ap calculus ab/bc unit 9 study guides

Key Concepts

• Parametric equations represent curves in the plane using a parameter, often denoted as $t$, instead of expressing $y$ as a function of $x$
• Polar coordinates define points in the plane using a distance from the origin (radius) and an angle from the positive $x$-axis (theta)
  • Polar coordinates are useful for curves that are more easily described by a radius and angle (spirals, circles, cardioids)
• Vector-valued functions represent curves in two or three dimensions using a parameter, with each coordinate expressed as a function of the parameter
  • Vector-valued functions can be used to model motion in space, with the parameter often representing time
• Converting between rectangular, parametric, and polar forms is an essential skill for working with these concepts
• Derivatives and integrals can be applied to parametric equations, polar coordinates, and vector-valued functions to analyze their properties and behavior

Parametric Equations

• Parametric equations are a set of equations that define $x$ and $y$ coordinates separately in terms of a parameter, usually $t$: $x = f(t)$ and $y = g(t)$
• To sketch the graph of parametric equations, create a table of values for $t$ and plot the corresponding $(x, y)$ points
• To find the rectangular equation from parametric equations, eliminate the parameter $t$ by solving one equation for $t$ and substituting it into the other equation
• The derivative of a parametric curve is given by $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, provided that $\frac{dx}{dt} \neq 0$
  • This is useful for finding tangent lines and determining the direction of motion along the curve
• To find the area under a parametric curve, use the formula $A = \int_a^b y(t) \frac{dx}{dt} dt$, where $a$ and $b$ are the parameter values corresponding to the desired interval

Polar Coordinates

• In polar coordinates, a point is defined by its distance from the origin (radius, $r$) and the angle from the positive $x$-axis (theta, $\theta$)
• The relationship between polar coordinates $(r, \theta)$ and rectangular coordinates $(x, y)$ is given by $x = r \cos(\theta)$ and $y = r \sin(\theta)$
• To convert from rectangular to polar coordinates, use the equations $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(\frac{y}{x})$, with appropriate adjustments based on the quadrant
• Polar equations can be graphed by creating a table of values for $\theta$ and plotting the corresponding $(r, \theta)$ points in the polar plane
• To find the area enclosed by a polar curve, use the formula $A = \frac{1}{2} \int_a^b r^2(\theta) d\theta$, where $a$ and $b$ are the angle values corresponding to the desired interval

Vector-Valued Functions

• A vector-valued function is a function that assigns a vector to each value of the parameter, usually $t$, and is written as $\vec{r}(t) = \langle f(t), g(t) \rangle$ in 2D or $\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ in 3D
• The domain of a vector-valued function is the set of all values of $t$ for which the function is defined
• To find the derivative of a vector-valued function, differentiate each component function separately: $\vec{r}'(t) = \langle f'(t), g'(t) \rangle$ in 2D or $\vec{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle$ in 3D
  • The derivative of a vector-valued function gives the tangent vector at each point, which can be used to determine the direction of motion and velocity
• To find the definite integral of a vector-valued function, integrate each component function separately over the given interval: $\int_a^b \vec{r}(t) dt = \langle \int_a^b f(t) dt, \int_a^b g(t) dt \rangle$ in 2D or $\int_a^b \vec{r}(t) dt = \langle \int_a^b f(t) dt, \int_a^b g(t) dt, \int_a^b h(t) dt \rangle$ in 3D

Applications and Real-World Examples

• Parametric equations can be used to model the path of a projectile, with $x(t)$ representing the horizontal position and $y(t)$ the vertical position at time $t$
  • For example, the equations $x(t) = v_0 \cos(\theta) t$ and $y(t) = v_0 \sin(\theta) t - \frac{1}{2}gt^2$ model the path of a projectile with initial velocity $v_0$, launch angle $\theta$, and acceleration due to gravity $g$
• Polar coordinates are useful for describing the position of objects in fields that exhibit radial symmetry, such as in astronomy (positions of stars and planets) or in electromagnetic fields
• Vector-valued functions can model the motion of objects in space, such as the path of a satellite orbiting Earth or the trajectory of a particle in a magnetic field
  • For instance, the motion of a particle with charge $q$ in a uniform magnetic field $\vec{B}$ can be described by the vector-valued function $\vec{r}(t) = \langle r_0 \cos(\omega t), r_0 \sin(\omega t), v_z t \rangle$, where $r_0$ is the radius of the circular motion, $\omega = \frac{qB}{m}$ is the angular frequency, and $v_z$ is the velocity in the $z$-direction
• Parametric equations and vector-valued functions are used in computer graphics and animation to create smooth, continuous motion paths for objects and characters

Common Pitfalls and Tips

• When working with parametric equations, be careful to use the correct parameter values when finding corresponding points or evaluating integrals
• In polar coordinates, remember that the angle $\theta$ is measured in radians, not degrees
  • To convert from degrees to radians, multiply the angle in degrees by $\frac{\pi}{180}$
• When sketching polar curves, pay attention to the signs of $r$ and $\theta$ to determine the correct quadrant and direction of the curve
• For vector-valued functions, ensure that the components are expressed in terms of the same parameter and that the parameter values are consistent when evaluating derivatives or integrals
• When finding the area enclosed by a polar curve, be mindful of the interval of integration and any symmetries in the curve that may simplify the calculation

Practice Problems

1. Find the rectangular equation of the parametric equations $x = 2t - 1$ and $y = t^2 + 3t$.
2. Convert the point $(2, \frac{\pi}{3})$ from polar coordinates to rectangular coordinates.
3. Find the derivative of the vector-valued function $\vec{r}(t) = \langle t^2 - 1, 2t + 3, \sin(t) \rangle$.
4. Determine the area enclosed by the polar curve $r = 2\sin(\theta)$ for $0 \leq \theta \leq \pi$.
5. A particle moves along the path described by the parametric equations $x = 3\cos(2t)$ and $y = 3\sin(2t)$. Find the velocity and acceleration of the particle at time $t = \frac{\pi}{4}$.

Additional Resources

• Khan Academy: Parametric equations, Polar coordinates, Vector-valued functions
  • Offers video lessons, articles, and practice problems for each topic
• Paul's Online Math Notes: Parametric Equations and Curves, Polar Coordinates, Calculus with Vector-Valued Functions
  • Provides detailed notes, examples, and practice problems for each concept
• MIT OpenCourseWare: Multivariable Calculus (Course 18.02)
  • Includes lecture videos, notes, and assignments covering parametric equations, polar coordinates, and vector-valued functions in the context of multivariable calculus
• 3Blue1Brown: Essence of linear algebra (video series)
  • Offers an intuitive, visual introduction to vectors and vector-valued functions, which can help build a strong foundation for understanding these concepts in calculus