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Chapter 6: Problem 58

Find the equation of the tangent line to the parabola at the given point. $$x^{2}=2 y,\left(-3, \frac{9}{2}\right)$$

Short Answer

Expert verified
The equation of the tangent line to the parabola at the given point is \(y = -3x + \frac{9}{2}\).

Step by step solution

01

Re-write the function

First, to work with the function in a more conventional format, rearrange the equation as a function of x, \(y=\frac{x^{2}}{2}\).
02

Find derivative of the function

The next step is to find the derivative of the function f(x). Use regular rules of differentiation: \(f'(x) = \frac{d}{dx}\left(\frac{x^{2}}{2}\right)=x\).
03

Substitute x-coordinate of the given point into the derivative

Now, to find the slope of the tangent line at the given point, substitute the x-value of the point (-3) into the derivative. \(f'(-3)= -3\). So, the slope of the tangent line at this point is -3.
04

Apply Point-Slope formula to find the equation of a line

Now that we have the slope of the tangent line and the coordinates of the point it touches, we can find the equation of the tangent line. To do this, we use the point-slope formula y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope. The equation of line is then y - \(\frac{9}{2}\) = -3(x - (-3)). Simplify to get y = -3x - \(\frac{9}{2}\) + 9.
05

Simplify the equation

Finally, after simplifying the equation we get y = -3x + \(\frac{9}{2}\).

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