The diagram shows the curve with equation $$$y=\frac{1}{(3 x-2)^{\frac{3}{2}}}$$$. The shaded region is bounded by the curve, the $$$x$$$-axis and the lines $$$x=1$$$ and $$$x=2$$$. The shaded region is rotated through $$$360^{\circ}$$$ about the $$$x$$$-axis. The normal to the curve at the point $$$(1,1)$$$ crosses the $$$y$$$-axis at the point $$$A$$$. Find the $$$y$$$-coordinate of $$$A$$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w21_qp_11 Year：2021 Question No：10(c)

Answer:

$
\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{3}{2} \times 3(3 x-2)^{-\frac{5}{2}}
$
At $x=1, \frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{9}{2}$
[Equation of normal is] $y-1=\frac{2}{9}(x-1)$ OR evaluates $c$
At $A, \quad y=\frac{7}{9}$

Knowledge points:

1.3.2 interpret and use any of the forms in solving problems (Including calculations of distances, gradients, midpoints, points of intersection and use of the relationship between the gradients of parallel and perpendicular lines.)

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

1.7.3 apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change (Including connected rates of change, e.g. given the rate of increase of the radius of a circle, find the rate of increase of the area for a specific value of one of the variables.)

Answer:

Knowledge points:

Solution:

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