FM Hyperbolic functions name _______________________

Objective Deadlines / Progress

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Know the identities for sinh x, cosh x and tanh x and their associated reciprocal functions Be able to sketch graphs of the hyperbolic functions and transformations of the graphs Solve equations involving hyperbolic functions; use the laws of logarithms to simplify answers; Define and use the inverse hyperbolic functions

Express inverse hyperbolic functions in terms of natural logarithms

Iden

tities

and

equ

ation

s

Know Osbornes rule and use to:

Find Identities for hyperbolic functions corresponding to Trig identities

Solve equations involving hyperbolic functions

Apply Identities and definitions to give proofs / solve show that problems

Calc

ulus

Differentiate hyperbolic functions including inverse hyperbolic functions

Find series expansions for hyperbolic functions using Maclaurin series formula

Integrate hyperbolic functions; integrate using given substitutions

FM Hyperbolic functions name _______________________

Notes

If you take a rope/chain, fix the two ends, and let it hang under the force of gravity, it will naturally form a hyperbolic cosine curve

Most curves that look parabolic are actually Catenaries, which is based in the hyperbolic cosine function. A good example of a Catenary would be the Gateway Arch in Saint Louis, Missouri

Hyperbolic functions have several properties in common with trigonometric functions, but they are defined in terms of exponential functions

sinh x≡ ex−e−x

2

cosh x≡ ex+e− x

2

tanh x ≡ sinh xcosh x

≡ e2x−1e2 x+1

There are corresponding reciprocal functions

These definitions can simply be stated but need to be memorised

cosech x ≡ 2ex−e−x

sech x≡ 2ex+e− x

coth x≡ 1tanh x

≡ e2 x+1e2 x−1

FM Hyperbolic functions name _______________________

WBA1 Find to 2dp the values of

a¿sinh 5

b¿cosh (ln 2 )

c ¿ tanh x2

FM Hyperbolic functions name _______________________

WB A2 Find the values of x for which

a¿ sinh x=5

b¿ tanh x=1517

FM Hyperbolic functions name _______________________

WB A3 sketch the graphs of sinh x ,cosh x∧¿ tanh x

FM Hyperbolic functions name _______________________

WB A4 Evaluate the following, leaving answers to 4 s fa) sinh 7 b) cosh(-5) c) tanh 0.2Solve the hyperbolic equations, leaving answers correct to 3sf d) sinh x = -2 e) cosh x = 3 f) tanh x = 0.8Try finding the exact solutions using the definitions for sinh[2,6 ]

FM Hyperbolic functions name _______________________

WB A5 sketch the graphs of cosech x ,sech x∧¿ coth x

FM Hyperbolic functions name _______________________

Inverse Hyperbolic functions

WB B1 Sketch the graphs of the inverse Hyperbolic functions

FM Hyperbolic functions name _______________________

WB B2 show that a¿ar sin h x=ln (x+√x2+1) b¿ar cosh x=ln (x+√ x2−1 ) , x ≥1

FM Hyperbolic functions name _______________________

Notes

ar sin h x= ln (x+√ x2+1) x∈ R

ar cosh x=ln (x+√ x2−1 ), x≥1

artan h x=12ln( 1+x1−x ) ,|x|<1

WB B3 Express as natural logarithms

a¿ar sinh 1

b¿arcosh 2

c ¿artanh 13

FM Hyperbolic functions name _______________________

FM Hyperbolic functions name _______________________

Identities and equations

WB C1 Prove that a¿cosh ¿2x−sinh2x ≡1b¿ sinh ( A+B )=sinh A cosh B+cosh A sinh B

Notes the addition formulae for hyperbolic functions are

sinh ( A±B )=sinh A coshB± cosh A sinhB

cosh (A ±B )=cosh A coshB∓sinh A sinhB

OSBORNS RULE sin x→sinh xcos x→cosh x

Replace any product of two sin terms by minus the product of sinh terms e.g. sinAsinB → - sinh A sinhBe.g. tanAtanB → - tanh A tanhB

FM Hyperbolic functions name _______________________

WB C2 a) prove that cosh 2x=1+2sinh2 xb) Write the hyperbolic identity corresponding to cos2 x=2cos2 x−1

WB C3Given that sinh x=34 find the exact value of a¿cosh xb¿ tanh x c ¿sinh2 x d¿cos h2 x

