AP Calculus · Unit 9
Parametric, Polar and Vector Functions
Slopes, speed, arc length and area for curves given by t or θ.
Topic 9.1Differentiate a parametric curve
BC only- Calculate the slope of a parametric curveDivide the y-component rate by the x-component rate where valid.
- Find the point and tangent lineEvaluate both coordinates and the slope at the same parameter value.
- Locate horizontal and vertical tangentsSolve the appropriate component-rate conditions and check the other component.
- Interpret direction and repeated pointsDistinguish the path's location from the direction in which the parameter traces it.
Topic 9.2Parametric second derivatives
BC only- Find the first derivative with respect to xForm the ratio of component rates correctly.
- Differentiate the slope with respect to the parameterApply the needed derivative rules to dy/dx.
- Convert to the second derivative with respect to xDivide the slope's parameter derivative by dx/dt.
- Interpret the result at a valid pointDetermine concavity and check where the formula is undefined or needs further analysis.
Topic 9.3Length of a parametric path
BC only- Find speed along a parametric pathTake the square root of the sum of squared component rates.
- Choose the parameter intervalUse the stated start and end parameter values. Check how much of the curve the parameter interval traces.
- Set up and evaluate lengthIntegrate speed rather than signed component change or dy/dx.
- Account for retracingDistinguish total distance traveled from the geometric length traced once.
Topic 9.4Differentiate vector-valued functions
BC only- Differentiate each vector componentProduce the derivative vector with components in the correct order.
- Find velocity and acceleration vectorsTake the first and second derivatives of the position vector.
- Evaluate vector derivatives at an instantSubstitute the same parameter value into every component.
- Distinguish a vector rate from a scalar rateRecognize that a component derivative, speed, and rate of distance from the origin are different quantities.
Topic 9.5Integrate vector-valued functions
BC only- Integrate each vector componentFind a vector antiderivative with a constant for each component.
- Use an initial position or velocityDetermine the constants from the supplied vector conditions.
- Calculate vector displacementIntegrate velocity component by component over the interval.
- Distinguish displacement from distanceRecognize that a zero vector displacement can still represent motion.
Topic 9.6Motion in the plane
BC only- Connect position, velocity, and accelerationDifferentiate or integrate the vector quantities and use initial conditions.
- Calculate speedFind the magnitude of the velocity vector.
- Determine whether speed increasesUse the velocity-acceleration dot product, handling zero-speed cases carefully.
- Calculate distance traveledIntegrate speed over the time interval.
- Interpret motion in the planeDistinguish direction changes, changes of speed, and changes of distance from the origin.
Topic 9.7Polar curves and tangent slopes
BC only- Convert between polar and Cartesian coordinatesUse the radial distance and angle correctly, including negative radii.
- Differentiate the Cartesian coordinatesApply the product and chain rules to r cosine theta and r sine theta.
- Find a polar tangent slopeDivide the coordinate derivatives where the ratio is valid.
- Locate horizontal and vertical tangentsCheck both coordinate derivatives and treat the pole or cusp separately.
Topic 9.8Area inside one polar curve
BC only- Identify the region and its angle intervalFind bounds for the requested sector, loop, or petal.
- Check whether the curve repeatsDo not count the same region more than once.
- Set up the polar area integralIntegrate half the squared radius over the correct angles.
- Calculate the requested areaUse symmetry only when justified and include the correct number of pieces.
Topic 9.9Area between polar curves
BC only- Find all relevant intersectionsSolve boundary crossings and check the pole, including points reached at different angles.
- Identify inner and outer boundariesDetermine radial order within each part of the requested region.
- Split the angle interval when neededAccount for changes in order or the boundary of a common interior.
- Set up and calculate the areaIntegrate half the difference of squared radii, or the appropriate common-interior expression.