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

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ENDURING UNDERSTANDING

CHA-3 Derivatives allow us to solve real-world problems involving rates of change.

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Topic Name

Essential Knowledge
9.1 Defining and Differentiation Parametric Equations   BC ONLY

LEARNING OBJECTIVE

CHA-3.G Calculate derivatives of parametric functions.  BC ONLY

 CHA-3.G.1 Methods for calculating derivatives of real-valued functions can be extended to parametric functions. BC ONLY
CHA-3.G.2 For a curve defined parametrically, the value of \frac{{dy}}{{dx}}  at a point on the curve is the slope of the line tangent to the curve at that point.\frac{{dy}}{{dx}}  the slope of the line tangent to a curve defined using parametric equations, can be determined by dividing \frac{{dy}}{{dt}}by , \frac{{dx}}{{dt}} provided \frac{{dx}}{{dt}}  does not equal zero. BC ONLY

Blog Posts

Implicit Differentiation of Parametric Equations

A Vector’s Derivatives  What they mean and how to find them. BC topic.

Units

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9.2 Second Derivatives of Parametric Equations BC ONLY

LEARNING OBJECTIVE

CHA-3.G Calculate derivatives of parametric functions.  BC ONLY

 CHA-3.G.3 \frac{{{{d}^{2}}y}}{{d{{x}^{2}}}}  can be calculated by dividing \frac{d}{{dt}}\left( {\frac{{dy}}{{dx}}} \right)  by \frac{{dx}}{{dt}}. BC ONLY

Blog Posts

Implicit Differentiation of Parametric Equations

A Vector’s Derivative

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ENDURING UNDERSTANDING

CHA-6 Definite integrals allow us to solve problems involving the accumulation of change in length over an interval.

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9.3 Finding Arc Lengths of Curves Given by Parametric Equations   BC ONLY

LEARNING OBJECTIVE

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

 CHA-6.B.1 The length of a parametrically defined curve can be calculated using a definite integral.  BC ONLY

Blog Posts

none

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ENDURING UNDERSTANDING

CHA-3 Derivatives allow us to solve real-world problems involving rates of change.

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9.4 Defining and Differentiating Vector-Valued Functions    BC ONLY

LEARNING OBJECTIVE

CHA-3.H Calculate derivatives of vector-valued functions. BC ONLY

 CHA-3.H.1 Methods for calculating derivatives of real-valued functions can be extended to vector-valued functions. BC ONLY

Blog Posts

Implicit Differentiation of Parametric Equations

A Vector’s Derivative

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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.

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9.5 Integrating Vector Valued Functions BC ONLY

LEARNING OBJECTIVE

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

 FUN-8.A.1 Methods for calculating integrals of real-valued functions can be extended to parametric or vector-valued functions. BC ONLY

Blog Posts

None

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9.6 Solving Motion Problems Using Parametric and Vector Valued Functions   BC ONLY

LEARNING OBJECTIVE

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

 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.   BC ONLY
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. BC ONLY

Blog Posts

None

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ENDURING UNDERSTANDING

FUN-3 Recognizing opportunities to apply derivative rules can simplify differentiation.

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9.7 Defining Polar Coordinates and Differentiating in Polar Form  BC ONLY

LEARNING OBJECTIVE

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

 FUN-3.G.1 Methods for calculating derivatives of real-valued functions can be extended to functions in polar coordinates. BC ONLY
FUN-3.G.2 For a curve given by a polar equation r=f\left( \theta  \right) , derivatives of r, x, and y with respect to \theta , and first and second derivatives of y with respect to  x can provide information about the curve.  BC ONLY

Blog Posts

None

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ENDURING UNDERSTANDING

CHA-5 Definite integrals allow us to solve problems involving the accumulation of change in area or volume over an interval.

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9.8 Finding Area of a Polar Region or the Area Bounded by a Single Polar Curve   BC ONLY

LEARNING OBJECTIVE

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

 CHA-5.D.1 The concept of calculating areas in rectangular coordinates can be extended to polar coordinates. BC ONLY

Blog Posts

None

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

LEARNING OBJECTIVE

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

 CHA-5.D.2 Areas of regions bounded by polar curves can be calculated with definite integrals. BC ONLY

Blog Posts

None

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Review Posts

Type 8: Parametric and Vector Equations (3-30-2018) Review Notes

Type 9: Polar Equation Questions (4-3-2018) Review Notes

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