How to use Euler formula to expand the functions exp (x) cosx and exp (x) SiNx into power series of X?
cosx=[e^ix+e^(-ix)]/2
e^x cosx=[e^(x+ix)+e^(x-ix)]/2
=1/2*∑[(x+ix)^n+(x-ix)^n]/n!
=1/2* ∑[x^n/n!*( (1+i)^n+(1-i)^n]
because
1+i=√2(cosπ/4+isinπ/4)
1-i=√2[cos(-π/4)+isin(-π/4)]
(1+i)^n+(1-i)^n=(√2)^n* 2cosnπ/4
So e ^ xcosx = ∑ [x ^ n / N! * (√ 2) ^ n Cosn π / 4]
Similarly:
sinx=[e^ix-e^(-ix)]/2i
e^x sinx=[e^(x+ix)-e^(x-ix)]/2i
=1/2*∑[(x+ix)^n-(x-ix)^n]/n!
=1/2* ∑[x^n/n!*( (1+i)^n+(1-i)^n]
because
1+i=√2(cosπ/4+isinπ/4)
1-i=√2[cos(-π/4)+isin(-π/4)]
(1+i)^n-(1-i)^n=(√2)^n* 2isin(nπ/4)
So e ^ xsinx = ∑ [x ^ n / N! * (√ 2) ^ n Sinn π / 4]
The function f (x) = SiNx is expanded into a power series of (x - π / 4) Do this: F (x) = SiNx = sin (x - π / 4 + π / 4) = root 2 / 2 (sin (x - π / 4) + cos (x - π / 4)) Let's talk about the direct substitution of X - π / 4 into the McLaughlin expansion of SiN x and COS x, OK?
Yes, because the McLaughlin formulas of SiNx and cosx hold true for all real numbers
SiNx = (e ^ x-e ^-x) / 2 is expanded into a power series of X
You mean SHX. Expand e ^ X and e ^ - x respectively and add them
e^x=1+x+x^2/2!+ x^3/3!+...+ x^n/n!+...
e^-x=1-x+x^2/2!- x^3/3!+...+ (-1)^nx^n/n!+...
shx=(e^x-e^-x)/2=x+x^3/3!+ x^5/5!+...+ x^(2n-1)/(2n-1)!+...
Help do some microeconomic problems Suppose that manufacturer AB and manufacturer B are competitors producing the same and different products in a market, the demand curve for manufacturer a is pa = 200-qa, and the demand curve for manufacturer B is Pb = 300-0.5qb. The current sales volumes of the two manufacturers are QA = 50 and QB = 100 respectively; (1) What are the price elasticity of demand EDA and EDB of manufacturers a and B? (2) If the price reduction of manufacturer B increases the demand of manufacturer B to 160 and reduces the demand of manufacturer a to 40, what is the cross price elasticity EAB of the demand of manufacturer a? (3) If B pursues the maximization of sales revenue, do you think manufacturer B's price reduction is a correct behavior choice?
1、edA=-3 edB=-5
2、11/6
3. Correct behavior
Microeconomic issues Q = 0.02m-2p, M = 7500, P = 30 yuan. Q. when the price rises to 40 yuan, what is the price effect of the price rise? How many bottles are the substitution effects? What is the income effect?
When p = 30 yuan, q = 0.02m-2p = 150-60 = 90 bottles
P = 40 yuan,
M = 7500, q = 0.02m-2p = 150-80 = 70 bottles, the price effect of price rise is 90-70 = 20 bottles
They are all substitution effects. The income has not changed, and the income effect is 0
Derivative problem f(x)=x^3/(x+1)
y=x^3/(x+1),
y'=(x^3)'/(x+1)+x^3 × [1/(x+1)]‘
=3x ²/ (x+1)+x^3[-1/(x+1) ²]= (2x^3+3x ²)/ (x+1) ²