Question

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=2 e^{1-x^{2}}} \quad \frac{\text { Point }}{\left(1,2\right)}\)

Short answer

The equation of the tangent line to the function \(y=2e^{1-x^{2}}\) at the point (1,2) is \(y=-4x+6\). The graphical solution and the derivative feature of the graphing utility confirm the accuracy of this result.

Step-by-step solution

Computing the Derivative

The first step is to compute the derivative of the given function \(y=2e^{1-x^{2}}\). Using the rules of derivatives; chain rule and power rule in particular, the derivative of the function \(f'(x)\) is obtained as \(-4x e^{1-x^{2}}\).

Calculate the slope of the tangent line at the point

Next, the derivative \(f'(x)\) is evaluated at the given point (1,2) which is the slope of the tangent line at that point. Substituting \(x=1\), in the derivative, the slope \(m\) is computed as \(m=f'(1)=-4 * 1 * e^{1-1^{2}}=-4e^{0}=-4\).

Determine the equation of the tangent line

With the slope \(m\) at the point (1,2), the equation of the tangent line can be obtained using the point slope form of a line \(y-y_{1}=m(x-x_{1})\). Substituting \(x_{1}=1\), \(y_{1}=2\) and \(m=-4\), the equation of the tangent at the point (1,2) is thus, \(y-2=-4(x-1)\) or \(y=-4x+6\).

Visualize the graph using a graphing utility

The next step is to graph the original function \(y=2e^{1-x^{2}}\) and the tangent line \(y=-4x+6\) using a graphing utility to confirm the equation of the tangent line.

Compare with a graphing utility’s derivative feature

Lastly, the derivative feature of the graphing utility is used to compute the derivative at the point (1,2), and this result is compared with the one derived analytically in step 2 to confirm the correctness of the previous steps.