Learning objects Differential calculus: derivatives and power functions

TLF ID L7827

Observe the non-linear distance-time graph of a rocket travelling at a changing velocity. Calculate the average velocity of the rocket over time intervals that become progressively shorter. Tabulate the results and derive a formula for finding the instantaneous velocity at a given point. In the second activity, observe the non-linear graphs of various powers of x and select the expressions for finding the gradient of the secant between small intervals of x (represented by Δx). Plot the derivatives of each function and look for patterns. This learning object is a combination of two objects in the same series.

Educational details

Key learning objectives
  • Students apply the concept of a limit in the context of the rate of change of a non-linear function.
  • Students interpret the gradient of the tangent to a curve as being the rate of change of the function.
  • Students apply first principles methods to differentiate the functions of t² and various powers of x.
  • Students interpret the concept of a derivative of a function, and identify the derivative for t² and for various power functions of x.
  • Students investigate the derivative to calculate the range of a function at a nominated point.
Educational value
  • Introduces students to the language, notation and methods of differential calculus.
  • Provides opportunities for students to find derivatives of functions from first principles.
  • Tabulates results to help students find patterns in the gradients at different points on the curve.
  • Allows students to construct tables for the derivatives of functions and plot the graphs.
  • Summarises key statements related to derivatives, average and instantaneous velocities, difference quotients and power functions.
  • Displays feedback for correct and incorrect answers.
Year level

11; 12

Learning area
  • mathematics
  • Mathematics/Algebra
Student activity
  • Interactives;
  • Experiment;
  • Analysis;
  • Modelling;
  • Multiple choice questions

Other details

  • Script writer
  • Educational validator
  • Name: Professor Graham Jones
  • Address: Broadbeach QLD 4218 Australia
  • Technical implementer
  • Subject matter expert
  • Name: Howard Reeves
  • Address: Lindisfarne TAS 7015 Australia
  • Educational validator
  • Name: Doctor Paul White
  • Address: Strathfield NSW 2135 Australia
  • Publisher
  • Date of contribution: 20 Sep 2013
  • Organisation: Education Services Australia
  • Address: Melbourne VIC 3000 Australia
  • URL: http://www.esa.edu.au/
Access profile
  • Colour independence
  • Device independence
  • Hearing independence
  • Generic
Learning resource type
  • Interactive Resource
  • Microsoft Internet Explorer - minimum version: 8.0 (MS-Windows) - maximum version: 9.0 (MS-Windows)
  • Firefox - minimum version: (MS-Windows)
  • Safari - minimum version: 5.1 (MacOS)
Operating systems
  • MacOS - minimum version: 10.6
  • MS-Windows - minimum version: XP - maximum version: 7
  • © Education Services Australia Ltd, 2013, except where indicated under Acknowledgements.