9.2.2Can I rewrite it in rectilinear form?

Converting Between Parametric and Rectilinear Form

Math Notes

Parametric Equations

Pete Moss’ play plan (problem 9-48) and the Battling Robots Qualification Round (problem 9-46) are both examples of the use of parametric equations. A parameter is an independent variable upon which other variables depend. It has come to signify a “helping” variable, in this case $t$ for time. Instead of $y$ being a function of $x$ directly, $x$ and $y$ are both functions of the parameter (helping variable).

Typically, parametric equations allow you to express two related motions separately, with respect to time, and then combine these separate equations into a single graph.

$x = x\left(t\right)$
$y = y\left(t\right)$

Random curve, starting in second quadrant, rising right then circling left then down turning in third quadrant, running almost horizontal to fourth quadrant, then curving up & left, then right into first quadrant, then turning & continuing left & up. $\$

$\huge \rightarrow$

$\$

9-58.

Consider the set of parametric equations $x(t) = 2t$ and $y(t) = t^2$ .

Your graphing calculator is also able to graph parametric equations. Use your graphing calculator to sketch a graph of the set of parametric equations above. In this case, assume $–3 ≤ t ≤ 3$ . Describe the shape of the graph.
Convert the set of parametric equations into rectilinear form. That is, find a way to eliminate the parameter and write one equation in terms of $x$ and $y$ only.
What if the domain of the set of parametric equations is $0 ≤ t ≤ 10$ ? Adjust your answer to part (b) to represent this situation.

9-59.

Use your graphing calculator (in parametric mode) to graph the curve generated by $x(t) = t \cos(t)$ and $y(t) = t \sin(t)$ for $0 ≤ t ≤ 4π$ .

What one word best describes the shape of the curve?
Is it possible to make this curve using a single function (in rectangular mode instead of the parametric mode) on your calculator? Why or why not?

9-60.

Let $x = t^2$ and $y = t^{4}$ .

Write an equation for $y$ as a function of $x$ . Describe the resulting function.
Is the graph of the rectangular function the same as the parametric curve? Determine this by graphing each equation for $-5 ≤ x ≤ 5$ . Give an explanation for the results.

9-61.

Unit circle, central angle in first quadrant, labeled theta, bottom ray on positive x axis, endpoints of arc corresponding to central angle labeled as ordered pairs, (R, comma 0), & (x, comma y). For parametric equations, the parameter, or “helping variable,” does not need to be time. In fact, you have already used parametric equations when studying unit circles.

A point with coordinates $(x, y)$ is located on a circle with radius $R$ centered at the origin. Use the diagram at right to write an equation for $x$ in terms of $θ$ . Then write a second equation for $y$ , also in terms of $θ$ .
Does your set of parametric equations work for other quadrants? Verify and adjust your equations if necessary.
Graph the parametric equations from part (a), using a radius of your choice. Name an interval of $θ$ that gives a complete circle.
Show algebraically that the parametric equations from part (a) are equivalent to the equation $x^2 + y^2 = R^2$ .

9-62.

Compare the graphs of the following sets of parametric equations.

$\left\{ \begin{array} { l } { x ( t ) = t ^ { 2 } + 2 t } \\ { y ( t ) = t + 1 } \end{array} \right.$

$\left\{ \begin{array} { l } { x ( t ) = \operatorname { tan } ^ { 2 } ( t ) - 1 } \\ { y ( t ) = \operatorname { tan } ( t ) } \end{array} \right.$

Record your observations about their similarities and differences.
For both sets of parametric equations, write an equation that directly relates $y$ and $x$ . How do the equations compare?
Create a different set of parametric equations that result in the same equation you found in part (b).

9-63.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method. Homework Help ✎

$\int \frac { 4 x } { \sqrt { 1 + x ^ { 2 } } } d x$

$\int \operatorname { cos } ( 4 \theta ) \operatorname { sin } ( 2 \theta ) d \theta$

$\int \frac { 4 } { \sqrt { 1 - x ^ { 2 } } } d x$

$\int 6 x ^ { 3 } \cdot 2 ^ { x ^ { 4 } } d x$

9-64.

No calculator! Determine the slope of the curve $x \cos(y) = xy + 1$ at the point $(1, 0)$ . Homework Help ✎ Compute without a calculator

9-65.

Compare the expressions $\frac { d } { d x }$ $\int_{ }^{ }f(x)dx$ and $\int ( \frac { d } { d x } f ( x ) ) d x$ . List any similarities and differences. Homework Help ✎

9-66.

Thoroughly investigate the graph of $f(x) = x^{5/3} + x^{2/3}$ . Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify point(s) of inflection, cusps, and intercepts. Be sure to justify all statements graphically and analytically. Homework Help ✎

9-67.

Examine the slope field of $\frac { d y } { d x }= x + y$ for $–2 ≤ x ≤ 2$ at right. Homework Help ✎

If $y(-2) = 2$ , use Euler’s Method to draw a solution curve for $y$ using $∆x = 1$ .
Draw a new solution using Euler’s Method if $y(-2) = 0$ .

9-68.

The velocity of a particle moving along the $x$ -axis is $v(t) = 6\sqrt { t }- 2t$ units per second. Homework Help ✎

What was the average velocity over $0 ≤ t ≤ 15$ seconds?
During the first $15$ seconds, what is the total distance the point traveled?
What was the particle’s total displacement over $0 ≤ t ≤ 15$ seconds?
What accounts for the large difference between the answers to parts (b) and (c)?

9-69.

Convert the following sets of parametric equations into rectangular form (in terms of $x$ and $y$ ). Homework Help ✎

$x = \cos(t)$ and $y = \sin(t)$
$x = \cos(2t)$ and $y = \sin(2t)$
$x = t^4 - 3t^2$ and $y = t^2$

9-70.

For each set of parametric equation in problem 9-69, describe what part of the curve is sketched in each case when $-2\le t\le2$ . Homework Help ✎

9-71.

A variant of Ying’s method (manipulating an infinite expression so that the expression appears as a part of itself) can be used in other situations. For example, it can be used to evaluate an infinite “continued fraction” such as $S = 2 + \frac { 1 } { 2 + \frac { 1 } { 2 + \frac { 1 } { 2 + \ldots } } }$ . Homework Help ✎ Explain why $S=2+\frac{1}{S}$ . Then solve for $S$ .

Calculus Third Edition

Parametric Equations