Learning objects Differential calculus: quartic function

TLF ID L7824

Observe the non-linear distance–time graph of a rocket travelling at a changing velocity. The distance, s, travelled by the rocket after t seconds is determined by the formula s(t) = t⁴ + t². Calculate the average velocity of the rocket over time intervals that become progressively shorter. Tabulate the results and look for a pattern. Use your knowledge of limits to derive a formula for finding the instantaneous velocity at a given point. This learning object is one in a series of ten objects. Some objects in the series are also packaged as combined learning objects.

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 function s(t) = t⁴ + t².
  • Students interpret the concept of a derivative of a function, and identify that for s(t) = t⁴ + t² the derivative is f'(t) = 4t³ + 2t.
Educational value
  • Introduces students to the language, notation and methods of differential calculus.
  • Provides opportunities for students to engage in first principles derivation of a basic polynomial function.
  • Tabulates results to help students find patterns between the gradients at different points on the curve.
  • Summarises key statements related to derivatives and average and instantaneous velocities.
  • 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
  • Organisation: 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
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.