1. ## equztion of tangent

(a) Find equation of the tangent line and the normal line to the curve
y
= (3x -2)e^-xat the point (0;-2);
(b) Find
dy/dx given that cos(x + y) + y = 0.

2. Originally Posted by trythis
(a) Find equation of the tangent line and the normal line to the curve
y
= (3x -2)e^-xat the point (0;-2);
(b) Find
dy/dx given that cos(x + y) + y = 0.

Tangents and normals

3. 1.)

$\displaystyle y=(3x-2)e^{-x}$

$\displaystyle \frac{dy}{dx}=3e^{-x}-(3x-2)e^{-x}=(5-3x)e^{-x}$

$\displaystyle y'(0)=5$

$\displaystyle y-y_1=m(x-x_1)$

$\displaystyle y+2=5(x-0)$

$\displaystyle y=5x-2$

$\displaystyle m_1=-\frac{1}{m}=-\frac{1}{5}$

$\displaystyle y-y_1=m_1(x-x_1)$

$\displaystyle y+2=-\frac{1}{5}(x-0)$

$\displaystyle y=-\frac{x}{5}-2$

2.)

$\displaystyle \cos(x+y)+y=0$

$\displaystyle y=-\cos(x+y)$

$\displaystyle y'=\frac{d}{dx}(-\cos(x+y))$

$\displaystyle y'=(1+y')\sin(x+y)=\sin(x+y)+y'\sin(x+y)$

$\displaystyle y'-y'\sin(x+y)=\sin(x+y)$

$\displaystyle y'[1-\sin(x+y)]=\sin(x+y)$

$\displaystyle y'=\frac{\sin(x+y)}{1-\sin(x+y)}$