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<h3><span style="color:#B22222;">Analysis & Approaches syllabus content (student notes)</span></h3><p style="margin-left: 40px;"><img alt="" src="../../../ib/mathanalysis/analysis/basics/syllabus-content2-img-1.jpg" style="float: right; width: 260px; height: 147px;">click on topic to skip down to that section of the syllabus</p><p style="margin-left: 40px;"><a class="scroll-to" data-target="T1"><u>Topic 1</u>: Number and Algebra</a></p><p style="margin-left: 40px;"><a class="scroll-to" data-target="T2"><u>Topic 2</u>: Functions</a></p><p style="margin-left: 40px;"><a class="scroll-to" data-target="T3"><u>Topic 3</u>: Geometry and trigonometry</a></p><p style="margin-left: 40px;"><a class="scroll-to" data-target="T4"><u>Topic 4</u>: Probability and statistics</a></p><p style="margin-left: 40px;"><a class="scroll-to" data-target="T5"><u>Topic 5</u>: Calculus</a></p><p><span style="color:#000000;"></span><a class="scroll-to" data-target="T5"><span style="color:#0000FF;"></span></a></p><hr><h4><span style="color:#0000FF;"><strong><a class="anchor" id="T1" name="T1"> </a>■ Topic 1: Number and algebra</strong><span style="color:#000000;"></span></span></h4><p><span style="color:#000000;"><strong><u>1. SL core</u></strong></span></p><p><strong>SL 1.1*</strong> Operations with numbers in the form <span class="math-tex">\(a \times {10^k}\)</span> where <span class="math-tex">\(1 \le a < 10\)</span> and <em>k</em> is an integer.</p><p><strong>SL 1.2*</strong> Arithmetic sequences and series; use of the formulae for the <em>n</em>th term and the sum of the first <em>n </em>terms of the sequence; use of sigma notation for sums of arithmetic sequences; applications (e.g. simple interest); interpretation and prediction where a model is not perfectly arithmetic in real life.</p><p><strong>SL 1.3*</strong> Geometric sequences and series; use of the formulae for the <em>n</em>th term and the sum of the first <em>n </em>terms of the sequence; use of sigma notation for the sums of geometric sequences; applications such as spread of disease, salary increase and decrease, population growth.</p><p><strong>SL 1.4* </strong>Financial applications of geo. sequences and series including compound interest, annual depreciation.</p><p><strong>SL 1.5*</strong> Laws of exponents with integer exponents; introduction to logarithms with base 10 and <em>e</em>; numerical evaluation of logarithms using technology.</p><p><span style="color:#000000;"><u><strong>1. SL Analysis</strong></u></span></p><p><strong>SL 1.6</strong> Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side proof; the symbols and notation for equality and identity.</p><p><strong>SL 1.7</strong> Laws of exponents with rational exponents; laws of logarithms; change of base of a logarithm; solving exponential equations, including using logarithms.</p><p><strong>SL 1.8</strong> Sum of infinite convergent geometric sequences.</p><p><strong>SL 1.9</strong> The binomial theorem; binomial expansions; use of Pascal’s triangle and <span class="math-tex">\({}_n{C_r}\)</span> (compute using both the formula and technology).</p><p><span style="color:#000000;"><u><strong>1. HL Analysis</strong></u></span></p><p><strong>AHL 1.10</strong> Counting principles, including permutations and combinations; extension of the binomial theorem to fractional and negative indices.</p><p><strong>AHL 1.11</strong> Partial fractions; maximum of two distinct linear terms in the denominator, with degree of numerator less than the degree of the denominator.</p><p><strong>AHL 1.12</strong> Complex numbers; Cartesian form <span class="math-tex">\(z = a + b{\textrm{i}}\)</span> ; the terms real part, imaginary part, conjugate, modulus and argument; the complex plane.</p><p><strong>AHL 1.13</strong> Modulus–argument (polar) form <span class="math-tex">\(z = r\left( {\cos \theta + {\textrm{i}}\sin \theta } \right)\)</span> ; Euler form <span class="math-tex">\(z = r{e^{{\textrm{i}}\theta }}\)</span> ; sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.</p><p><strong>AHL 1.14</strong> Complex conjugate roots of quadratic and polynomial equations with real coefficients; De Moivre’s theorem and its extension to rational exponents; powers and roots of complex numbers.