Question

Find the tangent to the witch of Agnesi. $$y=\frac{8}{4+x^{2}}$$ at the point \((2,1)\)

Short answer

The equation of the tangent to the witch of Agnesi at the point (2,1) is \(y = -x + 3\).

Step-by-step solution

Derivative of the function

To find the tangent line to a function at a certain point, we first need to find the derivative of given function. The given function is \(y = \frac{8}{4+x^2}\). By applying the power rule for differentiation which says that the derivative of \(x^n\) is \(nx^{n-1}\), the derivative of the function \(y\) can be expressed as \(y'=-\frac{16x}{(x^2+4)^2}\).

Finding the slope

Next, plug the x-value of the point of tangency, which is 2, into the derivative to find the slope of the tangent line. This will give \(m = -\frac{16*2}{(2^2+4)^2} = -1\). So, the tangent line has a slope of -1 at the point (2, 1).

Finding the equation of the tangent line

Now we use the point-slope formula which states that \(y - y1 = m (x - x1)\) where \((x1, y1) = (2, 1)\) and the slope \(m = -1\). Substituting in the known values gives the equation of the tangent line as \(y - 1 = -1 (x - 2)\), therefore, the equation of the tangent line at the point (2,1) is \(y = -x + 3\) .