Question

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=x(x-4)^{2},(4,0) $$

Short answer

The curvature at (4, 0) is 8, and the radius of curvature is \(\frac{1}{8}\).

Step-by-step solution

Sketch the Curve

The function given is \(y = x(x-4)^2\). To sketch the curve, first note that it is a cubic function. The roots of the equation, where \(y = 0\), are \(x = 0\) and \(x = 4\). These roots indicate intercepts on the x-axis. The point \((4,0)\) is one such intercept and part of graph. Check leading coefficient, +1, meaning end behavior as x approaches ±∞ is bond together. Plot various x-values to understand shape, including critical points from derivatives (next step).

Derivatives for Curvature

Find the first derivative \(y' = \frac{d}{dx}[x(x-4)^2]\) using the product rule: \(y' = (x-4)^2 + x (2)(x-4)\). Simplify to \(y' = 3x^2 - 16x + 16\). Then find the second derivative \(y'' = \frac{d^2}{dx^2}[3x^2 - 16x + 16]\): \(y'' = 6x - 16\).

Calculate Curvature at Point (4, 0)

Use the curvature formula \(\kappa = \frac{|y''|}{(1 + (y')^2)^{3/2}}\). First, compute \(y'\) and \(y''\) at \(x = 4\): \(y' = 3(4)^2 - 16(4) + 16 = 0\), \(y'' = 6(4) - 16 = 8\). Substitute into curvature formula: \(\kappa = \frac{|8|}{(1 + 0^2)^{3/2}} = 8\).

Find the Radius of Curvature

The radius of curvature \(R\) is the reciprocal of the curvature \(\kappa\): \(R = \frac{1}{\kappa}\). Given \(\kappa = 8\), \(R = \frac{1}{8}\).

Key concepts

Curve Sketching

To effectively sketch a curve, especially for a cubic equation like \(y = x(x-4)^2\), you start by identifying its roots. The roots, or x-intercepts, are found when \(y = 0\). For this equation, the roots are \(x = 0\) and \(x = 4\). These values indicate where the curve crosses the x-axis. In this case, the point \((4,0)\) is not just a root but also lies directly on the curve. Curve sketching also benefits from understanding the end behavior, which can be determined from the leading coefficient of the highest degree term. Here, the leading coefficient is +1, suggesting that as \(x\) moves towards positive or negative infinity, the curve stretches outwards at both ends.
Using derivatives, as you plot additional points, provides insights into the curve's shape changes. Critical points—where the first derivative is zero or undefined—indicate places where the curve may change direction. This analysis aids in drawing a more accurate curve by highlighting key features such as peaks, valleys, and inflections.

Curvature Calculation

Curvature gives us a measure of how "sharp" a curve is at a certain point, which can be quite insightful for understanding the curve's geometry. The general formula for curvature \(\kappa\) in terms of the first and second derivatives is: \[\kappa = \frac{|y''|}{(1 + (y')^2)^{3/2}}\] To calculate this curvature, we first need both the first and second derivatives of the function. For \(y = x(x-4)^2\), the first derivative calculated using the product rule is \(y' = 3x^2 - 16x + 16\), reflecting how the curve's slope changes.
Next, compute the second derivative, which is \(y'' = 6x - 16\). This second derivative shows how the slope itself changes, providing information necessary for calculating curvature at a specific point on the curve.

Radius of Curvature

Once you've calculated the curvature \(\kappa\) at a certain point, finding the radius of curvature \(R\) becomes a straightforward task. The radius of curvature is simply the reciprocal of curvature: \[R = \frac{1}{\kappa}\] At the point \((4,0)\), where the curvature \(\kappa = 8\), the radius of curvature is \(R = \frac{1}{8}\). This numerical value offers insight into the "tightness" of the curve at this point.
A smaller radius means a sharper curve, something you'd see in tighter turns on a graph or a physical road. Conversely, a larger radius indicates a gentler curve. Such understanding assists in fields ranging from physics to engineering, providing a sense of how objects might move along the path of the curve.

Differentiation

Differentiation is a powerful mathematical tool that describes how a function's value changes concerning its variable. For curve sketching and curvature analysis, derivatives are central. Starting with the function \(y = x(x-4)^2\), the use of differentiation through the product rule helps find how the rate of change itself varies:

• The first derivative \(y'\) indicates the slope of the tangent to the curve, unveiling how steeply the curve rises or falls at any point.
• The second derivative \(y''\), derived from the derivative of the first, informs on the curvature—essentially how the slope changes—highlighting concave up or down regions which are crucial in understanding curve behavior.

Differentiation not only aids in calculating slopes but also figures prominently in determining the curve's key features, enhancing the comprehension of its graphical representation and making it an indispensable tool in calculus.