# 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.

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