IBDP Maths Applications and Interpretation: IB Style Question Bank HL Paper-1

Paper 1

HL

  • Time: 150 minutes (150 marks)
  • No marks deducted from incorrect answers
  • A graphic display calculator is required for this paper.
  • A clean copy of the mathematics HL and further mathematics HL formula booklet is
    required for this paper

 New IBDP  Mathematics: applications and interpretation  HL Paper 1- Syllabus

Topic 1: Number and algebraSL content

  • Topic : SL 1.1
  • 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 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
  • 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 \(f(x)=\frac{{ax + b}}{{cx + d}}\) 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
  • Topic SL 3.4
  • 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

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Topic 4 : Statistics and probability-SL content

Topic 4 : Statistics and probability-AHL content

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
  • 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
  • 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.

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