Topic 5: Calculus

Concepts

  • Essential understandings:

    • Calculus describes rates of change between two variables and the accumulation of limiting areas. Understanding these rates of change and accumulations allow us to model, interpret and analyze real- world problems and situations. Calculus helps us to understand the behaviour of functions and allows us to interpret the features of their graphs.

  • Suggested concepts embedded in this topic:

    • Change, patterns, relationships, approximation, generalization, space, modelling.

    • AHL: Systems, quantity.

  • Content-specific conceptual understandings:

    • The derivative may be represented physically as a rate of change and geometrically as the gradient or slope function.

    • Areas under curves can be can be approximated by the sum of the areas of rectangles which may be calculated even more accurately using integration.

    • Examining rates of change close to turning points helps to identify intervals where the function increases/decreases, and identify the concavity of the function.

    • Numerical integration can be used to approximate areas in the physical world.

    • Mathematical modelling can provide effective solutions to real-life problems in optimization by maximizing or minimizing a quantity, such as cost or profit.

    • Derivatives and integrals describe real-world kinematics problems in two and three-dimensional space by examining displacement, velocity and acceleration.

    • AHL: Some functions may be continuous everywhere but not differentiable everywhere.

    • AHL: A finite number of terms of an infinite series can be a general approximation of a function over a limited domain.

    • AHL: Limits describe the output of a function as the input approaches a certain value and can represent convergence and divergence.

    • AHL: Examining limits of functions at a point can help determine continuity and differentiability at a point.

SL Content

  • SL 5.1

    • Introduction to the concept of a limit.

    • Derivative interpreted as gradient function and as rate of change.

  • SL 5.2

    • Increasing and decreasing functions.

    • Graphical interpretation of f(x)>0,f(x)=0,f(x)<0f'(x) > 0, f'(x)=0, f'(x) < 0

  • SL 5.3

    • Derivative of f(x)=axnf(x)=ax^n is f(x)=anxn1,nZf'(x)=anx^{n-1}, n \in \mathbb{Z}

    • The derivative of functions of the form f(x)=axn+bxn1+...f(x)=ax^n+bx^{n-1}+... where all exponents are integers.

  • SL 5.4

    • Tangents and normals at a given point, and their equations.

  • SL 5.5

    • Introduction to integration as anti-differentiation of functions of the form f(x)=axn+bxn1+...f(x)=ax^n+bx^{n-1}+... , where nZ,n1n \in \mathbb{Z}, n \neq -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)y=f(x) and the xx -axis, where f(x)>0f(x) > 0 .

  • SL 5.6

    • Derivative of xn(nQ),sinx,cosx,exx^n (n \in \mathbb{Q}), \sin x, \cos x, e^x and lnx\ln x

    • Differentiation of a sum and a multiple of these functions.

    • The chain rule for composite functions.

    • The product and quotient rules.

  • SL 5.7

    • The second derivative.

    • Graphical behaviour of functions, including the relationship between the graphs of ff , ff′ and ff″

  • SL 5.8

    • Local maximum and minimum points

    • Testing for maximum and minimum.

    • Optimization.

    • Points of inflexion with zero and non-zero gradients.

  • AHL 5.9

    • Kinematic problems involving displacement ss , velocity vv , acceleration aa and total distance travelled.

  • AHL 5.10

    • Indefinite integral of xn(nQx^n(n \in \mathbb{Q} , sinx\sin x , cosx\cos x , 1x\frac{1}{x} and exe^x

    • The composites of any of these with the linear function ax+bax+b .

    • Integration by inspection (reverse chain rule) or by substitution for expressions of the form: kg(x)f(g(x))dx\int kg' (x) f(g(x))dx

  • AHL 5.11

    • Definite integrals, including analytical approach.

    • Areas of a region enclosed by a curve y=f(x)y=f(x) and the f(x)f(x) -axis, where f(x)f(x) can be positive or negative, without the use of technology.

    • Areas between curves.

AHL Content

  • AHL 5.12

    • Informal understanding of continuity and differentiability of a function at a point.

    • Understanding of limits (convergence and divergence).

    • Definition of derivative from first principles f(x)=limh0f(x+h)f(x)hf'(x)=\lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h}

    • Higher derivatives.

  • AHL 5.13

    • The evaluation of limits of the form limxaf(x)g(x)\lim\limits_{x \to a} \frac{f(x)}{g(x)} and limxf(x)g(x)\lim\limits_{x \to \infty} \frac{f(x)}{g(x)} using l’Hôpital’s rule or the Maclaurin series.

    • Repeated use of l’Hôpital’s rule.

  • AHL 5.14

    • Implicit differentiation.

    • Related rates of change.

    • Optimisation problems.

  • AHL 5.15

    • Derivatives of tanx,secx,cscx,cotx,ax,logax,arcsinx,arccosx,arctanx\tan x, \sec x, \csc x, \cot x, a^x, \log_a x, \arcsin x, \arccos x, \arctan x

    • Indefinite integrals of the derivatives of any of the above functions.

    • The composites of any of these with a linear function.

    • Use of partial fractions to rearrange the integrand.

  • AHL 5.16

    • Integration by substitution.

    • Integration by parts.

    • Repeated integration by parts.

  • AHL 5.17

    • Area of the region enclosed by a curve and the yy -axis in a given interval.

    • Volumes of revolution about the xx -axis or yy -axis.

  • AHL 5.18

    • First order differential equations.

    • Numerical solution of dydx=f(x,y)\frac{dy}{dx}=f(x, y) using Euler's method.

    • Variables separable.

    • Homogeneous differential equation dydx=f(yx)\frac{dy}{dx}=f(\frac{y}{x}) using the subtitution y=vxy=vx .

    • Solution of y+P(x)y=Q(x)y'+P(x)y=Q(x) , using the integrating factor.

  • AHL 5.19

    • Maclaurin series to obtain expansions for ex,sinx,cosx,ln(1+x),(1+x)p,pQe^x, \sin x, \cos x, \ln (1+x), (1+x)^p, p \in \mathbb{Q}

    • Use of simple substitution, products, integration and differentiation to obtain other series.

    • Maclaurin series developed from differential equations.

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