Set the derivative to zero and check the second derivative to find optimum designs.
Critical points occur where f'=0 or is undefined. If f''>0 the point is a minimum, if f''<0 a maximum. Check endpoints on closed intervals. Optimisation finds the best design, e.g. minimum material for a volume.
A can of volume 330 mL uses least metal when height equals diameter.
Optimisation finds the best design, such as the lightest beam or the highest efficiency, and it depends on identifying where the derivative is zero.
Set the derivative to zero and check the second derivative to find optimum designs.
Optimisation finds the best design, such as the lightest beam or the highest efficiency, and it depends on identifying where the derivative is zero.
Check the sign of the second derivative. Negative means a maximum and positive means a minimum. If it is zero, use higher derivatives or check the values directly.
Not always. Endpoints and points where the function is not differentiable can also be extrema, so check them too.
Typical applications include choosing the diameter of a pipe that minimises cost, finding the angle that gives maximum range for a projectile and locating the operating point of maximum power transfer.
Study "Applications of derivatives" first (topic 30 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Global vs local extrema. Constraints reduce variables. Second derivative test.
A can of volume 330 mL uses least metal when height equals diameter.
The next topic is "Integrals", which builds directly on this one.
It is classed as Beginner in Engineering Mathematics (topic 31 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.
Want to practise? Open the interactive course for quick checks, quizzes, simulations and progress tracking.