A curve has parametric equations $$\( x=\mathrm{e}^{t}-2 \mathrm{e}^{-t}, \quad y=3 \mathrm{e}^{2 t}+1 . \)$$ Find the equation of the tangent to the curve at the point for which $$\(t=0\)$$. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
Exam No：9709_s20_qp_21 Year：2020 Question No：3
Answer:
State \(\frac{\mathrm{d} x}{\mathrm{~d} t}=\mathrm{e}^{t}+2 \mathrm{e}^{-t}, \quad \frac{\mathrm{d} y}{\mathrm{~d} t}=6 \mathrm{e}^{2 t}\)
Use \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} t} / \frac{\mathrm{d} x}{\mathrm{~d} t}\) either in terms of \(t\) or after substitution of \(t=0\)
Obtain gradient of tangent is 2
Attempt equation of tangent with numerical gradient and coordinates
Obtain \(y=2 x+6\) or equivalent
Use \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{d} y}{\mathrm{~d} t} / \frac{\mathrm{d} x}{\mathrm{~d} t}\) either in terms of \(t\) or after substitution of \(t=0\)
Obtain gradient of tangent is 2
Attempt equation of tangent with numerical gradient and coordinates
Obtain \(y=2 x+6\) or equivalent
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
2.4.3 find and use the first derivative of a function which is defined parametrically or implicitly.
Solution:
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