Code Scholars Tutoring · AP Calculus BC
9 topics across 9 lessons, with explanations, worked examples, and self-checks that develop complete mathematical reasoning.
Topics 9.1 · BC only
A parametric curve gives both coordinates as functions of a third variable, often time. The derivative compares the rates at which the two coordinates change. It is a slope in the coordinate plane, not the vertical coordinate's rate of change with respect to time.
Topics 9.2 · BC only
The second derivative measures how the Cartesian slope changes with the horizontal coordinate. After finding as a function of the parameter, differentiate that slope with respect to the parameter and divide by once more.
Topics 9.3 · BC only
For a parametric plane curve, both coordinates contribute to the distance traveled during a short parameter interval. The magnitude of the coordinate-derivative vector gives the length accumulated per unit parameter. Integrating that nonnegative quantity gives the length traced over the interval.
Topics 9.4 · BC only
A vector-valued function packages several coordinate functions into one position description. Differentiate each component to obtain velocity and differentiate again to obtain acceleration. These derivatives are vectors, while speed is the scalar magnitude of velocity.
Topics 9.5 · BC only
Integrate a vector-valued function one component at a time. Integrating velocity gives a family of position functions, and an initial position determines the separate constants. A definite integral of velocity gives displacement, which is a vector from the initial point to the final point.
Topics 9.6 · BC only
Planar motion combines position, velocity, and acceleration with scalar measurements of travel. Componentwise derivatives describe motion along each axis. Speed and total distance depend on the magnitude of velocity, so a particle can move quickly even when its net displacement is small.
Topics 9.7 · BC only
Polar coordinates describe a point by a signed radius and an angle. To differentiate a polar curve as a curve in the Cartesian plane, convert its coordinate functions to and , then use the parametric derivative ratio.
Topics 9.8 · BC only
A small polar sector has area approximately one half of radius squared times its angle width. Accumulating these sectors gives the polar area formula. Correct bounds must describe the intended region with the intended number of traversals.
Topics 9.9 · BC only
Polar area problems require identifying the allowed radial interval at each angle. For a region between two nonnegative radii, subtract the inner squared radius from the outer squared radius. For a region inside both curves and extending from the pole, the smaller radius determines the boundary.
