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)<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

    • Values of xx where the gradient of a curve is zero. Solution of xx .

    • 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 sinx,cosx,tanx,ex,lnx,xn\sin x, \cos x, \tan x, e^x, ln x, x_n where nQn \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 xnx^n where nQn \in \mathbf{Q} , including n=1,sinx,cosx,1cosx2n=-1, \sin x, \cos x, \frac{1}{\cos^2_x} and exe^x

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

  • AHL 5.12

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

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

  • AHL 5.13

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

  • 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 dydx=f(x,y)\frac{dy}{dx}=f(x, y)

    • Numerical solution of the coupled system dxdt=f1(x,y,t)\frac{dx}{dt}=f_1(x, y, t) and dydt=f2(x,y,t)\frac{dy}{dt}=f_2(x, y, t)

  • AHL 5.17

    • Phase portrait for the solutions of coupled differential equations of the form:

    • dxdt=ax+by\frac{dx}{dt}=ax+by

    • dydt=cx+dy\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 d2xdt2=f(x,dxdt,t)\frac{d^2 x}{dt^2}=f(x, \frac{dx}{dt}, t) by Euler's method.

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