</p><p><a href="../30095/proofs-mathematical-induction-115.html" target="_blank"><strong>AHL 1.15</strong></a> Proof by mathematical induction; proof by contradiction; use of a counterexample to show that a statement is not always true.</p><p><strong>AHL 1.16</strong> Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinite number of solutions or no solution.</p><hr><h4><span style="color:#0000FF;"><strong><a class="anchor" id="T2" name="T2"> </a>■</strong><strong> Topic 2: Functions </strong></span></h4><p><span style="color:#000000;"><u><strong>2. SL core</strong></u></span></p><p><strong>SL 2.1*</strong> Different forms of the equation of a straight line; gradient; intercepts; parallel & perpendicular lines.</p><p><strong>SL 2.2*</strong> Concept of a function, domain, range & graph; function notation; concept of a function as a math model; concept of inverse function; inverse function as a reflection in the line <span class="math-tex">\(y = x\)</span>; the notation <span class="math-tex">\({f^{ - 1}}\left( x \right)\)</span></p><p><strong>SL 2.3*</strong> Graph of a function; creating a sketch from info given or a context, including transferring a graph from screen to paper; using technology to graph functions including their sums and differences.</p><p><strong>SL 2.4*</strong> Determine key features of graphs; finding point of intersection of two curves or lines using technology.</p><p><span style="color:#000000;"><u><strong>2. SL Analysis</strong></u></span></p><p><strong>SL 2.5</strong> Composite functions; identity function; finding the inverse function <span class="math-tex">\({f^{ - 1}}\left( x \right)\)</span></p><p><strong>SL 2.6</strong> Quadratic functions: graphs, y-intercept; axis of symmetry; factored form <span class="math-tex">\(f\left( x \right) = a\left( {x - p} \right)\left( {x - q} \right)\)</span>; x-intercepts; the form <span class="math-tex">\(f\left( x \right) = a{\left( {x - h} \right)^2} + k\)</span> with vertex <span class="math-tex">\(\left( {h,k} \right)\)</span></p><p><strong>SL 2.7</strong> Solving quadratic equations and inequalities; using factorization, completing the square (vertex form); quadratic formula; discriminant and nature of roots: 2 distinct real roots, 2 equal real roots, no real roots.</p><p><strong>SL 2.8</strong> The reciprocal function 1/x: its graph and self-inverse nature; rational functions of the form <span class="math-tex">\(\frac{{{\textrm{linear}}}}{{{\textrm{linear}}}}\)</span> and their graphs; equations of vertical and horizontal asymptotes.</p><p><strong>SL 2.9</strong> Exponential functions and their graphs; logarithmic functions and their graphs.</p><p><strong>SL 2.10</strong> Solving equations, both graphically and analytically; use of technology to solve a variety of equations, including those where there is no appropriate analytic approach; applications of graphing skills and solving equations that relate to real-life situations.</p><p><strong>SL 2.11</strong> Transformations of graphs; translations, reflections (in both axes), vertical stretches, horizontal stretches, composite transformations.</p><p><u><strong>2. HL Analysis</strong></u></p><p><strong>AHL 2.12</strong> Polynomial functions, their graphs and equations; zeros, roots and factors; the factor and remainder theorems; sum and product of the roots of polynomial equations.</p><p><strong>AHL 2.13</strong> Rational functions of the forms <span class="math-tex">\(\frac{{{\textrm{linear}}}}{{{\textrm{quadratic}}}}\)</span> and <span class="math-tex">\(\frac{{{\textrm{quadratic}}}}{{{\textrm{linear}}}}\)</span>; horizontal, vertical & oblique asymptotes.</p><p><strong>AHL 2.14</strong> Odd & even functions; finding inverse function, including domain restriction; self-inverse functions.</p><p><strong>AHL 2.15</strong> Solutions of inequalities, both graphically and analytically.</p><p><strong>AHL 2.16</strong> Graphs of the functions <span class="math-tex">\(y = \left| {f\left( x \right)} \right|\)</span> , <span class="math-tex">\(y = f\left( {\left| x \right|} \right)\)</span>, <span class="math-tex">\(y = \frac{1}{{f\left( x \right)}}\)</span>, <span class="math-tex">\(y = f\left( {ax + b} \right)\)</span>, <span class="math-tex">\(y = {\left[ {f\left( x \right)} \right]^2}\)</span>; solution of modulus equations and inequalities.</p><hr><h4><span style="color:#0000FF;"><strong><a class="anchor" id="T3" name="T3"> </a>■</strong><strong> Topic 3: Geometry and trigonometry </strong></span></h4><p><u><strong>3. SL core</strong></u></p><p><strong>SL 3.1*</strong> 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.