Problem 48 - 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 path consisting of the line segment from \((-4,4)\) to \((0,8)\) followed by the segment of the parabola \(y=8-2 x^{2}\) from \((0,8)\) to \((2,0),\) using parameter values \(0 \leq t \leq 3\)

Step-by-Step Solution

Answer

Question: Write parametric equations for a line segment from \((-4,4)\) to \((0,8)\) and a parabolic segment with the equation \(y = 8 - 2x^2\) from \((0,8)\) to \((2,0)\). Graph the segments and specify the interval for the parameter t. Answer: For the line segment, the parametric equations are: \(x(t) = -4 + 4t\), \(y(t) = 4 + 4t\), where \(0 \leq t \leq 1\). For the parabolic segment, the parametric equations are: \(x(t) = \sqrt{4t - 4}\), \(y(t) = -4t + 12\), where \(1 \leq t \leq 3\). The graph of these segments will show the line segment going from \((-4,4)\) to \((0,8)\), and the parabolic segment going from \((0,8)\) to \((2,0)\), following the positive orientation. The interval for the parameter t is \(0 \leq t \leq 3\).

Step 1: Write parametric equations for the line segment

The line segment goes from \((-4,4)\) to \((0,8)\). We can use the standard parametric equation for a line segment from point A to point B over a given interval using a coefficient \(t\), on the interval \(0\leq t \leq 1\) as follows: \(x(t) = x_A + t(x_B - x_A)\) \(y(t) = y_A + t(y_B - y_A)\) Substitution the values of A \((-4,4)\) and B \((0,8)\): \(x(t) = -4 + t(0 - (-4))\) \(y(t) = 4 + t(8 - 4)\)

Step 2: Write parametric equations for the parabolic segment

The equation of the parabola is given by \(y = 8 - 2x^2\). We know we want to find a parametric equation from \((0,8)\) to \((2,0)\). One way of doing this is by solving for x in terms of y: \(x^2 = (8 - y)/2\) \(x = \sqrt{(8 - y)/2}\) Now we will express y in terms of a parameter \(t\) for the interval \(1 \leq t \leq 3\). Since y decreases from 8 to 0 along the parabola, we can examine the second interval to find the equation for y: Interval for t: \(1 \leq t \leq 3\) Value of y: \(8 = 8 - 2(0)^2\) \(0 = 8 - 2(2)^2\) We can use a linear function for y(t) in the second interval: \(y(t) = -4t + 12\) Now we can find x(t) by substituting y(t) into the equation for x: \(x(t) = \sqrt{(8 - (-4t + 12))/2}\) = \(\sqrt{4t - 4}\)

Step 3: Graph parametric equations

In this step, you'll graph the two parametric equations you derived in Steps 1 and 2. You can either use graphing software or graph it manually on graph paper. Make sure to clearly indicate the positive orientation on the graph. The positive orientation is specified by the intervals of parameter t: For the line segment, the positive orientation goes from \(t = 0\) to \(t = 1\), which means it goes from \((-4,4)\) to \((0,8)\). For the parabolic segment, the positive orientation goes from \(t = 1\) to \(t = 3\), which means it goes from \((0,8)\) to \((2,0)\).

Step 4: Specify parameter intervals

The interval for the parameter t is already given in the question: \(0 \leq t \leq 3\). If it was not given, we would use our understanding of the problem and the positive orientation to specify this interval, based on how we defined t in the two parametric equations.

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. Students often encounter the subject when exploring functions, limits, derivatives, and integrals. In the context of parametric equations, calculus can extend into evaluating derivatives of parametrically defined curves to understand the rate at which one parameter changes relative to another, or to determine the concavity and other aspects of curve behavior.
While the given exercise doesn't delve deeply into calculus concepts such as differentiation or integration directly, one indirectly applies the fundamental idea of a parameter varying over a certain interval, which is essential to understand when dealing with more advanced calculus problems involving parametric equations.

Parametric Representation of Curves

The parametric representation of curves provides a method to define a curve in the Cartesian plane by expressions for the x and y coordinates as functions of a third variable, often denoted as t (the parameter). This is different from a traditional function which relates y directly to x. Parametric curves allow for the more flexible modeling of movement and paths that might not be functions because they could fail the vertical line test.
In the given exercise, the parametric equations are used to describe a path consisting of a line segment and a segment of a parabola. By changing the value of the parameter t, the curve is traced out, starting from one point and ending at another. This technique is particularly powerful in scenarios where curves are defined by motion or other dynamic processes.

Linear and Quadratic Functions

Linear functions are the simplest type of algebraic function, characterized by a constant rate of change, which can be graphed as a straight line with the equation y = mx + b. Quadratic functions, on the other hand, have a variable rate of change and are represented by parabolas in the form y = ax^2 + bx + c.
In our exercise, the first segment of the path uses a linear function to determine the y-coordinate. The second segment features a quadratic function, represented by a parabolic arc. Parametric equations serve as a bridge between these two types of functions, representing various segments of the path as t, the parameter, varies and allows for the continuous tracing of the path composed of both linear and non-linear elements.

Graphing Parametric Equations

Graphing parametric equations involves plotting points on the Cartesian plane as the parameter t varies over a specified interval. Each value of t provides a set of corresponding x and y coordinates that help trace the curve's path. This approach can handle more complex curves than those representable by a simple y = f(x) equation, including circles, ellipses, and other intricate shapes that might loop or cross themselves.
For the exercises given, starting with the parametric equations for both the line segment and the parabola, you'll create a table of values for t, calculate the corresponding x(t) and y(t) for each value, and plot these points. Connecting the dots in the correct order, as per the positive orientation indicated by the interval of t, will reveal the entire path. Making certain to clearly indicate the direction on the graph enhances comprehension and visualizes the nature of the object's movement along the curves.