It is given that $y={{3}^{\left( {{x}^{2}}+2x \right)}}$ Show that \(\frac{{{\mathrm{d}}

Question

It is given that $y={{3}^{\left( {{x}^{2}}+2x \right)}}$ Show that $\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=\frac{1}{y}{{\left( \frac{\mathrm{d}y}{\mathrm{d}x} \right)}^{2}}+2y\ln 3$.[3] Find Maclaurin’s series for $y$, up to and including the term in ${{x}^{2}}$, leaving the coefficients in exact form.[3] Hence estimate the value of $\displaystyle\displaystyle \displaystyle\int_{0}^{1}{{{3}^{\left( {{x}^{2}}+2x \right)}}\,}\mathrm{d}x$, giving your answer in exact form. Explain how the accuracy of the estimation can be improved.[3]

Accepted Answer

$\text{(i)}\ \displaystyle y''=\frac{(y')^2}{y}+2y\ln3$ $\text{(ii)}\ \displaystyle 3^{x^2+2x}=1+2x\ln3+x^2\ln3(2\ln3+1)+O(x^3)$ $\text{(iii)}\ \displaystyle \int_0^1 3^{x^2+2x}\,dx\approx1+\ln3+\frac{\ln3(2\ln3+1)}{3}$; retain more terms to improve the accuracy.

2013 MI P1 Q2

2013 MI P1 Q2

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