Final Exam: 50-57 Parametric Equations

FINAL EXAM REVIEW:

Parametric Equations

• Parametric Form: $x = f(t),\ y = g(t)$, where $t$ varies over an interval.

• Applications: Used to describe motion, curves, and geometric shapes such as circles and ellipses.

Graphing and Analyzing

Plotting Parametric Curves

To graph a parametric curve, compute $(x, y)$ pairs for various values of $t$ and plot them in the coordinate plane. Labeling points with their corresponding $t$ values helps visualize the direction and orientation of the curve.

50. $x = t^2 + t$, $y = 2t - 1$

    For selected $t$ values (e.g., $t = -2, -1, 0, 1, 2$), compute $(x, y)$ and plot.

Converting Parametric to Cartesian Equations

To eliminate the parameter and find a Cartesian equation, solve one equation for $t$ and substitute into the other.

• Example: Given $x = t^2 + t$, $y = 2t - 1$:

• Solve $y = 2t - 1$ for $t$: $t = \frac{y + 1}{2}$

• Substitute into $x$: $x = \left(\frac{y + 1}{2}\right)^2 + \frac{y + 1}{2}$

Special Parametric Curves

Parametric Equations for Circles

A circle with center $(h, k)$ and radius $r$ can be represented parametrically as:

• $x = h + r\cos(t)$

• $y = k + r\sin(t)$

• For a circle centered at $(2, -3)$ with radius $7$:

• $x = 2 + 7\cos(t)$

• $y = -3 + 7\sin(t)$

• $t \in [0, 2\pi]$

Parametric Equations for Ellipses and Lissajous Curves

• Example: $x = 5\cos(3t)$, $y = 2\sin(3t)$, $t \in [0, 2\pi]$

• This describes an ellipse traced three times as $t$ goes from $0$to $2\pi$.

Tangent Lines to Parametric Curves

Finding the Equation of the Tangent Line

The slope of the tangent line to a parametric curve at a given $t$ is:

• $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

• Equation of the tangent line at $t = t_0$:

$y - y_0 = m(x - x_0)$, where $m = \left.\frac{dy}{dx}\right|_{t = t_0}$ and $(x_0, y_0) = (x(t_0), y(t_0))$

• Example: $x = t^3 - 3t$, $y = 3t^2 - 9$, $t = -2$

• Compute $x(-2)$, $y(-2)$, $\frac{dx}{dt}$, $\frac{dy}{dt}$, and $\frac{dy}{dx}$ at $t = -2$.

Vertical and Horizontal Tangents

• Horizontal Tangent: Occurs when $\frac{dy}{dt} = 0$ and $\frac{dx}{dt} \neq 0$.

• Vertical Tangent: Occurs when $\frac{dx}{dt} = 0$ and $\frac{dy}{dt} \neq 0$.

• Find $t$ values where these conditions are met, then compute corresponding points and tangent lines.

Applications of Parametric Equations

Area Enclosed by a Parametric Curve

The area $A$ enclosed by a parametric curve $(x(t), y(t))$ for $t$ in $[a, b]$ is:

$A = \int_a^b y(t) \frac{dx}{dt} dt$

• Example: $x = 6(t - \sin t)$, $y = 6(1 - \cos t)$

Arc Length of a Parametric Curve

The arc length $L$ of a parametric curve $(x(t), y(t))$ for $t$ in $[a, b]$ is:

$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

• Example: $x = 3\sin t$, $y = 3\cos t$, $0 \leq t \leq 2\pi$

Surface Area of Revolution (Parametric Form)

The surface area $S$ generated by rotating a parametric curve about the x-axis is:

$S = 2\pi \int_a^b y(t) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

• Example: $x = \cos^2 t$, $y = \sin^3 t$, $0 \leq t \leq \frac{\pi}{2}$

Summary Table: Key Parametric Formulas

Application	Formula	Description
Derivative	$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$	Slope of tangent line
Area	$A = \int_a^b y(t) \frac{dx}{dt} dt$	Area under parametric curve
Arc Length	$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$	Length of parametric curve
Surface Area (x-axis)	$S = 2\pi \int_a^b y(t) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$	Surface area by revolution

Additional info: These notes expand on the provided questions by including definitions, formulas, and examples relevant to parametric equations, as well as a summary table for quick reference.