Welcome to AP Calculus BC! This advanced course builds upon Calculus AB foundations, covering differential and integral calculus plus additional topics like parametric equations, polar coordinates, and infinite series. BC includes all AB content plus advanced integration techniques, series convergence tests, and Taylor polynomial approximations. The course typically progresses from September through May, preparing students for the comprehensive AP exam that tests both computational skills and conceptual understanding.
Advanced integration techniques are essential for AP Calculus BC. Integration by parts handles products like x times e to the x, using the formula u dv equals uv minus integral v du. Partial fractions decompose complex rational functions into simpler terms. For example, one over x squared minus one becomes a logarithmic expression. Improper integrals involve infinite limits or discontinuities. The integral of one over x squared from one to infinity converges to one, demonstrating how some infinite integrals have finite values.
Parametric equations express coordinates as functions of a parameter t, like x equals t squared and y equals t cubed. The derivative dy dx equals dy dt divided by dx dt. We can visualize parametric curves by tracing points as t varies. Polar coordinates use radius r and angle theta instead of x and y. For example, r equals 2 cosine theta creates a circle. Both parametric and polar curves have specific arc length formulas involving derivatives and integrals.
Infinite series are sums of infinitely many terms. A series converges if its partial sums approach a finite limit, otherwise it diverges. The ratio test examines the limit of consecutive term ratios, while the root test uses nth roots. The p-series sum of one over n squared converges to pi squared over six, but the harmonic series sum of one over n diverges to infinity. Geometric series with ratio less than one in absolute value converge to a over one minus r.
Taylor series represent functions as infinite polynomials using derivatives at a point. The general formula involves the nth derivative at point a. Maclaurin series are Taylor series centered at zero. For exponential function e to the x, we get one plus x plus x squared over two factorial plus higher terms. As we add more terms, the polynomial approximation becomes more accurate over a wider interval. Common Maclaurin series include sine x, cosine x, and natural log of one plus x. These series are essential for function approximation and solving differential equations in advanced calculus.