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 $f'(x) > 0, f'(x)=0, f'(x) < 0$
• SL 5.3

  • Derivative of $f(x)=ax^n$ is $f'(x)=anx^{n-1}, n \in \mathbb{Z}$
  • The derivative of functions of the form $f(x)=ax^n+bx^{n-1}+...$ 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 $f(x)=ax^n+bx^{n-1}+...$ , where $n \in \mathbb{Z}, n \neq -1$
  • Anti-differentiation with a boundary condition to determine the constant term.

  • Definite integrals using technology.

  • Area of a region enclosed by a curve $y=f(x)$ and the $x$ -axis, where $f(x) > 0$ .
• SL 5.6

  • Derivative of $x^n (n \in \mathbb{Q}), \sin x, \cos x, e^x$ and $\ln x$
  • 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 $f$ , $f′$ and $f″$
• 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 $s$ , velocity $v$ , acceleration $a$ and total distance travelled.
• AHL 5.10

  • Indefinite integral of $x^n(n \in \mathbb{Q}$ , $\sin x$ , $\cos x$ , $\frac{1}{x}$ and $e^x$
  • The composites of any of these with the linear function $ax+b$ .
  • Integration by inspection (reverse chain rule) or by substitution for expressions of the form: $\int kg' (x) f(g(x))dx$
• AHL 5.11

  • Definite integrals, including analytical approach.

  • Areas of a region enclosed by a curve $y=f(x)$ and the $f(x)$ -axis, where $f(x)$ 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 $f'(x)=\lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h}$
  • Higher derivatives.

• AHL 5.13

  • The evaluation of limits of the form $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ and $\lim\limits_{x \to \infty} \frac{f(x)}{g(x)}$ 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 $\tan x, \sec x, \csc x, \cot x, a^x, \log_a x, \arcsin x, \arccos x, \arctan x$
  • 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 $y$ -axis in a given interval.
  • Volumes of revolution about the $x$ -axis or $y$ -axis.
• AHL 5.18

  • First order differential equations.

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

  • Homogeneous differential equation $\frac{dy}{dx}=f(\frac{y}{x})$ using the subtitution $y=vx$ .
  • Solution of $y'+P(x)y=Q(x)$ , using the integrating factor.
• AHL 5.19

  • Maclaurin series to obtain expansions for $e^x, \sin x, \cos x, \ln (1+x), (1+x)^p, p \in \mathbb{Q}$
  • Use of simple substitution, products, integration and differentiation to obtain other series.

  • Maclaurin series developed from differential equations.