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.