The diagram shows the line $$\(x=\frac{5}{2}\)$$, part of the curve $$\(y=\frac{1}{2} x+\frac{7}{10}-\frac{1}{(x-2)^{\frac{1}{3}}}\)$$ and the normal to the curve at the point $$\(A\left(3, \frac{6}{5}\right)\)$$ Find the $$\(x\)$$-coordinate of the point where the normal to the curve meets the $$\(x\)$$-axis. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w21_qp_12 Year：2021 Question No：11(a)
Answer:
\(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{2}+\frac{1}{3(x-2)^{\frac{4}{3}}}\)
Attempt at evaluating their \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=3\left[\frac{1}{2}+\frac{1}{3(3-2)^{\frac{4}{3}}}=\frac{5}{6}\right]\)
Gradient of normal \(=\frac{-1}{\text { their } \frac{d y}{d x}}\left[=-\frac{6}{5}\right]\)
Equation of normal \(y-\frac{6}{5}=(\) their normal gradient \()(x-3)\)
\(\left[y=-\frac{6}{5} x+4.8 \Rightarrow 5 y=-6 x+24\right]\)
[When \(y=0,] x=4\)
Attempt at evaluating their \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=3\left[\frac{1}{2}+\frac{1}{3(3-2)^{\frac{4}{3}}}=\frac{5}{6}\right]\)
Gradient of normal \(=\frac{-1}{\text { their } \frac{d y}{d x}}\left[=-\frac{6}{5}\right]\)
Equation of normal \(y-\frac{6}{5}=(\) their normal gradient \()(x-3)\)
\(\left[y=-\frac{6}{5} x+4.8 \Rightarrow 5 y=-6 x+24\right]\)
[When \(y=0,] x=4\)
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.)
Solution:
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