FP2 (MEI) Hyperbolic functions -Introduction (part 1) Let Maths take you Further…

Introduction to hyperbolic functions Before you start: You need to be confident in manipulating exponential and logarithmic functions You need to be confident all the calculus techniques covered in Core 2 and 3 You need to have covered chapter 4 on Maclaurin series When you have finished… You should: Understand the definitions of hyperbolic functions and be able to sketch their graphs Be able to differentiate and integrate hyperbolic functions

Exploring with Autograph What does the graph look like if p=q=1? What happens if we change the values of p & q (where p & q are real constants)?

Cartesian and parametric forms Unit circle

Cartesian and parametric forms Rectangular hyperbola Difference of two squares:

let But notice the restriction that now t>0

Compare!

What do these hyperbolic functions look like?

What do these hyperbolic functions look like?

Cartesian and parametric forms Rectangular hyperbola These are not the standard parametric equations that are generally used, can you say why not? are used

Complex variables, z Replace z by iz Replace z by iz

Complex variables, z Replace z by iz Replace z by iz

Results cosh(iz) = cos z sinh(iz) = i sin z cos(iz) = cosh z sin(iz) = i sinh z

Circular trigonometric identities and hyperbolic trigonometric identities

Osborn’s rule “… change each trig ratio into the comparative hyperbolic function, whenever a product of two sines occurs, change the sign of that term…”

Differentiation

Integration

Calculus - Reminder

The usual techniques can be used….

Calculus - Reminder

The usual techniques can be used…

Introduction to hyperbolic functions When you have finished… You should: Understand the definitions of hyperbolic functions and be able to sketch their graphs Be able to differentiate and integrate hyperbolic functions

Independent study: Using the MEI online resources complete the study plan for Hyperbolic functions 1 Do the online multiple choice test for this and submit your answers online.