gmaths28.files.wordpress.com file · web viewobjective. deadlines / progress . hyperbolic functions...
TRANSCRIPT
FM Hyperbolic functions name _______________________
Objective Deadlines / Progress
Hype
rbol
ic fu
nctio
ns
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 _______________________
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 _______________________
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 _______________________
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 _______________________
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