</p><p><strong>SL 3.2*</strong> Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles; the sine rule (not including ambiguous case); the cosine rule; area of a triangle as <span class="math-tex">\({\textstyle{1 \over 2}}ab\sin C\)</span></p><p><strong>SL 3.3*</strong> Applications of right and non-right angled trigonometry, Pythagoras’ theorem (contexts may include use of bearings); angles of elevation & depression; construction of labelled diagrams from written statements.</p><p><u><strong>3. SL Analysis</strong></u></p><p><strong>SL 3.4</strong> The circle: radian measure of angles; length of an arc; area of a sector.</p><p><strong>SL 3.5</strong> Definition of sin<em>x</em> and cos<em>x</em> in terms of the unit circle; definition of tan<em>x</em> as sin<em>x</em> /cos<em>x</em>; exact values of trigonometric ratios of <span class="math-tex">\(0,{\textstyle{{\textrm{\pi }} \over 6}},{\textstyle{{\textrm{\pi }} \over 4}},{\textstyle{{\textrm{\pi }} \over 3}},{\textstyle{{\textrm{\pi }} \over 2}}\)</span> and their multiples; extension of the sine rule to the ambiguous case.</p><p><strong>SL 3.6</strong> The Pythagorean identity <span class="math-tex">\({\cos ^2}\theta + {\sin ^2}\theta = 1\)</span>; double angle identities for sine and cosine; the relationship between trigonometric ratios.</p><p><strong>SL 3.7</strong> The circular functions sin<em>x</em>, cos<em>x</em>, and tan<em>x</em>; amplitude, their periodic nature, and their graphs; composite functions of the form <span class="math-tex">\(f\left( x \right) = a\sin \left( {b\left( {x + c} \right)} \right) + d\)</span>; transformations; real-life contexts.</p><p><strong>SL 3.8</strong> Solving trigonometric equations in a finite interval, both graphically and analytically; equations leading to quadratic equations in sin<em>x</em>, cos<em>x</em>, or tan<em>x</em>.</p><p><u><strong>3. HL Analysis</strong></u></p><p><strong>AHL 3.9</strong> Definition of the reciprocal trigonometrical ratios; Pythagorean identities; the inverse trig functions; their domains and ranges; their graphs.</p><p><strong>AHL 3.10</strong> Compound angle identities; double angle identity for tan<em>x</em>.</p><p><strong>AHL 3.11</strong> Relationships between trigonometric functions and the symmetry properties of their graphs.</p><p><strong>AHL 3.12</strong> Concept of a vector; position vectors; displacement vectors; representation of vectors using directed line segments; base vectors <strong><em>i</em></strong>, <strong><em>j</em></strong>, <strong><em>k</em></strong>; components of a vector; algebraic and geometric approaches to following: sum and difference of two vectors, zero vector, multiplication by a scalar, parallel vectors, magnitude of a vector, unit vectors, position vectors, displacement vector; proofs of geometrical properties using vectors.</p><p><strong>AHL 3.13 </strong>The definition of the scalar product of two vectors; applications of the properties of the scalar product; the angle between two vectors; perpendicular vectors; parallel vectors.</p><p><strong>AHL 3.14</strong> Vector equation of a line in two and three dimensions; parametric form; Cartesian form; the angle between two lines; simple applications to kinematics.</p><p><strong>AHL 3.15</strong> Coincident, parallel, intersecting and skew lines, distinguishing between these cases; points of intersection.</p><p><strong>AHL 3.16</strong> The definition of the vector product of two vectors; properties of the vector product; geometric interpretations.</p><p><strong>AHL 3.17</strong> Vector equations of a plane: <span class="math-tex">\(r = a + \lambda b + \mu c\)</span> and <span class="math-tex">\(r \cdot n = a \cdot n\)</span>. Cartesian equation of a plane.</p><p><strong>AHL 3.18</strong> Intersections of: a line and a plane, 2 planes, 3 planes; angle between: a line and a plane, 2 planes.</p><h4><span style="color:#0000FF;"></span></h4><hr><h4><span style="color:#0000FF;"><strong><a class="anchor" id="T4" name="T4"> </a>■</strong><strong> Topic 4: Probability and statistics </strong></span></h4><p><u><strong>4. SL core</strong></u></p><p><strong>SL 4.1*</strong> Concepts of population, sample, random sample, discrete and continuous data; reliability of data sources and bias in sampling; interpretation of outliers; sampling techniques: simple random, convenience, systematic, quota and stratified methods.