| utf23-002 Let $F(x)= \int_1^x{\frac{dt }{ t}}, x>0.$ For the following $\underline {\text {do not use}}$ the fact that you know another name for this function.
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| utf23-007 Prove that Simpson's rule is exact when approximating the integral of a cubic polynomial function, and demonstrate the result for $$\int_{0}^{1}{{(x^3-3x^2+3x-1)}} dx, n=2.$$ |
| utf24-001 Solve the following differential equations subject to the prescribed initial conditions. \begin{alignat*}{6} &a) \frac{d y}{dx} = 4(x-7)^3 & &x=8, & &y=10 & &b) \frac{d y}{dx} =\frac{ x^2 + 1 }{ x^2} & &x=1, & &y=1 \\ &c) \frac{d y}{dx} = x\sqrt y & &x=0, & &y=1 & &b) \frac{d y}{dx} =\frac{ 4\sqrt{(1+y^2)^3} }{ y} & &x=0, & &y=1 \end{alignat*} |