AP®︎ Calculus BC: Unit 9 Practice Test
Prepare for your quiz, test, or the AP exam with focused practice questions on Unit 9 of AP Calculus BC – Parametric Equations, Polar Coordinates, and Vector-Valued Functions.
Questions List
Unit 9 (All Topics)
Question 1 Topic 9.1Easy
A curve is defined by and . What point is on the curve when ?
What You’re Being Tested On:
Explore the learning objectives taken directly from the College Board’s AP® Calculus BC Curriculum. Ensure you’re prepared for the exact topics covered on the AP® exam, in-class tests, and quizzes, and gain confidence in your mastery of the material.
Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions
Work with parametric, polar, and vector-valued functions using differentiation and integration.
Topic 9.1: Defining and Differentiating Parametric Equations
Learning Objective: CHA-3.G
Calculate derivatives of parametric functions.
Essential Knowledge: CHA-3.G.1
Methods for calculating derivatives of real-valued functions can be extended to parametric functions.
Essential Knowledge: CHA-3.G.2
For a curve defined parametrically, the value of dy/dx at a point on the curve is the slope of the line tangent to the curve at that point. dy/dx, the slope of the line tangent to a curve defined using parametric equations, can be determined by dividing dy/dt by dx/dt, provided dx/dt does not equal zero.
Topic 9.2: Second Derivatives of Parametric Equations
Learning Objective: CHA-3.G
Calculate derivatives of parametric functions.
Essential Knowledge: CHA-3.G.3
d^2y/dx^2 can be calculated by dividing d/dt(dy/dx) by dx/dt.
Topic 9.3: Finding Arc Lengths of Curves Given by Parametric Equations
Learning Objective: CHA-6.B
Determine the length of a curve in the plane defined by parametric functions, using a definite integral.
Essential Knowledge: CHA-6.B.1
The length of a parametrically defined curve can be calculated using a definite integral.
Topic 9.4: Defining and Differentiating Vector-Valued Functions
Learning Objective: CHA-3.H
Calculate derivatives of vector-valued functions.
Essential Knowledge: CHA-3.H.1
Methods for calculating derivatives of real-valued functions can be extended to vector-valued functions.
Topic 9.5: Integrating Vector-Valued Functions
Learning Objective: FUN-8.A
Determine a particular solution given a rate vector and initial conditions.
Essential Knowledge: FUN-8.A.1
Methods for calculating integrals of real-valued functions can be extended to parametric or vector-valued functions.
Topic 9.6: Solving Motion Problems Using Parametric and Vector-Valued Functions
Learning Objective: FUN-8.B
Determine values for positions and rates of change in problems involving planar motion.
Essential Knowledge: FUN-8.B.1
Derivatives can be used to determine velocity, speed, and acceleration for a particle moving along a curve in the plane defined using parametric or vector-valued functions.
Essential Knowledge: FUN-8.B.2
For a particle in planar motion over an interval of time, the definite integral of the velocity vector represents the particle’s displacement (net change in position) over the interval of time, from which we might determine its position. The definite integral of speed represents the particle’s total distance traveled over the interval of time.
Topic 9.7: Defining Polar Coordinates and Differentiating in Polar Form
Learning Objective: FUN-3.G
Calculate derivatives of functions written in polar coordinates.
Essential Knowledge: FUN-3.G.1
Methods for calculating derivatives of real-valued functions can be extended to functions in polar coordinates.
Essential Knowledge: FUN-3.G.2
For a curve given by a polar equation r = f(θ), derivatives of r, x, and y with respect to θ, and first and second derivatives of y with respect to x can provide information about the curve.
Topic 9.8: Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
Learning Objective: CHA-5.D
Calculate areas of regions defined by polar curves using definite integrals.
Essential Knowledge: CHA-5.D.1
The concept of calculating areas in rectangular coordinates can be extended to polar coordinates.
Topic 9.9: Finding the Area of the Region Bounded by Two Polar Curves
Learning Objective: CHA-5.D
Calculate areas of regions defined by polar curves using definite integrals.
Essential Knowledge: CHA-5.D.2
Areas of regions bounded by polar curves can be calculated with definite integrals.
