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.

Notation sin x arcsin x sinh x arsinh x cos x arccos x cosh x arcosh x trig. functions inverse trig. functions hyperbolic inverse hyperbolic 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

Sketch the graph of arcoshx and state its domain and range The domain needs to be refined to ensure the function is one to one

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

Summary

Differentiating inverse hyperbolic trig. functions Note: 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.

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.

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.