Y2, C6 - Hyperbolic Functions Flashcards

1
Q

What is coshx

A

1/2 (e^x + e^-x)

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2
Q

What is sinhx

A

1/2 (e^x - e^-x)

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3
Q

What is tanhx

A

(e^x - e^-x) / (e^x + e^-x)
(e^2x - 1) / (e^2x + 1)

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4
Q

What does it mean if a function is even

A

Reflection of symmetry in y axis f(x) = f(-x)

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5
Q

What does it mean if a function is odd

A

Rotational symmetry around origin f(x) = -f(-x)

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6
Q

How can you draw hyperbolic graphs

A

Use graphing calculator

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7
Q

What can the inverse of sinx and sinhx be written as

A

arcsinx
arsinhx (no ‘c’ in the ar)

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8
Q

Why are there only positve values for arcoshx

A

The inverse must have a one-to-one relationship

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9
Q

How would you prove that arcosh(x) = ln(x + root(x^2 - 1)), x>1

A

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))

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10
Q

How would you prove that cosh(2A) = 1 + 2sinh^2(A)

A

Sub in 0.5(e^A - e^-A) as sinh(A) and solve through

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11
Q

What is Osborne’s rule

A

1) Replace sin and cos with sinh and cosh
2) Negate any explicit or implied product of two sines

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12
Q

Using Osborne’s rule, would a sin^3x be negated

A

Yes = -sinh^3x

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13
Q

Using Osborne’s rule, would a sin^4x be negated

A

No = sinh^4x

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14
Q

When solving hyperbolic equations how do you know when to use hyperbolic functions or hyperbolic identities

A

If the equation would work with sinx, cosx, tanx instead then use indentities, if not use functions

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15
Q

How would you find exact and non exact solutions to hyperbolic calculations

A

Exact- use e’s to get exact answer
Non-exact- use calculator to get decimals

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16
Q

When is the function for arcoshx and arsinhx ±

A

When it has two solutions and we aren’t finding an inverse (just ± the root in the formula booklet)

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17
Q

What is d/dx (cothx)

A

-cosech^2(x)

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18
Q

What is d/dx (sechx)

A

-sechxtanhx

19
Q

What is d/dx (cosechx)

A

= -cosechxcothx

20
Q

What is d/dx (tanhx)

21
Q

What is d/dx (sinhx) and d/dx (coshx)

A

sinhx –> coshx
coshx –> sinhx

22
Q

What is cosh^2(x) equal to

A

1/2 + 1/2 cosh(2x)

23
Q

What is sinh^2(x) equal to

A

1/2 cosh(2x) - 1/2

24
Q

How do you prove differential results for hyperbolics

A

Use exponential form and differentiate normally

25
How do you find the differentials of arshinx, arcoshx, and artanhx?
Use formula booklet
26
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)
27
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)
28
What is the integral of cosech^2(x)
-cothx + c
29
What is the integral of sech(x)tanh(x)
-sechx + c
30
What is the integral of cosechxcothx
-cosechx + c
31
What is cosh(4x - 1) dx
1/4 * sinh(4x - 1) + c
32
What is 3 / root(1 + x^2) dx
3arsinh(x) + c
33
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
34
How to find tanhx dx
= sinhx / coshx dx = ln(coshx) + c
35
What is cosh^5(2x)sinh(2x) dx
1 / 12 * cosh^6(2x) + c
36
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
37
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
38
What do you do if you cannot get a hyperbolic integration to work
Rewrite in exponential form and attempt to solve normally
39
What substitution can you use when integrating hyperbolics in exponential form
u = e^x
40
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)
41
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))
42
What does cosh^2(u)sinh(u) integrate to
1/3 * cosh^3(u) + c
43