Question

Find the slope of the tangent and the normal to the following curve at the indicted point y = 2x² + 3 sin x at x = 0 ?

Answer 1

\[ y = 2 x^2 + 3 \sin x\]

\[ \Rightarrow \frac{dy}{dx} = 4x + 3 \cos x\]

When `x=0`

`y=2x^2+3sin x`

`=2(0)^2+3sin 0`

`=0`

\[\text { Now }, \]

\[\text { Slope of the tangent }= \left( \frac{dy}{dx} \right)_\left( 0, 0 \right)  =4\left( 0 \right)+ 3 \cos 0=3\]

\[\text { Slope of the normal }=\frac{- 1}{\left( \frac{dy}{dx} \right)_\left( 0, 0 \right)}=\frac{- 1}{3}\]