.
• Sketching trajectories and using phase portraits to identify key features such as equilibrium points, stable populations and saddle points.
• Topic: AHL 5.18
• Solutions of $$\frac{d^2x}{dt^2}=f(x,\frac{dx}{dt}$$,t) by Euler’s method.

### Paper 2

Topic 1 – Number and algebra

• Topic 1.0
• Basic use of the four operations of arithmetic, using integers, decimals and fractions, including order of operations
• Prime numbers, factors and multiples.
• Simple applications of ratio, percentage and proportion.
• Basic manipulation of simple algebraic expressions, including factorization and expansion
• Rearranging formulae
• Evaluating expressions by substitution.
• Solving linear equations in one variable.
• Solving systems of linear equations in two variables.
• Evaluating exponential expressions with integer values
• Use of inequalities $$<$$, $$\leqslant$$, $$>$$, $$\geqslant$$. Intervals on the real number line
• Intervals on the real number line
• Solving linear inequalities.
• Familiarity with commonly accepted world currencies
• Topic 1.1
• Natural numbers, $$\mathbb{N}$$ ; integers, $$\mathbb{Z}$$ ; rational numbers, $$\mathbb{Q}$$ ; and real numbers, $$\mathbb{R}$$ .
• Topic 1.2
• Topic 1.3
• Expressing numbers in the form $$a \times {10^k}$$ , where $$1 \le a < 10$$ and $$k$$ is an integer.
• Operations with numbers in this form.
• Topic 1.4
• SI (Système International) and other basic units of measurement: for example, kilogram ($${\text{kg}}$$), metre ($${\text{m}}$$), second ($${\text{s}}$$), litre ($${\text{l}}$$), metre per second ($${\text{m}}{{\text{s}}^{ – 1}}$$), Celsius scale.
• Topic 1.5
• Currency conversions.
• Topic 1.6
• Topic 1.7
• Topic 1.8
• Topic 1.9

Topic 2 – Descriptive statistics

Topic 3 – Logic, sets and probability

Topic 4 – Statistical applications

Topic 5 – Geometry and trigonometry

• Topic 5.0
• Basic geometric concepts: point, line, plane, angle
• Simple two-dimensional shapes and their properties, including perimeters and areas of circles, triangles, quadrilaterals and compound shapes.
• SI units for length and area.
• Pythagoras’ theorem
• Coordinates in two dimensions.
• Midpoints, distance between points
• Topic 5.1
• Equation of a line in two dimensions: the forms $$y = mx + c$$ and $$ax + by + d = 0$$ .Gradient; intercepts.
• Points of intersection of lines.
• Lines with gradients, $${m_1}$$ and $${m_2}$$ .Parallel lines $${m_1} = {m_2}$$.Perpendicular lines, $${m_1} \times {m_2} = – 1$$ .
• Topic 5.2
• Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
• Angles of elevation and depression.
• Topic 5.3
• Use of the sine rule: $$\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}}$$.Use of the cosine rule: $${a^2} = {b^2} + {c^2} – 2bc\cos A$$ ; $$\cos A = \frac{{{b^2} + {c^2} – {a^2}}}{{2bc}}$$.Use of the area of a triangle $$= \frac{1}{2}ab\sin C$$.Construction of labelled diagrams from verbal statements.
• Topic 5.4
• Geometry of three-dimensional solids: cuboid; right prism; right pyramid; right cone; cylinder; sphere; hemisphere; and combinations of these solids
• .The distance between two points; eg between two vertices or vertices with midpoints or midpoints with midpoints.
• The size of an angle between two lines or between a line and a plane.
• Topic 5.5
• Volume and surface areas of the three-dimensional solids defined in 5.4.

Topic 6 – Mathematical models

• Topic 6.1
• Concept of a function, domain, range and graph.
• Function notation, eg $$f\left( x \right)$$, $$v\left( t \right)$$, $$C\left( n \right)$$ .Concept of a function as a mathematical model.
• Topic 6.2
• Linear models.
• Linear functions and their graphs, $$f\left( x \right) = mx + c$$
• Topic 6.3
• Quadratic functions and their graphs (parabolas): $$f\left( x \right) = a{x^2} + bx + c$$ ; $$a \ne 0$$Properties of a parabola: symmetry; vertex; intercepts on the $$x$$-axis and $$y$$-axis.
• Equation of the axis of symmetry, $$x = \ – \frac{b}{{2a}}$$.
• Topic 6.4
• Exponential models.
• Exponential functions and their graphs: $$f\left( x \right) = k{a^x} + c$$; $$a \in {\mathbb{Q}^ + }$$, $$a \ne 1$$, $$k \ne 0$$ .Exponential functions and their graphs: $$f\left( x \right) = k{a^{ – x}} + c$$; $$a \in {\mathbb{Q}^ + }$$, $$a \ne 1$$, $$k \ne 0$$ .
• Concept and equation of a horizontal asymptote.
• Topic 6.5
• Models using functions of the form $$f\left( x \right) = a{x^m} + b{x^n} + \ldots$$; $$m,n \in \mathbb{Z}$$ .
• Functions of this type and their graphs.
• The $$y$$-axis as a vertical asymptote.
• Topic 6.6
• Drawing accurate graphs.
• Creating a sketch from information given.
• Transferring a graph from GDC to paper.
• Reading, interpreting and making predictions using graphs.
• Included all the functions above and additions and subtractions.
• Topic 6.7
• Use of a GDC to solve equations involving combinations of the functions above.

Topic 7 – Introduction to differential calculus

• Topic 7.1
• Concept of the derivative as a rate of change.
• Tangent to a curve.
• Topic 7.2
• The principle that $$f\left( x \right) = a{x^n} \Rightarrow f’\left( x \right) = an{x^{n – 1}}$$ .
• The derivative of functions of the form $$f\left( x \right) = a{x^n} + b{x^{n – 1}} + \ldots$$, where all exponents are integers.
• Topic 7.3
• Gradients of curves for given values of $$x$$.Values of $$x$$ where $$f’\left( x \right)$$ is given.
• Equation of the tangent at a given point.
• Equation of the line perpendicular to the tangent at a given point (normal).
• Topic 7.4
• Increasing and decreasing functions.
• Graphical interpretation of $$f’\left( x \right) > 0$$, $$f’\left( x \right) = 0$$ and $$f’\left( x \right) < 0$$.
• Topic 7.5
• Values of x where the gradient of a curve is zero.
• Solution of $$f’\left( x \right) = 0$$.
• Stationary points.
• Local maximum and minimum points.
• Topic 7.6
• Optimization problems.