</p><p><strong>SL 4.2*</strong> Presentation of data (discrete and continuous); frequency histograms with equal class intervals; cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR); production and understanding of box and whisker diagrams; use of box and whisker diagrams to compare two distributions, using symmetry, median, interquartile range or range; determining whether data may be normally distributed by consideration of the symmetry of the box and whiskers.</p><p><strong>SL 4.3*</strong> 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.</p><p><strong>SL 4.4*</strong> Linear correlation of bivariate data; Pearson’s product-moment correlation coefficient, <em>r </em>; 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, <em>a </em>and <em>b</em>, in a linear regression.</p><p><strong>SL 4.5*</strong> Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (<em>U </em>) and event; the probability of an event <em>A</em>; the complementary events; expected number of occurrences.</p><p><strong>SL 4.6*</strong> Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities; combined events; mutually exclusive events; conditional probability; independent events.</p><p><strong>SL 4.7*</strong> Concept of discrete random variables and their probability distributions; expected value (mean), E(<em>X</em>) for discrete data; applications.</p><p><strong>SL 4.8*</strong> Binomial distribution; situations where the binomial distribution is an appropriate model; mean and variance of the binomial distribution.</p><p><strong>SL 4.9*</strong> The normal distribution and curve; properties of normal distribution; diagrammatic representation; normal probability calculations; inverse normal calculations (no use of standardized normal variable <em>z</em>).</p><p><u><strong>4. SL Analysis</strong></u></p><p><strong>SL 4.10 </strong>Equation of the regression line of <em>x </em>on <em>y</em>; use of equation for prediction purposes (aware of reliability)</p><p><strong>SL 4.11</strong> Formal definition and use of the formulae: for conditional probabilities, and for independent events; testing for independence.</p><p><strong>SL 4.12</strong> Standardization of normal variables (<em>z-</em>values); inverse normal calculations where mean and standard deviation are unknown.</p><p><u><strong>4. HL Analysis</strong></u></p><p><strong>AHL 4.13</strong> Use of Bayes’ theorem for a maximum of three events.</p><p><strong>AHL 4.14</strong> Variance of a discrete random variable; continuous random variables and their probability density functions. including piecewise functions; mode and median of continuous random variables; mean, variance and standard deviation of discrete and continuous random variables; the effect of linear transformations.</p><hr><h4><span style="color:#0000FF;"><strong><a class="anchor" id="T5" name="T5"> </a>■</strong></span><strong><span style="color:#0000FF;"> Topic 5: Calculus</span> </strong></h4><p><u><strong>5. SL core</strong></u></p><p><strong>SL 5.1*</strong> Introduction to concept of a limit; derivative interpreted as gradient function and as rate of change.</p><p><strong>SL 5.2*</strong> Increasing and decreasing functions: graphical interpretation of <span class="math-tex">\(f'\left( x \right) > 0\)</span>, <span class="math-tex">\(f'\left( x \right) = 0\)</span>, <span class="math-tex">\(f'\left( x \right) < 0\)</span></p><p><strong>SL 5.3*</strong> Power rule for differentiation; derivative of functions of the form <span class="math-tex">\(f\left( x \right) = a{x^n} + b{x^{n - 1}} + \cdots \)</span></p><p><strong>SL 5.4*</strong> Tangents and normals at a given point, and their equations.</p><p><strong>SL 5.5*</strong> Introduction to integration as anti-differentiation; definite integrals using technology; areas between a curve and the x-axis; anti-differentiation with a boundary condition to determine the constant term.</p><p><u><strong>5. SL Analysis</strong></u></p><p><strong>SL 5.6</strong> Derivative of <span class="math-tex">\({x^n}\left( {n \in \mathbb{Q}} \right),\;\sin x,\;\cos x,\;{e^x}\)</span> and <span class="math-tex">\(\ln x\)</span>; differentiation of a sum and a multiple of these functions; the chain rule for composite functions; the product and quotient rules.