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Chapter 11: Problem 43

In Problems 43 and \(44,\) parametric equations of four plane curves are given. Graph each of them, indicating the orientation. \(\begin{array}{ll}C_{1}: & x(t)=t, \quad y(t)=t^{2} ; \quad-4 \leq t \leq 4 \\\ C_{2}: & x(t)=\cos t, \quad y(t)=1-\sin ^{2} t ; \quad 0 \leq t \leq \pi \\\ C_{3}: & x(t)=e^{t}, \quad y(t)=e^{2 t} ; \quad 0 \leq t \leq \ln 4 \\\ C_{4}: & x(t)=\sqrt{t}, \quad y(t)=t ; \quad 0 \leq t \leq 16\end{array}\)

Short Answer

Expert verified
Plot and indicate the orientation for \(C_1\), a parabola; \(C_2\), a cosine squared function; \(C_3\), an exponential curve; and \(C_4\), a parabola.

Step by step solution

01

Graph the first curve, \(C_1\)

The parametric equations for \(C_1\) are \(x(t) = t\) and \(y(t) = t^2\). This represents a parabola opening upwards. Plot points for \(-4 \leq t \leq 4\), such as \((-4, 16)\), \((-2, 4)\), \((0, 0)\), \((2, 4)\), \((4, 16)\). The orientation moves from left to right as \(t\) increases.
02

Graph the second curve, \(C_2\)

The parametric equations for \(C_2\) are \(x(t) = \cos t\) and \(y(t) = 1 - \sin^2 t\). Simplify \(1 - \sin^2 t\) to \ \cos^2 t\. This results in \(y = \cos^2 t\). Plot points for \(0 \leq t \leq \pi\), such as \((1, 1)\), \((0, 0)\), \((-1, 1)\). The curve is symmetric about the \(y\)-axis. The orientation moves from right to left as \(t\) increases.
03

Graph the third curve, \(C_3\)

The parametric equations for \(C_3\) are \(x(t) = e^t\) and \(y(t) = e^{2t}\). This represents an exponential curve. Plot points for \(0 \leq t \leq \ln 4\), such as \((1, 1)\), \((e, e^2)\), \((e^2, e^4)\). The orientation moves from left to right as \(t\) increases.
04

Graph the fourth curve, \(C_4\)

The parametric equations for \(C_4\) are \(x(t) = \sqrt{t}\) and \(y(t) = t\). This represents a function where \(y = x^2\). Plot points for \(0 \leq t \leq 16\), such as \((0, 0)\), \((1, 1)\), \((2, 4)\), \((3, 9)\), \((4, 16)\). The orientation moves from bottom to top as \(t\) increases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

graphing parametric equations
Parametric equations involve expressing both the x and y coordinates as functions of a third parameter, usually denoted by t. This can make plotting curves more intuitive and flexible.
For example, if given equations like \(x(t) = t\) and \(y(t) = t^2\), it means that for every t value, you can find corresponding x and y values.
Graphing parametric equations usually involves plotting points for various values of t and connecting them to reveal the curve. In this exercise, we plotted from \(t = -4\) to \(t = 4\). Some important points we plotted were: \((-4, 16)\), \((-2, 4)\), and \((0, 0)\). These points collectively created an upwards-opening parabola.
plane curves
Plane curves are the paths traced by parametric equations on a 2-dimensional plane. These curves can come in various shapes, including parabolas, circles, and more complex shapes.
In this exercise:
  • Curve \(C_1\) traced a parabolic shape as both x and y depended linearly and quadratically on t, respectively.
  • Curve \(C_2\) created a cosine-squared shape, a familiar form plotting points from \((1,1)\) to \((-1,1)\).
  • Curve \(C_3\) formed an exponential growth pattern because y grows faster than x for increasing t.
  • Curve \(C_4\) sketched a square-root curve fitting the form \(y = x^2\).
Each curve laid on a 2D plane but with unique characteristics given by their specific parametric forms.
orientation of curves
The orientation of a parametric curve tells you the direction in which the curve is traced as the parameter t increases.
For example, if t increases and the resulting points on the curve move to the right, the orientation is left to right. In the exercise:
  • Curve \(C_1\) showed an orientation from left to right as t went from \(-4\) to \(4\).
  • Curve \(C_2\) moved from right to left because as t increased from 0 to \(\pi\), the plot direction changed.
  • Curve \(C_3\) also moved from left to right due to the exponential function's growth.
  • Curve \(C_4\) went from bottom to top, moving upwards along the square root function.
Understanding the orientation helps in visualizing how a curve is plotted over a given interval.

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