Question

Find the equation of Tangent line to the curve ;

x=e^t , y=e^-t at t=1

Answer 1

Solution;

To find the equation of the tangent we need the slope;

$m=\frac{dy}{dx}$ At the point of tangency $(x_o,y_o)$

Then the equation of tangent is ;

$(y-y_o)=m(x-x_o)$

So we compute;

$\frac{dy}{dt}=-e^{-t}$

$\frac{dx}{dt}=e^t$

Therefore;

$\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=\frac{-e^{-t}}{e^t}=-e^{-2t}$

At t=1;

$m=-0.135$

$x_0=e^{1}=2.718$

$y_o=0.368$

Hence, equation of tangent is;

$(y-0.368)=-0.135(x-2.718)$

$y=3.086-0.135x$