Hyperbolic Functions

Hyperbolic Functions

The Hyperbolic Sine, Hyperbolic Cosine & Hyperbolic Tangent

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xx

xx

xx

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eex

eex

coshsinhtanh

2sinh

2cosh

The Inverse Hyperbolic Cotangent, Hyperbolic Secant &

Hyperbolic Cosecant

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xx

xx

xx

eexhx

eexhx

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2sinh1csc

2cosh1sec

sinhcosh

tanh1coth

Values

).2csc(.4)3coth(ln.3

)0tanh(.2)2cosh(ln.1

:followingtheofeachexactlyFind

12

12

122)2csc(3

45

810

313

313

)3coth(ln3

011110tanh2

45

2212

2)2cosh(ln.1

4

2

2

4

22

22

3ln3ln

3ln3ln

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00

2ln2ln

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Solution

Remember Logarithmic Identities ( Calculus I )

xaI xa log.

7.3

.2

75.1

:

7ln

ln

7log5

e

xe

Examples

x

Logarithmic Identities II

naa xxnII loglog.

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Examples

11

2512

2

32

33

lnlnln.4

ln5ln5ln2.3

25ln5ln5ln2.2

25log5log5log2.1

:

Identities

xhxh

xhx

xx

xx

xxx

xxxxx

22

22

2

2

22

22

csc1coth)7(

1sectanh)6(2

12coshsinh)5(

212coshcosh)4(

sinhcosh2cosh)3(

coshsinh22sinh)2(1sinhcosh)1(

ProofsIdentity (1)

1]4[41

)]2()2[(

)2

()2

(

sinhcosh

222241

22

22

xxxx

xxxx

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ProofsIdentity (2)

x

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xx

xx

xxxx

2sinh

)(21

)]([2

)2

()2

(2

coshsinh2

22

2241

ProofsIdentity (3)

x

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ee

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xx

xx

xx

xxxx

xxxx

2cosh

)(21

)22[(

)]2()2[(

)2

()2

(

sinhcosh

22

2241

222241

22

22

ProofsIdentity (4)

212coshcosh

1cosh2

)1(coshcosh

sinhcosh

2cosh

2

2

22

22

xx

x

xx

xx

x

ProofsIdentity (5)

212coshsinh

1sinh2

sinh)1(sinh

sinhcosh

2cosh

2

2

22

22

xx

x

xx

xx

x

ProofsIdentity (6)

xhxxx

xxx

xx

22

22

2

2

2

22

sectanh1cosh1

coshsinh

coshcosh

1sinhcosh

ProofsIdentity (7)

xhxxx

xxx

xx

22

22

2

2

2

22

csc1cothsinh1

sinhsinh

sinhcosh

1sinhcosh

Derivatives of Hyperbolic Functions

xhxhxxhxhx

xhx

xhx

xxxx

cothcsc)(csc)6(tanhsec)(sec)5(

csc)(coth)4(

sec)(tanh)3(

sinh)(cosh)2(cosh)(sinh)1(

2

2

Proofs(1) (sinhx)’ = coshx (2) (coshx)’ = sinhx

xee

ee

x

xx

xx

cosh)(

)2

(

)(sinh

21

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ee

x

xx

xx

sinh)[(

)2

(

)(cosh

21

Proofs(3) (tanhx)’ = sech2x

xhx

xxxx

xxxxxx

x

2

2

2

22

2

seccosh1cosh

sinhcoshcosh

sinhsinhcoshcosh

)coshsinh(

)(tanh

Proofs(4) (cothx)’ = -csch2x

xhxx

xxx

xxx

xxxxxx

x

2

22

22

2

22

2

csccosh

1cosh

)sinh(coshcosh

coshsinhsinh

coshcoshsinhsinh

)sinhcosh(

)(coth

Proofs(5) (sechx)’ = - sechx tanhx

xhx

x

xxx

xx

hx

tanhseccoshsinh

cosh1coshsinh

sinhcosh

)(cosh

)cosh1(

)(sec

2

2

1

Proofs(5) (cschx)’ = - cschx cothx

xhx

x

xxx

xx

hx

cothcscsinhcosh

sinh1sinhcosh

coshsinh

)(sinh

)sinh1(

)(csc

2

2

1

Integrals Involving hyperbolic Functions

chhxdxxhx

chhxdxxhx

cxdxxh

cxdxxh

cxdxx

cxdxx

csccothcsc.6

sectanhsec.5

cothcsc.4

tanhsec.3

coshsinh.2

sinhcosh.1

2

2

Examples I

x

x

x

hx

xy

hxyxxxxy

hxxy

y

xy

exxy

yFind

4tanh

1002coth

22

93

94

sec68

)3(cosh.7

)]7(cosharcsin5)in(sinh[sinh(arcs.6coscoshsinsinh.5

csccoth.4

)]2[cosh(sinarctan.3

)][ln(coshtanh.2

coshsinh.1

Examples II

dxxxhx

dxxx

dxxxhx

dxx

hx

dxxx

egralsfollowingtheofeachfollowingtheEvaluatet

)3cot(arcsin)3(arcsincsc91

1.5

)2nsinh(arcta411.4

)tanh()(sec1.3

)1(sec1.2

)cosh(ln1.1

:int

2

2

42

5

Examples III

dxxx

dxhx

dxx

dxx

dxxhxxxh

dxx

x

dxx

xh

egralsfollowingtheofeachEvaluatet

x7cosh76

2

2

2

5sinh7

csc.6

coth.5

tanh.45csc495coth5csc3

5sinh49

5cosh.2

5tanh495sec.1

:int

Examples IV

dxx

dxxh

dxxhx

dxxxh

dxx

dxxx

dxxx

dxx

egralsfollowingtheofeachfollowingtheEvaluatet

4

6

4100

101

4

22

5100

3

tanh.8

sec.7

csccoth.6

tanhsec.5

cosh.4

sinhcosh.3

sinhcosh.2

cosh.1

:int

Examples V

dxxe

dxxx

dxxx

dxxx

egralsfollowingtheofeachfollowingtheEvaluatet

x cosh.4

cosh.3

sinh.2

cosh.1

:int

2

3

2

Graphs of Hyperbolic Functions

Graphs of Exponential Functions Functions

Graphs of Exponential Functions

f(x) = Cosh x

f(x) = cosh x = (½)ex + (½) e-x

as x → ∞ the values f(x) → ∞ following (½)ex as x → - ∞ the values f(x) → ∞ following (½) e-x

f(x) = sinh x

f(x) = tanh x

f(x) = secx