Problem 82 - Calculus Early Transcendentals

Question

Which of the following parametric equations describe the same curve? a. \(x=2 t^{2}, y=4+t ;-4 \leq t \leq 4\) b. \(x=2 t^{4}, y=4+t^{2} ;-2 \leq t \leq 2\) c. \(x=2 t^{2 / 3}, y=4+t^{1 / 3} ;-64 \leq t \leq 64\)

Step-by-Step Solution

Answer

a. \(x = 2t^2\), \(y = 4 + t\), with \(-4 \leq t \leq 4\) b. \(x = 2t^4\), \(y = 4 + t^2\), with \(-2 \leq t \leq 2\) c. \(x = 2t^{\frac{2}{3}}\), \(y = 4 + t^{\frac{1}{3}}\), with \(-64 \leq t \leq 64\) Answer: None of the given parametric equations describe the same curve.

Step 1: Identify the functional relationships

In order to determine whether the given parametric equations describe the same curve, we need to compare the functional relationships between \(x\) and \(y\) for each set. a. \(x = 2t^2\), \(y = 4 + t\) b. \(x = 2t^4\), \(y = 4 + t^2\) c. \(x = 2t^{\frac{2}{3}}\), \(y = 4 + t^{\frac{1}{3}}\)

Step 2: Eliminate parameter t

Now, we will eliminate the parameter \(t\) in each set of equations and rewrite the equations in terms of \(x\) and \(y\) only. a. Solve for \(t\) in the first equation: \(t = \pm\sqrt{\frac{x}{2}}\). Substitute this into the second equation: \(y = 4 \pm \sqrt{\frac{x}{2}}\). b. Solve for \(t\) in the first equation: \(t = \pm\sqrt[4]{\frac{x}{2}}\). Substitute this into the second equation: \(y = 4 + \frac{x}{2}\). c. Solve for \(t\) in the first equation: \(t = (\frac{x}{2})^{\frac{3}{2}}\). Substitute this into the second equation: \(y = 4 + (\frac{x}{2})^{\frac{1}{2}}\).

Step 3: Compare functional relationships

Compare the relationships between \(x\) and \(y\) in the rewritten equations. a. \(y = 4 \pm \sqrt{\frac{x}{2}}\) b. \(y = 4 + \frac{x}{2}\) c. \(y = 4 + (\frac{x}{2})^{\frac{1}{2}}\) We can see that none of these relationships between \(x\) and \(y\) are the same, so none of the parametric equations describe the same curve.

Step 4: Consider the range of parameter t

Even though the curves are different in their functional relationships, we should also check the range of \(t\) for each set of parametric equations, as this could lead to different curve segments. a. \(-4 \leq t \leq 4\) b. \(-2 \leq t \leq 2\) c. \(-64 \leq t \leq 64\) These do not have the same range; thus, even if the equations had the same relationships between \(x\) and \(y\), the curves represented by the equations would be different segments.

Step 5: Conclusion

None of the given parametric equations describe the same curve, as neither the functional relationships nor the range of parameter \(t\) are the same for any pair of equations.