Functions and Series Flashcards
what is the power series expansion (PSE) of e^x (only include the first first 2 terms and the nth term
e^x = 1 + x + (x^n) / n!
what is the PSE of sinx
sinx = x - (x^3) / 3! +/- (x^2n-1) / (2n-1)!
what 2 conditions need to be met in order for PSE to be used
- x needs to be small AKA < 1
- the series needs to converge
what is the formula for taylors expansion
f(x) = f(0) + xf’(0) + (x^2)f’‘(0) / 2! +…+ (x^n / n!)*f(n-dashes)(0)
where does the binomial expansion come from
taylor expansion
what is the formula for the binomial expansion
(1 + x)^n = 1 + (nx) + [(n)(n-1)x^2] / 2! +…+ you know the rest
what is the formula for sinhx
sinhx = (e^x - e^-x) / 2
what is the formula for coshx
coshx = (e^x + e^-x) / 2
what is the formula for tanhx
tanhx = (e^x - e^-x) / (e^x + e^-x)
whats kinds of functions are sinhx, coshx and tanhx
- sinhx and tanhx are odd functions
- coshx are even functions
considering sinhx and tanhx are odd functions, what does sinh(-x) equal for example
sinh(-x) = -sinh(x)
considering coshx is an even function, what does cosh(-x) equal
cosh(-x) = cosh(x)
what do sinhx and coshx equal as x tends to infinity
e^x / 2
what do sinhx and coshx equal as x tends to -infinity
- sinhx = -e^x / 2
- coshx = e^x / 2
what does cosh^2x - sinh^2x equal
cosh^2x - sinh^2x = 1
what does cosh(2x) equal
cosh(2x) = cosh^2x + sinh^2x
what is the differentiation of sinhx and coshx
the differentiation of one is just the other one
how would you rewrite sinh^-1(x) = y to work with it properly
sinh^-1(x) = y is equal to sinhy = x
what is the formula for sinh^-1(x) in terms of ln
sinh^-1(x) = ln(x + sqrt(x^2 + 1))
what is the formula for cosh^-1(x) in terms of ln
cosh^1(x) = ln(x +/- sqrt(x^2 - 1))
why can we have the +/- for the cosh-1 but not the sinh-1
- 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
what does O(x^n) +/- O(x^n) equal
O(x^n)
what does O(x^n) * O(x^m) equal
O(x^n+m)
what does cO(x^n) +/- O(x^n) equal
O(x^n), constants dont matter
what does O(x^n) + O(x^m) equal where m < n
O(x^m)
what does x*O(x^n) equal
O(x^n+1)
what are the conditions for l’hopitals rule to be applied
if f(0) = g(0) = 0 and g’(0) DOES NOT = 0
what is the rule when those rules are applied
as x tends to 0: f(x) / g(x) = f’(0) / g’(0)
what if f’(0) and g’(0) also equal 0
use the rule and conditions for f’‘(0) and g’‘(0)
what are the main functions that youd need to look in the data book to find limits and approximations for
- (n^s)*(x^n)
- x^n / n!
- (1 + x/n)^n
- (x^s)*ln(x)
what is the power dynamic between exponentials, powers and logs
- exponentials win over powers
- everything wins over logs
how would you rearrange (p + qx)^n in order to perform the binomial expansion on it
(p + qx)^n = p^n(1 + qx/p)^n
what does it mean if a function is continuous in the region x = 0
it has no breaks in it
is f(x) = 1/x^2 continuous and why
no, because f(0) is undefined so there is a break
what other condition needs to be met in order for a function to be continuous
its differentiations also need to be continuous
how would you find a good approximation of sinx for example in the vicinity of a number not that small, like x = pi/4
- 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
what is the formula for finding the series expansion of a function at another point ‘a’
f(x) = f(a) + (x-a)*f’(a) + (x-a)^2/2! * f’‘(a) +…
