Problem 93 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(y' = x\).

Step 2: 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.

Step 3: 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 \(y = 4x - 8\).