the Further Mathematics network

www.fmnetwork.org.uk

the Further Mathematics network

www.fmnetwork.org.uk

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)?

122 qypx

Cartesian and parametric forms

122 yxUnit circle

Cartesian and parametric forms

122 yxRectangular hyperbola

Difference of two squares:

ttx1

2

1

tty1

2

1

uet let

But notice the restriction that now t>0

y

x

Compare!

ii ee 2

1cos

ii eei

2

1sin

uu eeu 2

1cosh

uu eeu 2

1sinh

What do these hyperbolicfunctions look like?

uu eeu 2

1cosh

What do these hyperbolic functions look like?

uu eeu 2

1sinh

Cartesian and parametric forms

122 yxRectangular hyperbola

ux coshuy sinh

These are not the standard parametric equations that are generally used, can you say why not?

secxtany

are used

Complex variables, z

zz eez 2

1cosh zz eez

2

1sinh

Replace z by iz Replace z by iz

Complex variables, z

iziz eez 2

1cos iziz ee

iz

2

1sin

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 1)(sin)(cos 22 iziz

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…”

1sincos 22 2cossincos 22

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.