AP Calculus BC: Unit 9 Notes

Unit 9: Parametrics, Polar Coordinates, and Vector-Valued Functions

9.1: Parametric Equations

• Definition: Parametric equations express multiple quantities as functions of another variable (parameter), typically used to describe x and y in terms of time.

• Graphing Steps:

  1. Create a table of values for the parameter.
  2. Plot the points corresponding to x and y.
  3. Connect the points, which may result in circles or non-function shapes (e.g., loops).

• Conversion to Rectangular Equations:

  1. Rearrange to solve for the parameter (e.g., t).
  2. Substitute t in both equations to express y in terms of x.
  3. Utilize trigonometric identities to simplify equations.

• Summary: Parametric equations are useful for describing scenarios involving multiple quantities, such as motion.

9.2: Differentiation of Parametric Equations

• Finding Derivative: To derive parametric equations, use
  $\frac{dy}{dx} = \frac{g'(t)}{f'(t)}$
  where g(t) corresponds to y and f(t) corresponds to x.

• Tangent Lines:

  1. Compute the derivative to find the slope.
  2. Substitute t-values into both functions to get points.
  3. Write the equation of the tangent line in point-slope form:
     $y - y<em>1 = m(x - x</em>1)$

• Velocity and Acceleration: Viewed as vectors:

  • Velocity Vector: $(f'(t), g'(t))$
  • Acceleration Vector: $(f''(t), g''(t))$
  • Speed: $\sqrt{[f'(t)]^2 + [g'(t)]^2}$

• Length of Parametric Curve: Given by
  $L = \int \sqrt{[f'(t)]^2 + [g'(t)]^2} dt$

• Summary: Differentiate parametric equations to determine velocity and acceleration. The length formula is unique for parametric curves.

9.3: Vector-Valued Functions

• Definition: A vector-valued function is defined by two real-valued functions, representing x and y directions:
  $\mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j}$
  or $\mathbf{r}(t) = \langle f(t), g(t) \rangle$

• Limit of Vector-Valued Function:
  $\lim<em>{t \to c} \mathbf{r}(t) = \lim</em>{t \to c} f(t) \mathbf{i} + \lim_{t \to c} g(t) \mathbf{j}$
  where the limits exist.

• Conditions for Continuity:

  1. The value of the vector function exists at point c.
  2. The limit of the vector function exists as t approaches c.
  3. The limit from both sides equals the function's value at c.

• Summary: Vector-valued functions are composed of component functions in the x and y directions, similar to parametric equations.

9.4: Differentiation and Integration of Vector-Valued Functions

• Derivative: The derivative of a vector-valued function describes a vector tangent to the curve traced:
  $\mathbf{r}'(t) = f'(t) \mathbf{i} + g'(t) \mathbf{j}$

• Integration: The integral of a vector-valued function is:
  $\int \mathbf{r}(t) dt = \int f(t) dt \mathbf{i} + \int g(t) dt \mathbf{j}$

  • Note: Include separate integration constants for each component.

• Summary: Derivatives and integrals of vector-valued functions derive from their respective components, with appropriate constants.

9.5: Velocity and Acceleration Using Vector-Valued Functions

• Speed: Given by:
  $\text{speed} = \sqrt{[f'(t)]^2 + [g'(t)]^2}$

• Velocity Vector: Indicates instantaneous change and direction.

• Acceleration: The derivative of the velocity vector.

• Summary: We can relate velocity and acceleration to vector-valued functions using their derivatives and integrals.

9.6: Polar Coordinates and Polar Graphs

• Definition: Polar coordinates specify points using a directed distance (r) from the origin and an angle (θ) from the positive x-axis, measured counter-clockwise.

• Uniqueness of Representation: Points can sometimes be represented in multiple ways, e.g.,

  • $(r, θ)$ and $(r, θ + 2\pi)$ represent the same point.

• Conversion Formulas:

  • $x = r \cos(θ)$
  • $y = r \sin(θ)$

• Graphing Polar Equations:

  1. Create a table of values for θ and r.
  2. Plot points (e.g., concentric circles) and connect for shapes (e.g., cardioids).

• Summary: Polar coordinates can represent points as distances and angles, and they can be graphed similar to Cartesian coordinates.

9.7: Polar Form of a Derivative

• Finding Derivative:

  • Change polar equations into parametric form to find the derivative.
  • Use:
    $\frac{dy}{dx} = f'(θ) \sin(θ) + g'(θ) \cos(θ)$

• Example Derivation: For the curve $r = 1 + \cos(θ)$:

  • Compute respective derivatives and simplify.

• Summary: The derivative of a polar curve can be obtained by converting to parametric equations or using formulas.

9.8: Areas in Polar Coordinates

• Area Formula: The area bounded by a polar curve is determined by the formula
  $A = \frac{1}{2} \int_α^β r^2 dθ$

• Example Calculation: For curve $r = 6 + 6 \cos(θ)$:

  • Set limits of integration based on curve and compute.
  • Convert to simpler integral forms to evaluate areas bounded.

• Summary: Area calculations in polar coordinates can use integration similar to Cartesian methods, focusing on circular sectors.