CPM Homework Help : APCALC Problem 8-125

Determine the point of intersection of the two lines tangent to $y = \frac { 1 } { 1 + x ^ { 2 } }$ at $x = -1$ and $x = 2$ .

Step 1:

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

Step 2:

Tangent at $x = -1$ :

$\text{Point: }y(-1)= \frac{1}{1+(-1)^{2}}=\frac{1}{2}$

$\text{Slope: }y'(-1)= -\frac{2(-1)}{(1+(-1)^{2})^{2}}=\frac{1}{2}$

$\text{Tangent line: }y-\frac{1}{2}=\frac{1}{2}(x+1)$

Step 3:

Write the equation of the tangent line at $x = 2$ .

Step 4:

Solve a system of equations to determine the point of intersection between the two tangent lines.