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Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

9 dot points across 9 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you differentiate a curve defined by parametric equations to find its slope?

Topic 9.1 Defining and Differentiating Parametric Equations: define a curve parametrically and find dy/dx as the ratio of the parametric derivatives (BC).
A focused answer to AP Calculus BC Topic 9.1, defining curves with parametric equations and finding the slope dy/dx as (dy/dt) over (dx/dt), with worked examples and the tangent line.
10 min answer →

How do you differentiate a vector-valued function to find velocity and acceleration?

Topic 9.4 Defining and Differentiating Vector-Valued Functions: define a vector-valued function and differentiate it component by component to find velocity and acceleration (BC).
A focused answer to AP Calculus BC Topic 9.4, defining a vector-valued function and differentiating it component by component to obtain the velocity and acceleration vectors, with worked examples.
9 min answer →

How do you describe a curve in polar coordinates and find its slope dy/dx?

Topic 9.7 Defining Polar Coordinates and Differentiating in Polar Form: convert between polar and Cartesian coordinates and find dy/dx for a polar curve r = f(theta) (BC).
A focused answer to AP Calculus BC Topic 9.7, defining polar coordinates, converting to and from Cartesian, and finding the slope dy/dx of a polar curve r = f(theta) by treating it parametrically, with worked examples.
10 min answer →

How do you find the length of a parametric curve over an interval of the parameter?

Topic 9.3 Finding Arc Lengths of Curves Given by Parametric Equations: compute the length of a parametric curve using the integral of the square root of (dx/dt)^2 + (dy/dt)^2 (BC).
A focused answer to AP Calculus BC Topic 9.3, computing the arc length of a parametric curve with the integral of the square root of (dx/dt)^2 + (dy/dt)^2 over the parameter interval, with worked examples.
9 min answer →

How do you find the area of a region bounded by two polar curves?

Topic 9.9 Finding the Area of the Region Bounded by Two Polar Curves: find the area between two polar curves by subtracting one-half r-squared integrals over the correct angle interval, after finding intersections (BC).
A focused answer to AP Calculus BC Topic 9.9, finding the area between two polar curves by subtracting one-half r-squared integrals over the correct angle interval, after locating the intersections, with worked examples.
10 min answer →

How do you find the area enclosed by a single polar curve using an integral?

Topic 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve: compute the area swept by a polar curve r = f(theta) using the one-half r-squared integral (BC).
A focused answer to AP Calculus BC Topic 9.8, computing the area enclosed by a single polar curve r = f(theta) using the integral of one-half r-squared over the correct angle interval, with worked examples.
10 min answer →

How do you integrate a vector-valued function to recover velocity and position?

Topic 9.5 Integrating Vector-Valued Functions: integrate a vector-valued function component by component to recover velocity from acceleration and position from velocity, using initial conditions (BC).
A focused answer to AP Calculus BC Topic 9.5, integrating a vector-valued function component by component to recover velocity from acceleration and position from velocity, applying initial conditions, with worked examples.
9 min answer →

How do you find the second derivative of a parametric curve to determine concavity?

Topic 9.2 Second Derivatives of Parametric Equations: find d^2y/dx^2 for a parametric curve by differentiating dy/dx with respect to t and dividing by dx/dt, and use it for concavity (BC).
A focused answer to AP Calculus BC Topic 9.2, finding the second derivative of a parametric curve by differentiating the first derivative with respect to t and dividing by dx/dt, and using it to determine concavity, with worked examples.
9 min answer →

How do you solve planar motion problems using parametric and vector-valued functions?

Topic 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions: find position, velocity, speed, acceleration, displacement and distance for a particle moving in the plane (BC).
A focused answer to AP Calculus BC Topic 9.6, solving planar motion problems with parametric and vector-valued functions, finding position, velocity, speed, acceleration, displacement and total distance, with worked examples.
10 min answer →