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Chapter 6: Problem 60

Find the equation of the tangent line to the parabola at the given point. $$y=-2 x^{2},(2,-8)$$

Short Answer

Expert verified
The equation of the tangent line to the parabola at the point (2,-8) is \(y = -8x + 16\).

Step by step solution

01

Find the derivative.

For it is necessary to find the derivative (slope) of the given parabola. The equation provided, \(y = -2x^{2}\), has to what is namely the power rule. The derivative, y', is given by taking the power (2) and multiplying it by the coefficient (-2), then decreasing the power in the new equation by 1, hence, \(y' = -4x\).
02

Find the slope at the given point.

The next step after finding the derivative equation is to plug in the x-value from the given point into the derivative equation to find the slope at that point. Here, at x = 2, the slope, \(m = -4 * 2 = -8\).
03

Use slope-intercept form to get tangent line equation.

Once the slope is determined, we can use the given point (2,-8) and the slope point form equation to find the equation of the tangent line. The slope point form equation is \(y - y1 = m(x - x1)\), where (x1,y1) is the given point and m is the slope. Thus, substituting the values gives \(y + 8 = -8(x - 2)\) which can be simplified to \(y = -8x + 16\).

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Most popular questions from this chapter

In Exercises \(91-116\), convert the polar equation to rectangular form. $$\theta=5 \pi / 3$$

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