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Chapter 6: Problem 37

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=t\\\ &y=\sqrt{t} \end{aligned}$$

Short Answer

Expert verified
The graph of the parametric equations \(x=t\) and \(y=\sqrt{t}\) starts at the origin and continues off to infinity, increasing slighter as 'x' increases. It is the shape of half of a parabola opening to the right side.

Step by step solution

01

Understand Parametric Equations

Parametric equations are a set of equations which define a quantity in terms of an independent variable called a parameter. In this problem, the parametric equations are \(x = t\) and \(y= \sqrt{t}\), where 't' is the parameter.
02

Understand the interval of t-values

In order to graph the parametric equations, it's important to identify the appropriate interval for the parameter 't'. Since the square root of a negative number is not a real number, 't' should be chosen such that \(t \geq 0\)
03

Use a Graphing Utility

To plot the pair of parametric equations, select a range of 't' values (in this case, non-negative real numbers) and compute the corresponding 'x' and 'y' values using the equations. Each pair ('x', 'y') represents a point in the Cartesian system, and connecting these points will provide the graph of the parametric equations.

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Most popular questions from this chapter

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$3 x-y+2=0$$

In Exercises \(91-116\), convert the polar equation to rectangular form. $$\theta=2 \pi / 3$$

Determine whether the statement is true or false. Justify your answer. Because the graphs of the parametric equations \(x=t^{2}, y=t^{2} \quad\) and \(\quad x=t, y=t\) both represent the line \(y=x,\) they are the same plane curve.

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$\theta=\pi / 6$$

Explain how the graph of each conic differs from the graph of \(\left.r=\frac{5}{1+\sin \theta} . \text { (See Exercise } 17 .\right)\) (a) \(r=\frac{5}{1-\cos \theta}\) (b) \(r=\frac{5}{1-\sin \theta}\) (c) \(r=\frac{5}{1+\cos \theta}\) (d) \(r=\frac{5}{1-\sin [\theta-(\pi / 4)]}\)
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