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. $$\mathbf{r}(t)=\left\langle t, t^{2}\right\rangle, \text { for }-2 \leq t \leq 2 \quad \text { (parabola) }$$

Short answer

Answer: The minimum curvature occurs at t = 0 and is given by \(\kappa(0) = \frac{2}{(1+4(0)^2)^{3/2}} = 2\).

Step-by-step solution

Graph the Curve

Plot the curve \(\mathbf{r}(t) = \left\langle t, t^2\right\rangle\) for \(-2 \leq t \leq 2\). This is a parabola in the xy-plane, with x = t and y = t^2.

Compute the Curvature

To compute the curvature, first find the first derivative \(\mathbf{r}'(t) = \left\langle 1, 2t\right\rangle\) and the second derivative \(\mathbf{r}''(t) = \left\langle 0, 2\right\rangle\). The curvature can be calculated as: $$\kappa(t) = \frac{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3}$$ Since \(\mathbf{r}'(t) \times \mathbf{r}''(t) = -2\), and \(\|\mathbf{r}'(t)\| = \sqrt{1 + 4t^2}\), the curvature will be: $$\kappa(t) = \frac{2}{(1+4t^2)^{3/2}}$$

Graph the Curvature Function

Plot the graph of the curvature \(\kappa(t) = \frac{2}{(1+4t^2)^{3/2}}\) as a function of the parameter t.

Identify Maximum or Minimum Curvature Points

To find maximum and minimum curvature points, find the critical points of the curvature function. Take the derivative of \(\kappa(t)\) with respect to t: $$\kappa'(t) = \frac{-12t}{(1+4t^2)^{5/2}}$$ Set \(\kappa'(t) = 0\) to find critical points, which gives: $$-12t = 0 \Rightarrow t = 0$$ The curvature has only one critical point at t = 0. Compute the second derivative of the curvature to determine whether it is a maximum or minimum: $$\kappa''(t) = \frac{48(1-4t^2)}{(1+4t^2)^{7/2}}$$ Since \(\kappa''(0) = 48 > 0\), the curvature has a minimum at t = 0.

Verify Consistency between Graphs

Compare the graph of the curvature function with the graph of the curve. Observe whether the curvature graph reflects the curvature properties of the curve especially at t = 0 where the curvature is minimum. This consistency means that our calculations are correct, and the results align with the given exercise.

Key concepts

Parametric Equations Graphing

Graphing parametric equations, such as \( \mathbf{r}(t) = \left\langle t, t^2\right\rangle \), is crucial for visualizing the behavior of curves in a plane. In this case, we plot each pair of (x, y) points corresponding to each value of the parameter \( t \) within the given interval, -2 to 2. The resulting curve is a parabola, which is a graphical representation of a quadratic functional relationship between \( t \) and the position vectors.Exercise Advice: When graphing parametric curves, it helps to select a range of values for \( t \) and to calculate corresponding \( x \) and \( y \) values systematically. For better understanding, students should note that the graph of this particular parametric equation will be a 'right-side up' parabola opening upwards since the coefficient of \( t^2 \) is positive.

Curvature Calculation

Curvature is a measure of how quickly a curve is changing direction at a given point. To calculate this for the parametric equation \( \mathbf{r}(t) = \left\langle t, t^2\right\rangle \) we use the formula \( \kappa(t) = \frac{\|\mathbf{r}'(t) \times \mathbf{r}''(t)\|}{\|\mathbf{r}'(t)\|^3} \). Here, \( \mathbf{r}'(t) = \left\langle 1, 2t\right\rangle \) represents the velocity vector and \( \mathbf{r}''(t) = \left\langle 0, 2\right\rangle \) the acceleration vector. Computing the cross product and taking norms gives us the curvature \( \kappa(t) = \frac{2}{(1+4t^2)^{3/2}} \).This formula reflects how the curve's direction rate is changing and is a direct way to study the 'bend' of the curve at any point. A higher curvature corresponds to a sharper bend.Exercise Advice: To master curvature calculations, students should practice derivative calculations and get comfortable with vector operations, as these are foundational steps in finding curvature.

Critical Points in Calculus

Critical points are where the first derivative of a function equals zero or doesn't exist, and they are vital for understanding the function's behavior, especially for identifying local maxima and minima. In the context of curvature, we look at \( \kappa'(t) \) to find critical points. For our curve, setting the derivative of the curvature function \( \kappa'(t) = \frac{-12t}{(1+4t^2)^{5/2}} \) to zero, we find that there is one critical point at \( t = 0 \).The second derivative test, \( \kappa''(t) = \frac{48(1-4t^2)}{(1+4t^2)^{7/2}} \) helps determine whether this point is a maximum or minimum - with a positive value indicating a concave up graph at this point, hence a minimum curvature.Exercise Advice: Students should reinforce their understanding of derivatives and their applications in identifying critical points for various functions, not just for curvature.

Curvature and Graph Consistency

A consistent relationship between the curvature of a curve and its graphical representation is essential for validating the correctness of the calculated curvature. For our parabola \( \mathbf{r}(t) = \left\langle t, t^2\right\rangle \), the curvature graph \( \kappa(t) = \frac{2}{(1+4t^2)^{3/2}} \) shows a minimum at \( t = 0 \), which makes sense when visually inspecting the parabola: the vertex of the parabola represents the flattest point, hence it has the minimum curvature.This correlation between the curvature's graph and the curve itself indicates that the curve is flattest at the vertex and becomes sharper as we move away from it.Exercise Advice: To understand the graph consistency, students are encouraged to sketch both the curve and its curvature on the same axes or adjacent to each other to observe how the curvature reflects the changes in the direction of the curve.