Y2, C6 - Hyperbolic Functions

Question 1

What is coshx

Answer 1

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

Question 2

What is sinhx

Answer 2

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

Question 3

What is tanhx

Answer 3

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

Question 4

What does it mean if a function is even

Answer 4

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

Question 5

What does it mean if a function is odd

Answer 5

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

Question 6

How can you draw hyperbolic graphs

Answer 6

Use graphing calculator

Question 7

What can the inverse of sinx and sinhx be written as

Answer 7

arcsinx
arsinhx (no ‘c’ in the ar)

Question 8

Why are there only positve values for arcoshx

Answer 8

The inverse must have a one-to-one relationship

Question 9

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

Answer 9

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))

Question 10

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

Answer 10

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

Question 11

What is Osborne’s rule

Answer 11

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

Question 12

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

Answer 12

Yes = -sinh^3x

Question 13

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

Answer 13

No = sinh^4x

Question 14

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

Answer 14

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

Question 15

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

Answer 15

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

Question 16

When is the function for arcoshx and arsinhx ±

Answer 16

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

Question 17

What is d/dx (cothx)

Answer 17

-cosech^2(x)

Question 18

What is d/dx (sechx)

Answer 18

-sechxtanhx

Question 19

What is d/dx (cosechx)

Answer 19

= -cosechxcothx

Question 20

What is d/dx (tanhx)

Answer 20

sech^2(x)

Question 21

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

Answer 21

sinhx –> coshx
coshx –> sinhx

Question 22

What is cosh^2(x) equal to

Answer 22

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

Question 23

What is sinh^2(x) equal to

Answer 23

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

Question 24

How do you prove differential results for hyperbolics

Answer 24

Use exponential form and differentiate normally

Question 25

How do you find the differentials of arshinx, arcoshx, and artanhx?

Answer 25

Use formula booklet

Question 26

Knowing from the formula booklet that d/dx * artanhx = 1 / (1-x^2), what is d/dx * artanh(3x)

Answer 26

Chain rule
1 / (1-9x^2) * 3
ans = 3 / (1-9x^2)

Question 27

y = arsinhx, find d/dx * arsinhx

Answer 27

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)

Question 28

What is the integral of cosech^2(x)

Answer 28

-cothx + c

Question 29

What is the integral of sech(x)tanh(x)

Answer 29

-sechx + c

Question 30

What is the integral of cosechxcothx

Answer 30

-cosechx + c

Question 31

What is cosh(4x - 1) dx

Answer 31

1/4 * sinh(4x - 1) + c

Question 32

What is 3 / root(1 + x^2) dx

Answer 32

3arsinh(x) + c

Question 33

How do you find (2 + 5x) / (root(x^2 + 1)) dx

Answer 33

Split into two fractions
2 / root(x^2 + 1) + 5x / root(x^2 + 1)
ans = 2sinh(x) + 5(x^2 + 1)^1/2 + c

Question 34

How to find tanhx dx

Answer 34

= sinhx / coshx dx
= ln(coshx) + c

Question 35

What is cosh^5(2x)sinh(2x) dx

Answer 35

1 / 12 * cosh^6(2x) + c

Question 36

What are the identities for cosh^2(x) and sinh^2(x)

Answer 36

cosh^2(x) = 1/2 + 1/2cosh(2x)
sinh^2(x) = 1/2cosh(2x) - 1/2

Question 37

How do you find cosh^2(3x) dx

Answer 37

Use identity
cosh^2(3x) = 1/2 + 1/2 cosh(6x)
ans = 1/2 * x + 1/12 * cosh(6x) + c

Question 38

What do you do if you cannot get a hyperbolic integration to work

Answer 38

Rewrite in exponential form and attempt to solve normally

Question 39

What substitution can you use when integrating hyperbolics in exponential form

Answer 39

u = e^x

Question 40

What substitutions should you use when dealing with:
a) 1 / root(a^2 + x^2)
b) 1 / root(x^2 - a^2)

Answer 40

a) x = a * sinh(u)
b) x = a * cosh(u)

Question 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

Answer 41

x = sinh(u)
Hint is on the RHS (arsinh(x))

Question 42

What does cosh^2(u)sinh(u) integrate to

Answer 42

1/3 * cosh^3(u) + c