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Chapter 10: Problem 93

Find an equation of the tangent line to the parabola at the given point. $$x^{2}=2 y,(4,8)$$

Short Answer

Expert verified
The equation of the tangent line to the parabola \(x^{2}=2 y\) at the point (4,8) is \(y = 4x - 8\).

Step by step solution

01

Deriving Parabola's Equation

The derivative of the function \(x^{2}=2 y\) is obtained by applying the chain rule. Taking the derivative with respect to \(x\) of both sides gives \(\frac{d}{dx}(x^{2}) = 2 \frac{d}{dx}( y)\). This simplifies to \(2x = 2 y'\), where \(y'\) is the derivative of \(y\) with respect to \(x\). Solving for \(y'\) (the slope of the tangent line), we get \(y' = x\).
02

Find the Slope at a Specific Point

To find the slope of the tangent line at the point (4,8), substitute \(x = 4\) into the derivative equation \(y' = x\). This will give \(y' = 4\). That's the slope of the tangent line at the given point.
03

Finding the Equation of the Tangent Line

Assuming \(y1\) and \(x1\) represent the point of tangency on the parabola (4,8), the equation of the tangent line using the point-slope form equation (y - \(y1'\)) = m (x - \(x1\')) can be written as: y - 8 = 4(x - 4). Distributing through the parentheses and then simplifying, the equation becomes \(y = 4x - 8\).

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