FP2 (MEI) Inverse hyperbolic functions

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FP2 (MEI)Inverse hyperbolic functions

• Let Maths take you Further

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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 finishedYou 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.

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Notation
trig. functions inverse trig. functions hyperbolic 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
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Latin for arc
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Graphs
Use the graph of sinhx to sketch the graph of
arsinhx
Hint use the line yx to help!
Remember for a function to have an inverse it has
to be a one-to-one function
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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
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Logarithmic form of the inverse hyperbolic
functions

• yarsinh x so xsinh y

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Summary
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Differentiating inverse hyperbolic trig. functions
Note this can be done using the same technique
that was used for differentiating inverse trig.
functions
yarcosh x
x cosh y
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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.
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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
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Inverse hyperbolic functions

• When you have finishedYou 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.

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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.