Problem 47 - Calculus Early Transcendentals

Question

Give a set of parametric equations that describes the following curves. Graph the curve and indicate the positive orientation. If not given, specify the interval over which the parameter varies. The piecewise linear path from \(P(-2,3)\) to \(Q(2,-3)\) to \(R(3,5)\) using parameter values \(0 \leq t \leq 2\)

Step-by-Step Solution

Answer

Answer: The parametric equations for the path are given by: \(x(t) = \begin{cases} -2 + 4t, & 0 \le t \le 1 \\ 2 + (t-1), & 1 \le t \le 2 \end{cases}\) \(y(t) =\begin{cases} -\frac{3}{2}(-2 + 4t) + 3, & 0 \le t \le 1 \\ 8(t-1) - 3, & 1 \le t \le 2 \end{cases}\)

Step 1: Find the line connecting Point P to Point Q

We have points \(P(-2,3)\) and \(Q(2,-3)\). To find the line connecting these two points, first find the slope \(m\) and use the point-slope form of a line: \(m_PQ = \frac{-3 - 3}{2 - (-2)} = \frac{-6}{4} = -\frac{3}{2}\) Using point-slope form, the equation of line PQ is: \(y - 3 = -\frac{3}{2}(x + 2)\) Now, we need to convert this equation to parametric form when \(0 \leq t \leq 1\).

Step 2: Convert the line PQ to parametric form

To convert the line to parametric equations, we can let \(x(t) = -2 + 4t\) since we know that when \(t=0\), the x-coordinate is -2 and when \(t=1\), the x-coordinate is 2. Now, we plug the \(x(t)\) into the equation we found in Step 1 and solve for \(y(t)\). \(y - 3 = -\frac{3}{2}(-2 + 4t)\) \(y = -\frac{3}{2}(-2 + 4t) + 3\) Thus, the parametric equations for PQ are: \(x(t) = -2 + 4t\) \(y(t) = -\frac{3}{2}(-2 + 4t) + 3\), for \(0 \leq t \leq 1\)

Step 3: Find the line connecting Point Q to Point R

We have points \(Q(2,-3)\) and \(R(3,5)\). To find the line connecting these two points, first get the slope \(m\) and use the point-slope form of a line: \(m_QR = \frac{5 - (-3)}{3 - 2} = \frac{8}{1} = 8\) Using point-slope form, the equation of line QR is: \(y + 3 = 8(x - 2)\) Now, we need to convert this equation to parametric form when \(1 \leq t \leq 2\).

Step 4: Convert the line QR to parametric form

To convert the line to parametric equations, we can let \(x(t) = 2 + (t-1)\) since we know that when \(t=1\), the x-coordinate is 2 and when \(t=2\), the x-coordinate is 3. Now, we plug the \(x(t)\) into the equation we found in Step 3 and solve for \(y(t)\): \(y + 3 = 8(2 + (t-1) - 2)\) \(y = 8(t-1) - 3\) Thus, the parametric equations for QR are: \(x(t) = 2 + (t-1)\) \(y(t) = 8(t-1) - 3\), for \(1 \leq t \leq 2\) Finally, we have the set of parametric equations for the given problem: \(x(t) = \begin{cases} -2 + 4t, & 0 \le t \le 1 \\ 2 + (t-1), & 1 \le t \le 2 \end{cases}\) \(y(t) =\begin{cases} -\frac{3}{2}(-2 + 4t) + 3, & 0 \le t \le 1 \\ 8(t-1) - 3, & 1 \le t \le 2 \end{cases}\)

Key Concepts

Piecewise Linear Path

A piecewise linear path is a series of straight-line segments connected end to end, creating a broken line through a sequence of points. The path often describes a route a particle takes as it moves from one location to another. When defining such paths using parametric equations, each segment is represented by a separate equation for a specific interval of the parameter, typically denoted by t.

For instance, in our exercise, the path from point P to Q and then R is described by two line segments. Efficient modeling of these segments allows users to understand how the position of an object changes with time. In creating parametric equations for these paths, we focus on how the horizontal (x) and vertical (y) coordinates change in relation to the parameter t.

Point-Slope Form

The point-slope form of a linear equation is a way to describe a line when you know one point on the line and its slope. The standard form of this equation is \(y - y_1 = m(x - x_1)\), where m represents the slope, and \(x_1, y_1\) are the coordinates of the known point.

In the exercise, point-slope form is used to find the equations of the lines connecting P to Q and Q to R. This form is particularly helpful as it makes the derivation of parametric equations straightforward. By knowing the slope and a point, we can express how x and y change together as functions of the parameter t.

Positive Orientation

The concept of positive orientation on a curve is related to the direction a point moves along the curve as the parameter t increases. In general, a positive orientation follows the conventional way of traversing a path, such as moving from left to right or from down to up on the Cartesian plane.

For graphing purposes, an arrow is used to indicate the positive orientation on the curve. The exercise specifies to graph the curve and indicate the positive orientation, which infers the direction we expect the path from P to Q to R should be followed. This orientation is important, especially when dealing with more complex mathematical concepts like line integrals and vector fields.

Graphing Parametric Curves

When it comes to graphing parametric curves, we plot the set of points that are described by the parametric equations x(t) and y(t) for a range of values of t. Each value of t provides a coordinate on the plane, and these points are connected to show the path of the curve.

In the context of our exercise, we graph two parametric equations representing two distinct segments of the path. By considering each segment within its interval, we accurately depict the piecewise linear path. In addition, ensuring we include the correct positive orientation aids in visualizing the direction of motion along the path. Graphing these parametric curves is not only a pivotal part of understanding the movements of objects but also a foundational tool in fields such as physics, engineering, and computer graphics.