Functions and Series

Question 1

what is the power series expansion (PSE) of e^x (only include the first first 2 terms and the nth term

Answer 1

e^x = 1 + x + (x^n) / n!

Question 2

what is the PSE of sinx

Answer 2

sinx = x - (x^3) / 3! +/- (x^2n-1) / (2n-1)!

Question 3

what 2 conditions need to be met in order for PSE to be used

Answer 3

• x needs to be small AKA < 1

- the series needs to converge

Question 4

what is the formula for taylors expansion

Answer 4

f(x) = f(0) + xf’(0) + (x^2)f’‘(0) / 2! +…+ (x^n / n!)*f(n-dashes)(0)

Question 5

where does the binomial expansion come from

Answer 5

taylor expansion

Question 6

what is the formula for the binomial expansion

Answer 6

(1 + x)^n = 1 + (nx) + [(n)(n-1)x^2] / 2! +…+ you know the rest

Question 7

what is the formula for sinhx

Answer 7

sinhx = (e^x - e^-x) / 2

Question 8

what is the formula for coshx

Answer 8

coshx = (e^x + e^-x) / 2

Question 9

what is the formula for tanhx

Answer 9

tanhx = (e^x - e^-x) / (e^x + e^-x)

Question 10

whats kinds of functions are sinhx, coshx and tanhx

Answer 10

• sinhx and tanhx are odd functions

- coshx are even functions

Question 11

considering sinhx and tanhx are odd functions, what does sinh(-x) equal for example

Answer 11

sinh(-x) = -sinh(x)

Question 12

considering coshx is an even function, what does cosh(-x) equal

Answer 12

cosh(-x) = cosh(x)

Question 13

what do sinhx and coshx equal as x tends to infinity

Answer 13

e^x / 2

Question 14

what do sinhx and coshx equal as x tends to -infinity

Answer 14

• sinhx = -e^x / 2

- coshx = e^x / 2

Question 15

what does cosh^2x - sinh^2x equal

Answer 15

cosh^2x - sinh^2x = 1

Question 16

what does cosh(2x) equal

Answer 16

cosh(2x) = cosh^2x + sinh^2x

Question 17

what is the differentiation of sinhx and coshx

Answer 17

the differentiation of one is just the other one

Question 18

how would you rewrite sinh^-1(x) = y to work with it properly

Answer 18

sinh^-1(x) = y is equal to sinhy = x

Question 19

what is the formula for sinh^-1(x) in terms of ln

Answer 19

sinh^-1(x) = ln(x + sqrt(x^2 + 1))

Question 20

what is the formula for cosh^-1(x) in terms of ln

Answer 20

cosh^1(x) = ln(x +/- sqrt(x^2 - 1))

Question 21

why can we have the +/- for the cosh-1 but not the sinh-1

Answer 21

• because sqrt(x^2 -1) will be smaller than x
• meaning the number in the ln will always be +ve anyway
• this is not the case for sinh-1

Question 22

what does O(x^n) +/- O(x^n) equal

Answer 22

O(x^n)

Question 23

what does O(x^n) * O(x^m) equal

Answer 23

O(x^n+m)

Question 24

what does cO(x^n) +/- O(x^n) equal

Answer 24

O(x^n), constants dont matter

Question 25

what does O(x^n) + O(x^m) equal where m < n

Answer 25

O(x^m)

Question 26

what does x*O(x^n) equal

Answer 26

O(x^n+1)

Question 27

what are the conditions for l’hopitals rule to be applied

Answer 27

if f(0) = g(0) = 0 and g’(0) DOES NOT = 0

Question 28

what is the rule when those rules are applied

Answer 28

as x tends to 0: f(x) / g(x) = f’(0) / g’(0)

Question 29

what if f’(0) and g’(0) also equal 0

Answer 29

use the rule and conditions for f’‘(0) and g’‘(0)

Question 30

what are the main functions that youd need to look in the data book to find limits and approximations for

Answer 30

• (n^s)*(x^n)
• x^n / n!
• (1 + x/n)^n
• (x^s)*ln(x)

Question 31

what is the power dynamic between exponentials, powers and logs

Answer 31

• exponentials win over powers

- everything wins over logs

Question 32

how would you rearrange (p + qx)^n in order to perform the binomial expansion on it

Answer 32

(p + qx)^n = p^n(1 + qx/p)^n

Question 33

what does it mean if a function is continuous in the region x = 0

Answer 33

it has no breaks in it

Question 34

is f(x) = 1/x^2 continuous and why

Answer 34

no, because f(0) is undefined so there is a break

Question 35

what other condition needs to be met in order for a function to be continuous

Answer 35

its differentiations also need to be continuous

Question 36

how would you find a good approximation of sinx for example in the vicinity of a number not that small, like x = pi/4

Answer 36

• youd ‘shift the origin’
• basically write a y = equation that would make y small with the vicinity range you have
• in this case y = x - pi/4
• rearrange for x = y + pi/4
• sinx = sin(y + pi/4), can be expanded
• now youd have a siny and a cosy, and because y is small you can use PSE on them

Question 37

what is the formula for finding the series expansion of a function at another point ‘a’

Answer 37

f(x) = f(a) + (x-a)*f’(a) + (x-a)^2/2! * f’‘(a) +…