the Further Mathematics network

www.fmnetwork.org.uk

the Further Mathematics network

www.fmnetwork.org.uk

FP2 (MEI)Inverse hyperbolic

functions

Let Maths take you Further…

Inverse hyperbolic functions

Before you start:

You need to be confident in manipulating exponential and logarithmic functions. You need to have covered the work on Maclaurin series from chapter 4. You need to have covered Calculus from chapter 1 (integration using inverse trig

functions)

When you have finished…You should:

Understand and be able to use the definitions of the inverse hyperbolic functions.

Be able to use the logarithmic forms of the inverse hyperbolic functions. Be able to integrate

and and related functions.

22

1

ax

22

1

ax

Notationtrig. functions inverse trig.

functionshyperbolic

trig. functions

inverse hyperbolic

trig. functions

sin x arcsin x sinh x arsinh x

cos x arccos x cosh x arcosh x

tan x arctan x tanh x artanh x

cosec x arccosec x cosech x arcosech x

sec x arcsec x sech x arsech x

cot x arccot x coth x arcoth x

Latin for arc

Graphs

Use the graph of sinhx to sketch the graph of arsinhx

Hint: use the line y=x to help!

Remember for a function to have an inverse it has to be a one-to-one function

The domain needs to be refined to ensure the function is one to one

Sketch the graph of arcoshx and state its domain and range

Logarithmic form of the inverse hyperbolic functions y=arsinh x so x=sinh y

Summary

Differentiating inverse hyperbolic trig. functionsNote: this can be done using the same technique that was used for differentiating inverse trig. functions

y=arcosh x x= cosh y

Results

We can now integrate expressions of these forms!

We can also differentiate composite functions involving inverse hyperbolic functions using the chain rule e.g.

1)2(

2)2sinh(

2

xxar

dx

d

Using the previous results, together with the results we established by considering inverse trig. Functions, we should now be able to integrate functions of the form:

Inverse hyperbolic functions

When you have finished…You should:

Understand and be able to use the definitions of the inverse hyperbolic functions.

Be able to use the logarithmic forms of the inverse hyperbolic functions. Be able to integrate

and and related functions.

22

1

ax

22

1

ax

Independent study:

Using the MEI online resources complete the study plan for Hyperbolic functions 2

Do the online multiple choice test for this and submit your answers online.