Y2, C6 - Hyperbolic Functions Flashcards
What is coshx
1/2 (e^x + e^-x)
What is sinhx
1/2 (e^x - e^-x)
What is tanhx
(e^x - e^-x) / (e^x + e^-x)
(e^2x - 1) / (e^2x + 1)
What does it mean if a function is even
Reflection of symmetry in y axis f(x) = f(-x)
What does it mean if a function is odd
Rotational symmetry around origin f(x) = -f(-x)
How can you draw hyperbolic graphs
Use graphing calculator
What can the inverse of sinx and sinhx be written as
arcsinx
arsinhx (no ‘c’ in the ar)
Why are there only positve values for arcoshx
The inverse must have a one-to-one relationship
How would you prove that arcosh(x) = ln(x + root(x^2 - 1)), x>1
Let f(x) = coshx
Then f^-1(x) = arcosh(x)
y = coshx
x = coshy
x = 0.5(e^y + e^-y)
2x = e^y + e^-y (then multiply by e^y)
2xe^y = e^2y + 1
0 = 2xe^y - e^2y - 1
0 = (e^y - x)^2 - x^2 + 1
±root(x^2 - 1) = e^y - x
e^y = x ±root(x^2 - 1)
y = ln(x + root(x^2 - 1))
How would you prove that cosh(2A) = 1 + 2sinh^2(A)
Sub in 0.5(e^A - e^-A) as sinh(A) and solve through
What is Osborne’s rule
1) Replace sin and cos with sinh and cosh
2) Negate any explicit or implied product of two sines
Using Osborne’s rule, would a sin^3x be negated
Yes = -sinh^3x
Using Osborne’s rule, would a sin^4x be negated
No = sinh^4x
When solving hyperbolic equations how do you know when to use hyperbolic functions or hyperbolic identities
If the equation would work with sinx, cosx, tanx instead then use indentities, if not use functions
How would you find exact and non exact solutions to hyperbolic calculations
Exact- use e’s to get exact answer
Non-exact- use calculator to get decimals
When is the function for arcoshx and arsinhx ±
When it has two solutions and we aren’t finding an inverse (just ± the root in the formula booklet)
What is d/dx (cothx)
-cosech^2(x)
What is d/dx (sechx)
-sechxtanhx
What is d/dx (cosechx)
= -cosechxcothx
What is d/dx (tanhx)
sech^2(x)
What is d/dx (sinhx) and d/dx (coshx)
sinhx –> coshx
coshx –> sinhx
What is cosh^2(x) equal to
1/2 + 1/2 cosh(2x)
What is sinh^2(x) equal to
1/2 cosh(2x) - 1/2
How do you prove differential results for hyperbolics
Use exponential form and differentiate normally
How do you find the differentials of arshinx, arcoshx, and artanhx?
Use formula booklet
Knowing from the formula booklet that d/dx * artanhx = 1 / (1-x^2), what is d/dx * artanh(3x)
Chain rule
1 / (1-9x^2) * 3
ans = 3 / (1-9x^2)
y = arsinhx, find d/dx * arsinhx
y = arsinhx
coshy * dy/dx = 1
dy/dx = 1 / coshy
dy / dx = 1/root(1+sinh^2(y))
dy/dx = 1 / root(1+x^2)
What is the integral of cosech^2(x)
-cothx + c
What is the integral of sech(x)tanh(x)
-sechx + c
What is the integral of cosechxcothx
-cosechx + c
What is cosh(4x - 1) dx
1/4 * sinh(4x - 1) + c
What is 3 / root(1 + x^2) dx
3arsinh(x) + c
How do you find (2 + 5x) / (root(x^2 + 1)) dx
Split into two fractions
2 / root(x^2 + 1) + 5x / root(x^2 + 1)
ans = 2sinh(x) + 5(x^2 + 1)^1/2 + c
How to find tanhx dx
= sinhx / coshx dx
= ln(coshx) + c
What is cosh^5(2x)sinh(2x) dx
1 / 12 * cosh^6(2x) + c
What are the identities for cosh^2(x) and sinh^2(x)
cosh^2(x) = 1/2 + 1/2cosh(2x)
sinh^2(x) = 1/2cosh(2x) - 1/2
How do you find cosh^2(3x) dx
Use identity
cosh^2(3x) = 1/2 + 1/2 cosh(6x)
ans = 1/2 * x + 1/12 * cosh(6x) + c
What do you do if you cannot get a hyperbolic integration to work
Rewrite in exponential form and attempt to solve normally
What substitution can you use when integrating hyperbolics in exponential form
u = e^x
What substitutions should you use when dealing with:
a) 1 / root(a^2 + x^2)
b) 1 / root(x^2 - a^2)
a) x = a * sinh(u)
b) x = a * cosh(u)
What would the substitution be to show that root(1 + x^2) dx = 1/2 * arsinh(x) + 1/2 * x * root(1+x^2) + c
x = sinh(u)
Hint is on the RHS (arsinh(x))
What does cosh^2(u)sinh(u) integrate to
1/3 * cosh^3(u) + c
