Problem 29 - Calculus of a Single Variable

Question

Find the slope of the tangent line to the graph at the given point. Witch of Agnesi: \(\left(x^{2}+4\right) y=8\) Point: \((2,1)\)

Step-by-Step Solution

Answer

The slope of the tangent line to the graph of the given function at the point (2,1) is -1/2

Step 1: Rewrite The Equation

We rewrite the given equation \(\left(x^{2}+4\right) y=8\) in implicit form in terms of y and isolate y to obtain: \[ y = \frac{8}{{x^2 + 4}} \]

Step 2: Compute the Derivative

In order to find the slope of tangent at a point, we need to find the derivative of the function. The derivative of this function is found using power rule and chain rule of differentiation. \[y' = \frac{{-16x}}{{(x^2+4)^2}}\]

Step 3: Substitute X-Value

Now substitute x = 2, the x-coordinate of the given point, into the derivative for y to get the slope: \[ m = y'(2) = \frac{{-16*2}}{{(2^2+4)^2}} = -\frac{1}{2}\]