The diagram shows part of the curve $$\(y=\frac{2}{(3-2 x)^{2}}-x\)$$ and its minimum point $$\(M\)$$, which lies on the $$\(x\)$$-axis. Find expressions for $$\(\frac{\mathrm{d} y}{\mathrm{~d} x}, \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\)$$ and $$\(\int y \mathrm{~d} x\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w20_qp_12 Year：2020 Question No：10(a)
Answer:
\(\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)=[8] \times\left[(3-2 x)^{-3}\right]+[-1] \quad\left(=\frac{8}{(3-2 x)^{3}}-1\right)\)
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=-3 \times 8 \times(3-2 x)^{-4} \times(-2) \quad\left(=\frac{48}{(3-2 x)^{4}}\right)\)
\(\int y \mathrm{~d} x=\left[(3-2 x)^{-1}\right][2 \div(-1 \times-2)]\left[-1 / 2 x^{2}\right](+\mathrm{c}) \quad\left(=\frac{1}{3-2 x}-\frac{1}{2} x^{2}+c\right)\)
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=-3 \times 8 \times(3-2 x)^{-4} \times(-2) \quad\left(=\frac{48}{(3-2 x)^{4}}\right)\)
\(\int y \mathrm{~d} x=\left[(3-2 x)^{-1}\right][2 \div(-1 \times-2)]\left[-1 / 2 x^{2}\right](+\mathrm{c}) \quad\left(=\frac{1}{3-2 x}-\frac{1}{2} x^{2}+c\right)\)
Knowledge points:
1.7.1 understand the gradient of a curve at a point as the limit of the gradients of a suitable sequence of chords, and use the notations for first and second derivatives (Only an informal understanding of the idea of a limit is expected.)
1.7.2 use the derivative of (for any rational ), together with constant multiples, sums and differences of functions, and of composite functions using the chain rule
1.8.1 understand integration as the reverse process of differentiation, and integrate (for any rational n except-1 , together with constant multiples, sums and differences
Solution:
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