Question

In Exercises \(9-12,\) at the indicated point find (a) the slope of the curve, (b) an equation of the tangent, and (c) an equation of the tangent. (d) Then draw a graph of the curve, tangent line, and normal line in the same square viewing window. $$v=x^{2} \quad\( at \)\quad x=-2$$

Short answer

The slope of the curve at \(x=-2\) is -4. The equation of the tangent line is \(y=-4x-4\). The equation of the normal line is \(y = (1/4)x +4.5\).

Step-by-step solution

Finding the derivative

The derivative of a function gives the slope of the line tangent to the function at a given point. In this context, the derivative of the function \(v = x^2\) is \(v'=2x\). This gives the slope of the tangent line to the curve at any point \(x\).

Finding the slope at \(x=-2\)

Plug \(x=-2\) into the derivative \(v'=2x\) to find the slope of the tangent line at that point. On doing so, we get \(v'(\-2) = 2*-2 = -4\). So the slope of the tangent line at \(x=-2\) is -4.

Finding the equation of the tangent line

A line tangent to a point on a curve has the form y = mx + b, where m is the slope and b is the y-intercept. We already know our slope is -4, and since the line is tangent to the curve at \(x=-2,\) the y-coordinate for this point on \(v=x^2\) is \((-2)^2 = 4.\) Therefore, the equation of the tangent line is \(y = -4x +b.\) Substituting the point (-2,4) in to solve for \(b\), we find \(b = 4 - (-4*-2) = -4,\) and our equation of the tangent line becomes \(y = -4x - 4.\)

Finding the equation of the normal line

The normal line to a curve at a given point is the line perpendicular to the tangent at that point. The slope of the normal line is the negative inverse of the slope of the tangent line. Therefore, the slope of the normal line is \(-1/(-4) = 1/4\). Thus, the equation of the normal line follows the same steps as in step 3, substituting the new slope: \(y = (1/4)x + b.\) Solving for \(b\) using the point \(-2,4,\) we find \(b=4-(1/4*-2) = 4.5.\), so the equation of the normal line becomes \(y = (1/4)x + 4.5.\)

Drawing a graph of the curve, tangent line, and normal line

Usually, this step would use a graphing tool or software. Make sure to accurately graph the function, \(v = x^2,\) and the tangent line, \(y = -4x -4,\) and the normal line, \(y = (1/4)x +4.5,\) on the same graph. The viewing window should be set in a way that all important points and lines are clearly visible.