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# 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
Strand
• Mathematics/Algebra
Student activity
• Interactives;
• Experiment;
• Analysis;
• Modelling;
• Multiple choice questions

#### Other details

Contributors
• Script writer
• Educational validator
• Name: Professor Graham Jones
• 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
Browsers
• 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
Rights
• © Education Services Australia Ltd, 2013, except where indicated under Acknowledgements.