FP2 (MEI) Hyperbolic functions -Introduction (part 1)

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FP2 (MEI)Hyperbolic functions -Introduction
(part 1)

• Let Maths take you Further

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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 finishedYou should
• Understand the definitions of hyperbolic
  functions and be able to sketch their graphs
• Be able to differentiate and integrate hyperbolic
  functions

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Exploring with Autograph

• What does the graph look like if pq1?
• What happens if we change the values of
• p q (where p q are real constants)?

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Cartesian and parametric forms
Unit circle
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Cartesian and parametric forms
Rectangular hyperbola
Difference of two squares
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let
But notice the restriction that now tgt0
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Compare!
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What do these hyperbolicfunctions look like?
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What do these hyperbolic functions look like?
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Cartesian and parametric forms
Rectangular hyperbola
These are not the standard parametric equations
that are generally used, can you say why not?
are used
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Complex variables, z
Replace z by iz
Replace z by iz
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Complex variables, z
Replace z by iz
Replace z by iz
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Results
cosh(iz) cos z sinh(iz) i sin z
cos(iz) cosh z sin(iz) i sinh z
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Circular trigonometric identities and hyperbolic
trigonometric identities
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Osborns rule

• change each trig ratio into the comparative
  hyperbolic function, whenever a product of two
  sines occurs, change the sign of that term

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Differentiation
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Integration
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Calculus - Reminder
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The usual techniques can be used.
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Calculus - Reminder
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The usual techniques can be used
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Introduction to hyperbolic functions

• When you have finishedYou should
• Understand the definitions of hyperbolic
  functions and be able to sketch their graphs
• Be able to differentiate and integrate hyperbolic
  functions

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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.