An infinite series adds the terms of a sequence forever: a1 + a2 + a3 + …. Its nth partial sum, Sn, adds just the first n terms. If the partial sums approach one finite number S, the series converges and S is its sum. If not, the series diverges.
A sequence converges if its terms get closer and closer to one number L as n grows. If the terms grow without bound or never settle on one value, the sequence diverges. To find the limit, treat n like x and use limits at infinity, L’Hôpital’s rule or the squeeze theorem.
The arc length of a polar curve r = f(θ) from θ = α to θ = β is L = ∫ √(r² + (dr/dθ)²) dθ with limits α and β. It comes from the parametric arc length formula with x = r cos θ and y = r sin θ. As with area, choose limits that trace the curve exactly once.
The area between polar curves is A = ½∫ (R² − r²) dθ, where R is the outer curve, r is the inner curve, and the limits are the angles where the curves meet. Find those angles by setting the two equations equal. Square each radius before you subtract.
The area in polar coordinates inside a curve r = f(θ), from θ = α to θ = β, is A = ½∫ r² dθ with limits α and β. The formula adds up thin wedges, each shaped like a sector of a circle with area ½r²Δθ. Choose the limits so the curve is traced exactly once.
To find the derivative of a polar function r = f(θ), treat θ as a parameter: x = r cos θ and y = r sin θ. Then the slope is dy/dx = (dy/dθ)/(dx/dθ), where dy/dθ = r′ sin θ + r cos θ and dx/dθ = r′ cos θ − r sin θ. The derivative dr/dθ alone is not the slope.
A vector-valued function r(t) = ⟨x(t), y(t)⟩ gives the position of a moving point at each time t. Its derivative is the velocity vector v(t) = ⟨x′(t), y′(t)⟩. The speed is the length of velocity, |v(t)| = √(x′(t)² + y′(t)²), and the derivative of velocity is the acceleration vector a(t).
The arc length of a parametric curve x = x(t), y = y(t) from t = a to t = b is L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt. The expression under the integral is the speed of a point moving along the curve, so the arc length is the distance the point travels if it traces the curve once.
To find the derivative of parametric equations x = x(t) and y = y(t), divide the two rates: dy/dx = (dy/dt)/(dx/dt), as long as dx/dt is not zero. For the second derivative, differentiate dy/dx with respect to t, then divide by dx/dt again.
A logistic differential equation, dP/dt = kP(1 − P/L), models growth that slows down as it nears a carrying capacity L. Its solution is P = L/(1 + Ae^(−kt)), an S-shaped curve. The population grows fastest when it reaches half the carrying capacity, P = L/2, and it approaches L over time.