The diagram shows the curve with parametric equations $$\[ x=4 t+\mathrm{e}^{2 t}, \quad y=6 t \sin 2 t, \]$$ for $$$0 \leqslant t \leqslant 1$$$. The point $$$P$$$ on the curve has parameter $$$p$$$ and $$$y$$$-coordinate 3 . Find the gradient of the curve at $$$P$$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s21_qp_23 Year：2021 Question No：7(d)

Answer:

$\operatorname{Obtain} \frac{\mathrm{d} x}{\mathrm{~d} t}=4+2 \mathrm{e}^{2 t}$
Use product rule to find $\frac{\mathrm{d} y}{\mathrm{~d} t}$
Obtain $6 \sin 2 t+12 t \cos 2 t$
Divide to obtain $\frac{\mathrm{d} y}{\mathrm{~d} x}$ using their $\frac{\mathrm{d} y}{\mathrm{~d} t}$ and $\frac{\mathrm{d} x}{\mathrm{~d} t}$ correctly
Obtain $0.826$

Knowledge points:

2.4.1 use the derivatives of ,ln x,sin x,cos x,tan x, together with constant multiples, sums, differences and composites

2.4.2 differentiate products and quotients

2.4.3 find and use the first derivative of a function which is defined parametrically or implicitly.

Answer:

Knowledge points:

Solution:

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