</p><p><strong>SL 5.7</strong> The second derivative; graphical behaviour of functions, including the relationship between the graphs of a function, its 1<sup>st</sup> derivative and its 2<sup>nd</sup> derivative.</p><p><strong>SL 5.8</strong> Local maximum and minimum points; testing for maximum and minimum; optimization; points of inflexion with zero and non-zero gradients.</p><p><strong>SL 5.9</strong> Kinematic problems involving displacement, velocity, acceleration and total distance travelled.</p><p><strong>SL 5.10</strong> Indefinite integral of <span class="math-tex">\({x^n}\left( {n \in \mathbb{Q}} \right),\;\sin x,\;\cos x,\;\frac{1}{x}\)</span> and <span class="math-tex">\({e^x}\)</span>; composites of any of these with the linear function <span class="math-tex">\(ax + b\)</span>; integration by inspection (reverse chain rule) or by substitution for expressions of the form <span class="math-tex">\(\int {k\,g'\left( x \right)} \,f\left( {g\left( x \right)} \right)dx\)</span></p><p><strong>SL 5.11</strong> Definite integrals, including analytical approach; areas of a region enclosed by a curve and the -axis, where the curve can be positive or negative, without the use of technology; areas between curves.</p><p><u><strong>5. HL Analysis</strong></u></p><p><strong>AHL 5.12</strong> Informal understanding of continuity and differentiability of a function at a point; understanding of limits (convergence and divergence); definition of derivative from first principles; higher derivatives.</p><p><strong>AHL 5.13</strong> The evaluation of limits of the form and using l’Hôpital’s rule; repeated use of l’Hôpital’s rule.</p><p><strong>AHL 5.14</strong> Implicit differentiation; related rates of change; optimisation problems.</p><p><strong>AHL 5.15</strong> Derivatives of all six trig functions and their inverses; indefinite integrals of the derivatives of any of these functions; composites of any of these with a linear function; use partial fractions to rearrange integrand</p><p><strong>AHL 5.16</strong> Integration by substitution; integration by parts; repeated integration by parts.</p><p><strong>AHL 5.17</strong> Area of the region enclosed by a curve and the <em>y</em>-axis in a given interval; volumes of revolution about the <em>x </em>-axis or <em>y </em>-axis.</p><p><strong>AHL 5.18</strong> First order differential equations; numerical solution of using Euler’s method; by separation of variables; homogeneous differential equations; solutions using the integrating factor method.</p><p><strong>AHL 5.19</strong> Maclaurin series; use of simple substitution, products, integration and differentiation to obtain other series; Maclaurin series developed from differential equations.</p><script>document.querySelectorAll('.tib-teacher-only').forEach(e => e.remove());</script>
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Please allow Pop Ups in your browser settings.</p></div>').insertBefore("body")),newWindow}if($(".open-student-access-help").click((function(e){e.preventDefault();var h=$(window).height()-20,w=$(window).width()<1280?$(window).width():1280;popupHelp(helpURL,"Student Access Help",w,h)})),$(".pop-up-help").click((function(e){e.preventDefault();var url=$(this).attr("href"),title=$(this).data("title"),h=$(window).height()-20,w;popupHelp(url,title,$(window).width()<1280?$(window).width():1280,h)})),$("section.tib-hiddenbox").length){var count=0;$($("section.tib-hiddenbox").get().reverse()).each((function(){var box=$(this),revealButton;$("<a />").attr("class","btn showhider").attr("rel","hiddenBoxContent"+count).attr("style","margin-bottom: 0;").html('<i class="fa fa-eye"></i>').insertBefore(box);var newContainer=$("<div />").attr("class","hidden-content").attr("id","hiddenBoxContent"+count).html(box.html());newContainer.hide(),newContainer.insertBefore(box),box.remove(),count++})),$("a.showhider").on("click",(function(e){var container=$("#"+$(this).attr("rel"));container.is(":hidden")?(container.fadeIn("fast"),$(this).html('<i class="fa fa-eye-slash"></i>')):(container.fadeOut("fast"),$(this).html('<i class="fa fa-eye"></i>'))}))}
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$('a.btn.showhider').click(function(e) {
var showHiderId = $( this ).attr('rel');
if( $('#'+showHiderId).find('iframe').length > 0 ) {
$('#'+showHiderId+' iframe').each(function() {
if ( $(this).attr('src').indexOf('.pdf') > 0 ) {
this.contentWindow.location.reload(true);
}
});
}
});
});
</script>
</body>
</html>