Hello Professor Baker's Math Class!

This post is your go-to resource for Chapter 10, where we explore the fascinating world of Parametric Equations and Polar Coordinates. Get ready to describe curves in exciting new ways! Remember to attend our Zoom meetings on April 14th and 19th.

Here are your class notes and resources for the chapter:

Zoom Meeting Information

Join us for our live Zoom sessions where we'll discuss these topics and answer any questions you may have!

Class Notes

Access PDF notes from our class discussions. You can follow along, take notes, and review them as you study.

Chapter 10 Topics

Below is a breakdown of the topics we will be covering in Chapter 10, along with accompanying resources. Let's get started!

Section 10-1: Curves Defined by Parametric Equations

In this section, we explore parametric equations, which offer a powerful way to describe curves. Instead of defining $y$ as a function of $x$, we define both $x$ and $y$ as functions of a third variable, often denoted as $t$ (the parameter). This allows us to describe curves that are not functions in the traditional sense.

For instance, consider the equations:

$$x = f(t)$$ $$y = g(t)$$

As $t$ varies, the point $(x, y) = (f(t), g(t))$ traces out a curve in the coordinate plane. The attached notes provide examples of how to sketch these curves by plotting points and eliminating the parameter.

Section 10-2: Calculus with Parametric Curves

Now that we know how to define a tangent line at a point on a parametric curve $x = f(t), y = g(t)$, where $y$ is a differentiable function of $x$. Then, the Chain Rule gives $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$.

The slope of the tangent line can be found using the formula:

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

We'll also explore how to find the area under a parametric curve and the arc length of a parametric curve. The attached resources will be useful.

Section 10-3: Polar Coordinates

In this section, we introduce a new coordinate system called the polar coordinate system. Instead of using Cartesian coordinates $(x, y)$, we use polar coordinates $(r, \theta)$, where $r$ is the distance from the origin (or pole) and $\theta$ is the angle from the positive x-axis (or polar axis).

The relationship between polar and Cartesian coordinates is given by:

$$x = r \cos(\theta)$$ $$y = r \sin(\theta)$$ $$r^2 = x^2 + y^2$$ $$\tan(\theta) = \frac{y}{x}$$

We'll learn how to convert between polar and Cartesian coordinates and how to graph equations in polar coordinates.

Section 10-4: Areas and Lengths in Polar Coordinates

Finally, we'll combine our knowledge of polar coordinates and calculus to find areas and arc lengths of regions defined by polar equations. The area of a region bounded by the polar curve $r = f(\theta)$ and the rays $\theta = a$ and $\theta = b$ is given by:

$$A = \frac{1}{2} \int_{a}^{b} [f(\theta)]^2 d\theta$$

We'll also derive a formula for the arc length of a polar curve.

Remember, math is a journey, not a destination. Keep practicing, keep asking questions, and you'll master these concepts in no time! Good luck with Chapter 10!