FM Hyperbolic functions name _______________________

WB C4 solve each equation for all real values of x, give answers as natural logarithms where appropriate

a¿6sinh x−2cosh x=7

b¿2cosh2 x−5sinh x=5

c ¿cosh2 x−5cosh x+4=0

FM Hyperbolic functions name _______________________

Notes the standard derivatives are ddxsinh x=cosh x

ddxcosh x=sinh x note +ve sign

ddxtanh x=sech2 x

ddxar sinh x= 1

√x2+1

ddxar cosh x= 1

√ x2−1, x>1

ddxar tanh x= 1

1−x2, |x|<1

WB D1 Prove each result: a¿ ddx

sinh x=cosh x

b¿ ddxcosh x=sinh x c ¿ ddx

tanh x=sech2

a¿ ddxsinh x=cosh x

b¿ ddxcosh x=sinh x

c ¿ ddxtanh x=sech2

FM Hyperbolic functions name _______________________

WB D2 a) show that

ddx

(arsinh x )= 1√ x2+1

b) Find the derivative of y=arsinh(3 x+2)

WB D3 Differentiate

a) cosh 3x

b) x2cosh 4 x

c) x arcosh x

FM Hyperbolic functions name _______________________

WB D4 Given that y=A cosh 3 x+B sinh 3 x , where A and B are constants

Prove that d2 yd x2

=9 y

WB D5 Given that y= (arcosh x )2 ,

Prove that (x2−1 )( dydx )2

=4 y

FM Hyperbolic functions name _______________________

FM Hyperbolic functions name _______________________

WB D6 a) find the first two non-zero terms in the series expansion of arsinh x

The general term for the series expansion of arsinh x is given by

arsinh x=∑r=0

∞

( (−1 )n (2n )!22n (n !)2 ) x2n+12n+1

b) find , in simplest terms, the terms of the coefficient of x5

c) Use your approximation up to the term in x5 to find an approximate value for arsinh 0.5d) Calculate the % error for c)

FM Hyperbolic functions name _______________________

FM Hyperbolic functions name _______________________

WB D7 y=sin x sinh x

a) Show that d4 yd x4

=−4 y

b) Hence find the first three non-zero terms of the Maclaurin series for y, giving each coefficient in its simplest form

c) Find an expression for the nth non-zero term of the Maclaurin series for y

FM Hyperbolic functions name _______________________

WB E1 find these integrals

a¿ ∫cosh (4 x−1)dx b¿∫ 2+5 x√x2+1

dx

WB E2 find these integralsa¿ ∫cosh52x sinh 2x dx b¿∫ tanh x dx

FM Hyperbolic functions name _______________________

FM Hyperbolic functions name _______________________

WB E3 find these integralsa¿ ∫cosh23 xdx b¿∫sinh33 xdx

WB E4 find these integrals

a¿ ∫ e2 xsinh xdx b¿ ∫−1 /2

1/2 cosh xex

dx

FM Hyperbolic functions name _______________________

WB E5 a¿ find ∫ 1

√ x2−a2dx b¿ show that∫

5

8 1√ x2−16

dx=ln( 2+√32 )

WB E6 show that∫ √1+x2dx=12 arsinh x+

12x √1+x2+C

FM Hyperbolic functions name _______________________

WB E7By using a hyperbolic substitution, evaluate ∫

0

6 x3

√ x2+9dx

WB E8 Find ∫ 1

√12 x+2 x2dx

FM Hyperbolic functions name _______________________

Notes

The standard derivatives are ddxsinh x=cosh x

ddxcosh x=sinh x

ddxtanh x=sech2 x

ddxar sinh x= 1

√x2+1

ddxar cosh x= 1

√ x2−1, x>1

The standard integrals are

∫sinh x dx=cosh x+C

∫cosh x dx=s∈hx+C

∫ tanh xdx=ln (cosh x )+C

∫ 1√ x2+1

dx=arsinh x+C

∫ 1√ x2−1

, dx=arcosh x+C, x>1

FM Hyperbolic functions name _______________________

ddxar tanh x= 1

1−x2, |x|<1

Also sinh x≡ e

x−e−x

2

cosh x≡ ex+e− x

2

tanh x ≡ sinh xcosh x

≡ e2x−1e2 x+1

ar sinh x=ln (x+√ x2+1)

ar cosh x=ln (x+√ x2−1 )

∫ 1√ x2+a2

dx ¿arsinh xa+C

∫ 1√ x2−a2

dx ¿arcosh xa+C x>a

Also

sinh2 x=cosh2 x−1

a2sinh2 x=a2 cosh2 x−a2

cosh 2x=2cosh2 x−1

sinh 2 x=2sinh x cosh x

WB E9Use the substitution x=12

(3+4coshu ) to find ∫ 1√4 x2−12 x−7

dx

FM Hyperbolic functions name _______________________