Question

What is the equation of the tangent line of `f(x)=1/x+1` at `x=2`?

Answer 1

`y=-1/4x+2`

Explanation:

To find the equation of the tangent we require to know m . it's gradient and a point .
f'(2) will give us the value of m and evaluating f(2) a point on the line.

express `f(x)=1/x+1=x^-1+1`

`rArrf'(x)=-x^-2=-1/x^2`

`rArrf'(2)=-1/(2)^2=-1/4=m`

and `f(2)=1/2+1=3/2rArr(2,3/2)" is a point on tangent line"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(y-b=m(x-a))color(white)(a/a)|)))`
is the equation of line where m is gradient and (a ,b) a point.

using `m=-1/4" and " (a,b)=(2,3/2)`

`y-3/2=-1/4(x-2)`

`rArry=-1/4x+2" is equation of tangent line at x = 2"`
graph{(y-1/x-1)(y+1/4x-2)=0 [-10, 10, -5, 5]}