I find a problem that have no idea for it. in calculating $ \int^{1}_{0} (x^6-mx^5)dx $ we know Trapezoidal Error is lower than Simpson Error. what is the range of $m$?
Solution: $\frac {217}{210} < m< \frac {263}{238}$
Option:
1) $\frac {15}{14} < m< \frac {85}{74}$
2) $\frac {23}{48} < m< \frac {57}{63}$
3) $\frac {237}{215} < m< \frac {279}{193}$
4) $\frac {217}{210} < m< \frac {263}{238}$
How this calculated?
I think you are expected to compute the integral three ways: exactly, using a single interval of the trapezoidal rule, and using a single application of Simpsion's rule. The exact value is $E=\frac 17-\frac m6$. The trapezoidal value is $T=\frac12(f(0)+f(1))=\frac 12(1-m)$. The Simpson's value is $S=\frac 16(f(0+4f(\frac 12)+f(1))=\frac 16(\frac 1{16}-\frac m8+1-m)=\frac {17}{96}-\frac {9m}{48}$ The error of the trapezoidal method is $|E-T|$ and the error of Simpson's method is $|E-S|$ You are expected to find the range of $m$ such that $|E-T| \lt |E-S|$. Alpha reports the range as $\frac {31}{30} \lt m \lt \frac {263}{238}$, which is choice $4$. Mark your book down for not reducing $\frac {217}{210}$ to lowest terms.
Added: In the inequality $|\frac m3 -\frac 5{14}|\lt |\frac m{48}-\frac {23}{672}|$ each side represents a V shaped graph where the point of the V is the place where it equals zero. The Vs will typically have two points of intersection and the solution will either be the part of the line between them or the part outside. To solve it, we need to find the points of intersection. With two absolute value signs, we have four choices of the signs of the contents. We make each assumption, solve the resulting linear equation, and check if the signs work according to our assumption. So if the contents are both positive, we are solving $\frac m3 -\frac 5{14}=\frac m{48}-\frac {23}{672}$, which gives $m=\frac {31}{30}$. You can plug that in and find both sides are positive as assumed, so this is one point of intersection. Try the other three combinations and you will find the other intersection point. Then check a value for $m$ that is between the intersection points to find the sense of the inequality. The graph in the Alpha page I linked to should help.