# 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 allows us to model, interpret and analyze real-world problems and situations. Calculus helps us 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, space, generalization.

AHL: Systems, quantity.

Content-specific conceptual understandings:

Students will understand the links between the derivative and the rate of change and interpret the meaning of this in context.

Students will understand the relationship between the integral and area and interpret the meaning of this in context.

Finding patterns in the derivatives of polynomials and their behavior, such as increasing or decreasing, allows a deeper appreciation of the properties of the function at any given point or instant.

Calculus is a concise form of communication used to approximate nature.

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

Optimization of a function allows us to find the largest or smallest value that a function can take in general and can be applied to a specific set of conditions to solve problems.

Maximum and minimum points help to solve optimization problems.

The area under a function on a graph has a meaning and has applications in space and time.

AHL: Kinematics allows us to describe the motion and direction of objects in closed systems in terms of displacement, velocity, and acceleration.

AHL: Many physical phenomena can be modelled using differential equations and analytic and numeric methods can be used to calculate optimum quantities.

AHL: Phase portraits enable us to visualize the behavior of dynamic systems.

## 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

Values of $x$ where the gradient of a curve is zero. Solution of $x$ .

Local maximum and minimum points.

SL 5.7

Optimisation problems in context.

SL 5.8

Approximating areas using the trapezoidal rule.

## AHL Content

AHL 5.9

The derivatives of $\sin x, \cos x, \tan x, e^x, ln x, x_n$ where $n \in \mathbf{Q}$

The chain rule, product rule and quotient rules.

Related rates of change.

AHL 5.10

The second derivative.

Use of second derivative test to distinguish between a maximum and a minimum point.

AHL 5.11

Definite and indefinite integration of $x^n$ where $n \in \mathbf{Q}$ , including $n=-1, \sin x, \cos x, \frac{1}{\cos^2_x}$ and $e^x$

Integration by inspection, or substitution of the form $\int f(g(x))g'(x)dx$

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.

AHL 5.13

Kinematic problems involving displacement $s$ , velocity $v$ and acceleration $a$ .

AHL 5.14

Setting up a model/differential equation from a context.

Solving by separation of variables.

AHL 5.15

Slope fields and their diagrams.

AHL 5.16

Euler’s method for finding the approximate solution to first order differential equations.

Numerical solution of $\frac{dy}{dx}=f(x, y)$

Numerical solution of the coupled system $\frac{dx}{dt}=f_1(x, y, t)$ and $\frac{dy}{dt}=f_2(x, y, t)$

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.

AHL 5.18

Solutions of $\frac{d^2 x}{dt^2}=f(x, \frac{dx}{dt}, t)$ by Euler's method.

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