IBDP Maths Applications and Interpretation: Syllabus, Study Notes

 New IBDP  Mathematics: applications and interpretation courses- Syllabus

Paper 2

Topic 1: Number and algebraSL 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: FunctionsSL content

  • Topic: SL 2.1
    • Different forms of the equation of a straight line.
    • Gradient; intercepts.
    • 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.
    • Quadratic models.
      • 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
    • Solution of quadratic equations and inequalities. The quadratic formula.
    • 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: FunctionsAHL 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
    • Adjacency matrices.
      • Walks.
      • Number of k -length walks (or less than k -length walks) between two vertices.
    • Weighted adjacency tables.
      • 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.

 Old DP mathematics Studies courses-Syllabus

Paper 2

Topic 1 – Number and algebra

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 models.
    • 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.

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