Q BankQuestion BankDocsDocuments

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

Syllabus
2020
Section
—
Level
—

Exam analysis

No tagged past-paper evidence yet

Published Concept pages under this syllabus area do not have tagged past-paper appearances in the selected level yet.

Recent 5 years

In this section

Topic 9.1

9.1 Defining and Differentiating Parametric Equations

Objectives in this topic

CHA-3.G—Calculate derivatives of parametric functions

  • CHA-3.G Calculate derivatives of parametric functions.
  • CHA-3.G.1 Methods for calculating derivatives of real-valued functions can be extended to parametric functions.
  • 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.
  • Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

Topic 9.2

9.2 Second Derivatives of Parametric Equations

Objectives in this topic

CHA-3.G—Calculate derivatives of parametric functions—Topic 9.2

  • CHA-3.G Calculate derivatives of parametric functions.
  • CHA-3.G.3 d²y/dx² can be calculated by dividing (d/dt)(dy/dx) by dx/dt.
  • Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

Topic 9.3

9.3 Finding Arc Lengths of Curves Given by Parametric Equations

Objectives in this topic

CHA-6.B—Determine the length of a curve in the plane defined by parametric functions, using a definite integral

  • CHA-6.B Determine the length of a curve in the plane defined by parametric functions, using a definite integral.
  • CHA-6.B.1 The length of a parametrically defined curve can be calculated using a definite integral.
  • Enduring understanding CHA-6: Definite integrals allow us to solve problems involving the accumulation of change in length over an interval.

Topic 9.4

9.4 Defining and Differentiating Vector-Valued Functions

Objectives in this topic

CHA-3.H—Calculate derivatives of vector-valued functions

  • CHA-3.H Calculate derivatives of vector-valued functions.
  • CHA-3.H.1 Methods for calculating derivatives of real-valued functions can be extended to vector-valued functions.
  • Enduring understanding CHA-3: Derivatives allow us to solve real-world problems involving rates of change.

Topic 9.5

9.5 Integrating Vector-Valued Functions

Objectives in this topic

FUN-8.A—Determine a particular solution given a rate vector and initial conditions

  • FUN-8.A Determine a particular solution given a rate vector and initial conditions.
  • FUN-8.A.1 Methods for calculating integrals of real-valued functions can be extended to parametric or vector-valued functions.
  • Enduring understanding FUN-8: Solving an initial value problem allows us to determine an expression for the position of a particle moving in the plane.

Topic 9.6

9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions

Objectives in this topic

FUN-8.B—Determine values for positions and rates of change in problems involving planar motion

  • FUN-8.B Determine values for positions and rates of change in problems involving planar motion.
  • 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.
  • 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.
  • Enduring understanding FUN-8: Solving an initial value problem allows us to determine an expression for the position of a particle moving in the plane.

Topic 9.7

9.7 Defining Polar Coordinates and Differentiating in Polar Form

Objectives in this topic

FUN-3.G—Calculate derivatives of functions written in polar coordinates

  • FUN-3.G Calculate derivatives of functions written in polar coordinates.
  • FUN-3.G.1 Methods for calculating derivatives of real-valued functions can be extended to functions in polar coordinates.
  • 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.
  • Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

Topic 9.8

9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve

Objectives in this topic

CHA-5.D—Calculate areas of regions defined by polar curves using definite integrals

  • CHA-5.D Calculate areas of regions defined by polar curves using definite integrals.
  • CHA-5.D.1 The concept of calculating areas in rectangular coordinates can be extended to polar coordinates.
  • Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

Topic 9.9

9.9 Finding the Area of the Region Bounded by Two Polar Curves

Objectives in this topic

CHA-5.D—Calculate areas of regions defined by polar curves using definite integrals—Topic 9.9

  • CHA-5.D Calculate areas of regions defined by polar curves using definite integrals.
  • CHA-5.D.2 Areas of regions bounded by polar curves can be calculated with definite integrals.
  • Enduring understanding CHA-5: Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.
ConceptAP Calculus BC