Question

Consider the following curves. a. Graph the curve. b. Compute the curvature. c. Graph the curvature as a ficnction of the parameter. d. Identify the points (if any) at which the curve has a maximum or minimum curvature. e. Verify that the graph of the curvature is consistent with the graph of the curve. $$\mathbf{r}(t)=\langle t, \sin t\rangle, \text { for } 0 \leq t \leq \pi \quad \text { (sine curve) }$$

Short answer

Based on the given information, a. Graph the curve representing the parametric equation $$\mathbf{r}(t) = \langle t, \sin t \rangle \text { for } 0 \leq t \leq \pi$$. b. Compute the curvature and find the expression for the curvature $$\kappa(t) = \frac{|\sin t|}{(1 + \cos^2 t)^\frac{3}{2}}$$. c. Graph the curvature as a function of the parameter, which involves plotting \(\kappa(t)\) on the vertical axis versus \(t\) on the horizontal axis, for \(0 \leq t \leq \pi\). d. Identify points of max/min curvature by finding critical points in the interval \(0 \leq t \leq \pi\). This involves differentiating the curvature \(\kappa(t)\) with respect to \(t\) and setting it to zero. e. Verify the consistency between the curvature graph and the curve graph by comparing high and low curvature regions in both graphs to confirm the calculated curvature is consistent with the actual behavior of the curve.

Step-by-step solution

Graph the curve

To graph the curve, we need to create a plot of the parametric equations $$x(t) = t$$ and $$y(t) = \sin t$$, for \(0 \leq t \leq \pi\). This curve is a segment of the sine function embedded in the first and second quadrants.

Compute the curvature

First, we have to find the first and second derivatives of the given parametric equations w.r.t. \(t\). For the first derivative: $$\frac{dx}{dt} = 1$$ $$\frac{dy}{dt} = \cos t$$ So, \(\mathbf{r'}(t) = \langle 1, \cos t \rangle\). Now, for the second derivative: $$\frac{d^2 x}{dt^2} = 0$$ $$\frac{d^2 y}{dt^2} = -\sin t$$ So, \(\mathbf{r''}(t) = \langle 0, -\sin t \rangle\). Now, we compute the curvature using the formula: $$\kappa(t) = \frac{\| \mathbf{r'}(t) \times \mathbf{r''}(t) \| }{\| \mathbf{r'}(t) \| ^3}$$ Computing the cross product: $$\mathbf{r'}(t) \times \mathbf{r''}(t) = (1)(-\sin t) - (0)(\cos t) = -\sin t$$ Computing the norms: $$\| \mathbf{r'}(t) \| = \sqrt{(1)^2 + (\cos t)^2} = \sqrt{1 + \cos^2 t}$$ $$\| \mathbf{r'}(t) \| ^3 = (1 + \cos^2 t)^\frac{3}{2}$$ Finally, we plug these values into the curvature formula: $$\kappa(t) = \frac{|\sin t|}{(1 + \cos^2 t)^\frac{3}{2}}$$

Graph the curvature

To graph the curvature, use the formula we found above, \(\kappa(t)\), and plot it on the vertical axis versus \(t\) on the horizontal axis, for \(0 \leq t \leq \pi\).

Identify points of max/min curvature

To find any points of max/min curvature, we have to differentiate the curvature \(\kappa(t)\) w.r.t. \(t\) and find its critical points. Then, we'll analyze the interval containing these points for maximum or minimum curvature. Taking the derivative of \(\kappa(t)\) and setting it equal to zero, we can find any critical points in the domain \(0 \leq t \leq \pi\).

Verify the consistency between the curvature graph and the curve graph

To verify the consistency between the curvature graph and the curve graph, we can look for any visible relationships between the two. For example, high curvature regions on the curvature graph should correspond to the places where the curve is bending the most, while low curvature regions should correspond to the less bent areas of the curve. By comparing both graphs, we can confirm whether the calculated curvature is consistent with the actual behavior of the given curve.

Key concepts

Graphing Parametric Equations

Understanding the graph of parametric equations is central to visualizing the motion described by these equations over time. Parametric equations express a set of related quantities as explicit functions of an independent variable, often referred to as the parameter. For example, in the parametric equations \( x(t) = t \) and \( y(t) = \sin t \), \( t \) serves as the parameter that traces the curve over the interval \( 0 \leq t \leq \pi \).

To graph parametric equations, one typically plots the values of \( x(t) \) and \( y(t) \) on the horizontal and vertical axes, respectively. The resulting graph for the given equations is a segment of the sine wave stretching through the first and second quadrants, offering a visual representation of the curve's progression over time. This visual aspect is crucial for students to connect the algebraic representation with the geometric shape of the curve.

Calculating Curvature

Curvature is a measure that describes how sharply a curve bends at a given point. To calculate the curvature of a parametric curve, \( \mathbf{r}(t) \), we rely on the derivatives of the curve with respect to \( t \). Specifically, you need the first derivative \( \mathbf{r'}(t) \) and the second derivative \( \mathbf{r''}(t) \).

The curvature \( \kappa(t) \) is given by the formula \(\kappa(t) = \frac{\| \mathbf{r'}(t) \times \mathbf{r''}(t) \| }{\| \mathbf{r'}(t) \| ^3}\), involving the cross product of these derivatives and the norm of the first derivative. The absolute value ensures that curvature is always a non-negative quantity, signifying the rate of change of the tangent vector's direction: it is high where the curve bends sharply and low where the curve is relatively straight. This concept is an essential element of differential geometry and has numerous applications in physics and engineering.

Critical Points in Calculus

In calculus, critical points are where the derivative of a function is either zero or undefined. These points are interesting because they can indicate local maxima, minima, or points of inflection on a curve. When calculating curvature, we find the critical points by setting the derivative of the curvature function \( \kappa'(t) \) to zero and solving for \( t \).

Identifying these critical points helps us understand more about the curvature's behavior without graphing it. When you differentiate the curvature and equate it to zero, the solutions indicate the parameter values at which the curve's bending changes most significantly. These critical points, when analyzed, can reveal maximum or minimum curvature on the curve, further informing us about the curve's geometric properties.

Relationship Between Curvature and Graph of Curve

The relationship between the curvature of a curve and its graph is profound since it directly links the geometric bend of the curve to a numerical measure. High curvature corresponds to regions of the curve that bend or turn quickly, while low curvature indicates flatter areas.

For instance, when we look at the graph of \( \mathbf{r}(t) = \langle t, \sin t \rangle \) alongside its curvature function, we can anticipate that points of maximum curvature will align with the peaks and troughs of the sine wave, where the direction changes most abruptly. Conversely, points of minimum curvature will occur where the sine wave approaches its inflection points, indicating a smoother transition in direction.

By comparing the graph of the curve with its curvature plot, students can better visualize how curvature reflects the physical shape of the curve. This comparison is not just a mathematical exercise but a fundamental concept in understanding the interplay of shape and motion in varied contexts, from the design of roads to the study of planetary orbits.