Tangent Lines and Area for Parametric Equations

10.2 Tangent Lines and Area for Parametric Equations Notes

Introduction

This section explores how to find derivatives, tangent lines, and areas under curves when equations are given in parametric form. Parametric equations express both x and y as functions of a third variable, typically t or θ.

Derivatives for Parametric Equations

• Parametric Equations: These are equations where both x and y are defined in terms of a parameter, such as t:

  • $x = f(t)$

  • $y = g(t)$

• Derivative Formula: The derivative of y with respect to x is found using the chain rule:

$\frac{dy}{dx} = \frac{g'(t)}{f'(t)}$ where $f'(t) \neq 0$.

• This formula allows us to compute slopes and analyze the behavior of curves defined parametrically.

Equation of the Tangent Line to a Parametric Curve

To find the equation of the tangent line at a specific value of the parameter t:

1. Find the point on the curve: $(x_0, y_0) = (f(t_0), g(t_0))$.

2. Compute the slope using the derivative formula above at $t = t_0$.

3. Write the tangent line in point-slope form:

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

• Example: For $c(t) = (t^2 + 1, t^2 - 4t)$, find the tangent line at $t = 3$.

  • Compute $x_0 = 3^2 + 1 = 10$, $y_0 = 3^2 - 4 \times 3 = 9 - 12 = -3$.

  • $\frac{dx}{dt} = 2t$, $\frac{dy}{dt} = 2t - 4$.

  • At $t = 3$, $\frac{dx}{dt} = 6$, $\frac{dy}{dt} = 2$.

  • Slope: $m = \frac{2}{6} = \frac{1}{3}$.

  • Tangent line: $y + 3 = \frac{1}{3}(x - 10)$.

Area Under Parametric Curves

The area under a parametric curve from $t = a$ to $t = b$ is given by:

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

• This formula is derived from the substitution $dx = \frac{dx}{dt} dt$ in the standard area integral.

• Example: For $c(t) = (5 \cos t, 3 \sin t)$, $-\pi \leq t \leq \pi$:

  • $\frac{dx}{dt} = -5 \sin t$

  • Area: $A = \int_{-\pi}^{\pi} 3 \sin t \cdot (-5 \sin t) dt = -15 \int_{-\pi}^{\pi} \sin^2 t dt$

  • Evaluate using trigonometric identities as needed.

Area Under a Cycloid

A cycloid is the curve traced by a point on the rim of a rolling circle. Its parametric equations are:

• $x = f(\theta) = 5(\theta - \sin \theta)$

• $y = g(\theta) = 5(1 - \cos \theta)$

To find the area under one arch (as $\theta$ goes from $0$to $2\pi$):

$A = \int_{0}^{2\pi} y(\theta) \frac{dx}{d\theta} d\theta$

• Compute $\frac{dx}{d\theta} = 5(1 - \cos \theta)$.

• Plug into the area formula and integrate over $[0, 2\pi]$.

Summary Table: Key Formulas for Parametric Equations

Concept	Formula	Description
Derivative	$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$	Slope of the curve at a given parameter value
Tangent Line	$y - y_0 = m(x - x_0)$	Equation of tangent line at $(x_0, y_0)$ with slope $m$
Area Under Curve	$A = \int_{a}^{b} y(t) \frac{dx}{dt} dt$	Area between curve and x-axis from $t = a$ to $t = b$

Additional info: The notes reference Desmos graph links for visualization, which are useful for understanding the geometric interpretation of tangent lines and areas under parametric curves.