Problem 25 - Thomas Calculus

Question

The formula $$ \kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left[1+\left(f^{\prime}(x)\right)^{2}\right]^{3 / 2}} $$ derived in Exercise \(5,\) expresses the curvature \(\kappa(x)\) of a twice- differentiable plane curve \(y=f(x)\) as a function of \(x\) . Find the curvature function of each of the curves. Then graph \(f(x)\) together with \(\kappa(x)\) over the given interval. You will find some surprises. $$ y=\sin x, \quad 0 \leq x \leq 2 \pi $$

Step-by-Step Solution

Answer

The curvature function is \(\kappa(x) = \frac{\sin |x|}{\sqrt{8}}\).

Step 1: Calculate the first derivative

Start by finding the first derivative of the function \(y = \sin x\). The derivative, \(f'(x)\), is given by \(\cos x\).

Step 2: Calculate the second derivative

Next, calculate the second derivative of the function. The derivative, \(f''(x)\), is \(-\sin x\), as the derivative of \(\cos x\) is \(-\sin x\).

Step 3: Substitute into the curvature formula

Now substitute \(f'(x) = \cos x\) and \(f''(x) = -\sin x\) into the curvature formula:\[\kappa(x) = \frac{|f''(x)|}{[1 + (f'(x))^2]^{3/2}}\]This becomes:\[\kappa(x) = \frac{| -\sin x |}{[1 + (\cos x)^2]^{3/2}}\]

Step 4: Simplify the curvature expression

Simplify the expression by recognizing that \(1 + \cos^2 x = 2 - \sin^2 x\), but also note that \(1 + \cos^2 x = 1 + (1 - \sin^2 x) = 2\), due to the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Thus, \[\kappa(x) = \frac{| -\sin x |}{2^{3/2}}\]And further simplifies to:\[\kappa(x) = \frac{\sin |x|}{\sqrt{8}}\]

Step 5: Graph \(y = \sin x\) with \(\kappa(x)\)

Sketch the graphs of \(y = \sin x\) and \(\kappa(x) = \frac{\sin |x|}{\sqrt{8}}\) over the interval \(0 \leq x \leq 2\pi\).- The \(\sin x\) graph is a standard sine wave.- The \(\kappa(x)\) graph indicates the curvature and reaches maximum values when \(\sin x\) is maximized or minimized. It shows the most curvature around \(x = \frac{\pi}{2}\) and \(x = \frac{3\pi}{2}\).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that describes how a function changes at any given point. It involves finding the derivative of a function, which is a new function that gives the slope of the original function at each point. This slope or rate of change helps us understand the behavior and shape of the graph of the function.In the exercise provided, the function we are working with is the sine function, \(y = \sin x\). The first step is to find its first derivative, \(f'(x)\). For \(y = \sin x\), the first derivative is \(\cos x\). This means at any point \(x\), the slope of the tangent to the curve \(y = \sin x\) is given by \(\cos x\).The next step involves finding the second derivative, \(f''(x)\). Differentiating \(\cos x\) gives us \(-\sin x\). The second derivative provides information about the curvature of the graph, indicating how the slope of a function is changing at a particular point.

• The first derivative \(f'(x)\) indicates the rate of change of \(y\) with respect to \(x\).
• The second derivative \(f''(x)\) helps determine points of inflection and concavity of the graph.

Understanding these derivatives is essential for analyzing the behavior of functions in calculus.

Plane Curves

A plane curve is a curve that lies entirely on a single plane. It can be represented using a function, like the sine function \(y = \sin x\) in the given exercise. Plane curves are often studied in calculus to understand geometric properties such as curvature, which is a measure of how sharply a curve bends at a particular point.The curvature, denoted by \(\kappa(x)\), provides insights into the geometry of a plane curve. It is derived for functions that are at least twice differentiable. For a given function \(y = f(x)\), the formula for curvature is:\[\kappa(x)=\frac{|f''(x)|}{[1+(f'(x))^2]^{3/2}}.\]This formula uses the first and second derivatives to calculate curvature, showing how much the curve deviates from being straight.

• Curvature is zero for a straight line because there is no bending.
• High curvature values indicate sharp turns in the graph.
• Plane curves can demonstrate different bending behaviors, visible in the variation of \(\kappa(x)\) along the curve.

In the context of our exercise, you see that the curvature \(\kappa(x)\) is especially high at the points where the sine wave reaches its maximum or minimum. This is where the plane curve bends most sharply.

Graphing Functions

Graphing functions is a key skill in understanding calculus. It allows you to visually interpret the relationship between variables. For the function \(y = \sin x\), graphing means plotting a sine wave, which oscillates between -1 and 1. This periodic behavior repeats every \(2\pi\) units.When we are asked to graph alongside the curvature \(\kappa(x)\), we gain further insight into the geometric features of the sine wave. The curvature function \(\kappa(x) = \frac{\sin |x|}{\sqrt{8}}\) describes how sharply the curve \(y = \sin x\) bends at each point.

• The graph of \(y = \sin x\) is a smooth, continuous wave.
• Plotting \(\kappa(x)\) helps you see areas of highest curvature, corresponding to peaks and troughs of the sine wave.
• Comparing these graphs helps to visualize the interplay between the function and its geometric properties.

Graphing multiple functions together, like \(y = \sin x\) and \(\kappa(x)\), enables the synthesis of algebraic and geometric understanding, which is crucial in advanced mathematical studies.