10.2 Tangent Lines and Area for Parametric Equations

Introduction

Parametric equations describe curves in the plane using a parameter, typically denoted as t. This section explores how to find derivatives, tangent lines, and areas under curves defined parametrically.

Derivatives for Parametric Equations

• Definition: If a curve is defined by x = f(t) and y = g(t), the derivative dy/dx is found using the chain rule.

• Formula:

$ \frac{dy}{dx} = \frac{g'(t)}{f'(t)} \quad \text{where} \ x = f(t), \ y = g(t), \ f'(t) \neq 0 $

• Interpretation: This formula gives the slope of the tangent to the curve at a given value of t.

Equation of the Tangent Line to a Parametric Curve

• To find the tangent line at a specific parameter value t = t_0:

  1. Compute the point: (x_0, y_0) = (f(t_0), g(t_0)).

  2. Find the slope: $m = \frac{g'(t_0)}{f'(t_0)}$.

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

$y - y_0 = m(x - x_0)$

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

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

  • Compute derivatives: $f'(t) = 2t$, $g'(t) = 2t - 4$.

  • At $t = 3$, $f'(3) = 6$, $g'(3) = 2$.

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

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

Area Under Parametric Curves

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

$A = \int_{a}^{b} y(t) \cdot x'(t) \, dt$

• Interpretation: This formula generalizes the area under a curve to parametric equations.

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

  • $x(t) = 5 \cos t$, $y(t) = 3 \sin t$

  • $x'(t) = -5 \sin t$

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

Area Under a Cycloid

• Definition: A cycloid is the curve traced by a point on the rim of a rolling circle.

• Parametric Equations:

$x = f(\theta) = 5(\theta - \sin \theta)$ y = g(\theta) = 5(1 - \cos \theta)$

• Area under one arch: Use the area formula for parametric curves, integrating over one period of $\theta$ (typically $0$ to $2\pi$ for a full arch):

$A = \int_{0}^{2\pi} y(\theta) \cdot x'(\theta) \, d\theta$

• Application: This method can be used to find the area under more complex parametric curves, such as the cycloid.