1. ## Calculus/tangent line

Find all values of x at which the tangent line to the given curve passes through the origin for y = 1/(x + 4)

2. Originally Posted by pantera
Find all values of x at which the tangent line to the given curve passes through the origin for y = 1/(x + 4)
The tangent is a line of slope "m".

The derivative of the function gives this slope.

As the tangent passes through the origin, it's equation is y=mx, as c=0.

Hence, the point of intersection of this tangent and the curve gives us x.

$f'(x)=\frac{(x+4)(0)-1(1)}{(x+4)^2}$

The equation of the tangent(s) through (0,0) is $y=mx=\frac{-x}{(x+4)^2}$

Solving f(x) for the curve = mx for the line finds x.

$\frac{1}{x+4}=\frac{-x}{(x+4)^2}$

Solving this, bearing in mind that x=-4 is not part of the domain of f(x) or f'(x) discovers x.

Hence multiply both fractions by (x+4) before solving for x.

3. Lines look like this: $(y-y_{0}) = m(x-x_{0})$

We know we need the Origin: $y_{0} = m \cdot x_{0}$

The derivative gives the slope: $m = -\frac{1}{(x_{0}+4)^{2}}$

We know: $y_{0} = \frac{1}{x_{0}+4}$

I'm a little surprised to find only one solution. I expected two solutions when I started the problem. I was wrong.

Let's see what you get.

4. Here's why there is only one solution