Question

What is the slope of the line tangent to the curve y+2 = (x^2/2) - 2siny at the point (2,0)?

Answer 1

The slope of the line tangent to the given curve at the point (2,0) is 2/3.

Explanation:

The given equation is

`y+2=((x^2)/(2))-2\sin y`

we need to find the slope of the line tangent to the given curve at the point (2,0).

`slope=(dy)/(dx)_((2,0))`

Differentiate given equation with respect to x.

`(dy)/(dx)+0=(2x)/(2)-2(\cos y)(dy)/(dx)`

`(dy)/(dx)=x-2\cos y(dy)/(dx)`

`(dy)/(dx)+2\cos y(dy)/(dx)=x`

`(1+2\cos y)(dy)/(dx)=x`

Divide both sides by(1+2cos y).

`(dy)/(dx)=(x)/((1+2\cos y))`

Substitute x=2 and y=0.

`(dy)/(dx)_((2,0))=(2)/(1+2\cos (0))`

`(dy)/(dx)_((2,0))=(2)/(1+2(1))`

`(dy)/(dx)_((2,0))=(2)/(3)`

Therefore the slope of the line tangent to the given curve at the point (2,0) is 2/3.