Question and Answers Forum

All Question   Topic List

Question Number 163300 by amin96 last updated on 05/Jan/22

∫_0 ^2 f(x)dx⋍f(a)+f(2−a)

$$\int_{\mathrm{0}} ^{\mathrm{2}} \boldsymbol{{f}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}\backsimeq\boldsymbol{{f}}\left(\boldsymbol{{a}}\right)+\boldsymbol{{f}}\left(\mathrm{2}−\boldsymbol{{a}}\right) \\ $$ $$ \\ $$ For what values ​​of a is the following formula accurate for polynomials of degree 3?\\n

Answered by MJS_new last updated on 05/Jan/22

f(x)=c_3 x^3 +c_2 x^2 +c_1 x+c_0 ∫_0 ^2 f(x)dx=f(a)+f(2−a) 4c_3 +(8/3)c_2 +2c_1 +2c_0 =2(3c_3 +c_2 )a^2 −4(3c_3 +c_2 )a+2(4c_3 +2c_2 +c_1 +c_0 ) a^2 −2a+(2/3)=0 a=1±((√3)/3)

$${f}\left({x}\right)={c}_{\mathrm{3}} {x}^{\mathrm{3}} +{c}_{\mathrm{2}} {x}^{\mathrm{2}} +{c}_{\mathrm{1}} {x}+{c}_{\mathrm{0}} \\ $$ $$\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}={f}\left({a}\right)+{f}\left(\mathrm{2}−{a}\right) \\ $$ $$\mathrm{4}{c}_{\mathrm{3}} +\frac{\mathrm{8}}{\mathrm{3}}{c}_{\mathrm{2}} +\mathrm{2}{c}_{\mathrm{1}} +\mathrm{2}{c}_{\mathrm{0}} =\mathrm{2}\left(\mathrm{3}{c}_{\mathrm{3}} +{c}_{\mathrm{2}} \right){a}^{\mathrm{2}} −\mathrm{4}\left(\mathrm{3}{c}_{\mathrm{3}} +{c}_{\mathrm{2}} \right){a}+\mathrm{2}\left(\mathrm{4}{c}_{\mathrm{3}} +\mathrm{2}{c}_{\mathrm{2}} +{c}_{\mathrm{1}} +{c}_{\mathrm{0}} \right) \\ $$ $${a}^{\mathrm{2}} −\mathrm{2}{a}+\frac{\mathrm{2}}{\mathrm{3}}=\mathrm{0} \\ $$ $${a}=\mathrm{1}\pm\frac{\sqrt{\mathrm{3}}}{\mathrm{3}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com