Class Cancelled: Important Information Regarding Chapter 10 Test

Hello everyone,

Due to a severe migraine, I have to cancel class tonight, November 1, 2018. I apologize for any inconvenience this may cause. Your well-being and understanding of the material are my priorities.

Attached you will find the Chapter 10 test. This is the same test you would have taken in class tonight, so it covers the material we've been working on recently, including parametric equations, polar coordinates, and applications of calculus. Please submit the test at the beginning of class on Tuesday.

I am trusting that you will complete this test independently and honestly. This is a great opportunity to assess your own understanding of the material. Remember, true learning comes from grappling with the concepts yourself! Use your notes, textbook, and previously worked problems to guide you, but ensure the work you submit is entirely your own.

Key Concepts Covered in the Test

The test focuses on Chapter 10, which includes several key concepts. Here's a quick recap to help you prepare:

  • Parametric Equations: These equations define $x$ and $y$ in terms of a third variable, often $t$ or $\theta$. You'll need to understand how to find derivatives ($dy/dx$) and integrals in parametric form. For example, you may need to find the equation of a tangent line given parametric equations $x = f(t)$ and $y = g(t)$. Remember that $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.
  • Horizontal and Vertical Tangents: To find horizontal tangents, set $dy/dt = 0$. To find vertical tangents, set $dx/dt = 0$. Solve for the parameter $t$ (or $\theta$) and then plug back into the original parametric equations to find the $(x, y)$ coordinates.
  • Area Enclosed by Parametric Curves: If a curve $C$ is described by the parametric equations $x=f(t)$ and $y=g(t)$, $a \le t \le b$, and where $y$ is continuous and $y \ge 0$ on the interval $[a,b]$, then the area bounded by the curve and the x-axis is given by $$A = \int_a^b y \, dx = \int_a^b g(t)f'(t) \, dt$$
  • Arc Length of Parametric Curves: The arc length $L$ of a parametric curve defined by $x = f(t)$ and $y = g(t)$ from $t = a$ to $t = b$ is given by $$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$
  • Surface Area: When rotating a parametric curve around the x-axis, the surface area is given by $$S = 2\pi \int_a^b y(t) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$$
  • Polar Coordinates: Be comfortable converting between Cartesian ($x, y$) and polar ($r, \theta$) coordinates. Remember $x = r \cos(\theta)$ and $y = r \sin(\theta)$, and $r^2 = x^2 + y^2$, $\tan(\theta) = y/x$.
  • Tangent Lines in Polar Coordinates: To find the slope of a tangent line to a polar curve $r = f(\theta)$, use the formula:$$\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}$$
  • Area in Polar Coordinates: The area enclosed by a polar curve $r = f(\theta)$ from $\theta = a$ to $\theta = b$ is given by$$A = \frac{1}{2} \int_a^b [f(\theta)]^2 \, d\theta = \frac{1}{2} \int_a^b r^2 \, d\theta$$
  • Arc Length of Polar Curves: The arc length $L$ of a polar curve $r = f(\theta)$ from $\theta = a$ to $\theta = b$ is given by $$L = \int_a^b \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta$$

Review these concepts carefully. If you have any clarifying questions before Tuesday, feel free to email me, and I will do my best to answer them when my headache subsides.

Thank you for your understanding. I appreciate your hard work and dedication to this course. I look forward to seeing you all on Tuesday!

Best regards,

Professor Baker