AP Calculus

Summary: Integral is area under curve, derivative is an anti-integral

1: Limits and Continuity

Big Idea: Limits describe how a function behaves near a point, not just at it. They’re the foundation of calculus.
Limit of f(x) as x approaches a: lim(x→a) f(x) = where left and right approach at a
One-sided limits: lim(x→a⁻) f(x) (left approach) and lim(x→a⁺) f(x) (right)
A function is continuous at x = a if:
1. f(a) is defined
2. lim(x→a) f(x) exists
3. lim(x→a) f(x) = f(a)
Discontinuities: removable (hole), jump, infinite (asymptote)

Big Idea: Derivatives measure instantaneous change, basically rate of change
Derivative definition: f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Power Rule: d/dx[x^n] = n*x^(n-1)
Constant Rule: d/dx[c] = 0
Constant Multiple Rule: d/dx[c*f(x)] = c*f'(x)
Sum Rule: d/dx[f(x) + g(x)] = f'(x) + g'(x)
Product Rule: d/dx[f(x)*g(x)] = f'(x)*g(x) + f(x)*g'(x)
Quotient Rule: d/dx[f(x)/g(x)] = [f'(x)*g(x) - f(x)*g'(x)] / [g(x)]^2

Big Idea: Chain rule differentiates complex and nested functions.
Chain Rule: If y = f(g(x)), then: dy/dx = f'(g(x)) * g'(x)
Implicit Differentiation: Differentiate both sides, treat y as a function of x, solve for dy/dx.
Inverse Functions: If y = f⁻¹(x), then: d/dx[f⁻¹(x)] = 1 / f'(f⁻¹(x))

Big Idea: Derivatives model real-world rates like velocity, growth, decay, and more.
Instantaneous rate = derivative at a point
Related Rates: When you derive treat letters as variables, dh/dt for example
L’Hôpital’s Rule: For indeterminate forms like 0/0 or ∞/∞: Derive numerator and denominator separately and replace original numerator and denominator

Big Idea: Derivatives reveal where a function increases, decreases, peaks, and flattens.
Critical points: Where f'(x) = 0 or undefined
First Derivative Test: Sign of f'(x) changes → local min/max
Second Derivative Test: f”(x) > 0 → min, f”(x) < 0 → max
Points of inflection: f”(x) changes sign
Optimization: Use derivatives to maximize/minimize under constraints

Big Idea: Integration is reverse of differentiation it accumulates change over an interval.
FUNDAMENTAL THEOREM: ∫[a to b] f'(x) dx = f(b) - f(a)
Indefinite integral: ∫ f(x) dx = F(x) + C
Properties: Integrals respect linearity and intervals can be split or reversed

Big Idea: These equations describe how quantities change and evolve over time.
General form: dy/dx = f(x, y)
Separation of Variables: Rearranged to integrate both sides
Exponential Growth/Decay: dy/dt = ky → y = Ce^(kt)
Logistic Growth: dy/dt = ky(1 - y/L) → growth slows, y approaches L (carrying capacity)

Big Idea: Integration gives you total accumulation — area, volume, distance.
Area between curves: ∫[a to b] [top - bottom] dx
Volume (disk method): V = π∫[a to b] [f(x)]^2 dx
Volume (washer method): V = π∫[a to b] ([outer radius]^2 - [inner radius]^2) dx
Arc length: L = ∫[a to b] √(1 + (dy/dx)^2) dx

Big Idea: Functions can be defined in terms of motion, direction, and rotation.
Parametric: x = f(t), y = g(t)
- dx/dt and dy/dt → slope: dy/dx = (dy/dt)/(dx/dt)
Arc length (parametric):
L = ∫[a to b] √[(dx/dt)^2 + (dy/dt)^2] dt
Polar: r = f(θ)
- Area: A = 1/2 ∫[a to b] [r(θ)]^2 dθ
Vectors: Position, velocity, acceleration via vector functions in form of (x, y)

Big Idea: Some infinite processes add up to finite values — others don’t.
Sequence limit: lim(n→∞) a_n = L
Series sum: Σ a_n
Convergence Tests:
- Geometric: converges if |r| < 1
- p-Series: converges if p > 1
- Alternating Series Test, Ratio Test, Integral Test, etc.
Taylor Series:
f(x) = Σ [f⁽ⁿ⁾(a) / n!]*(x - a)^n
Error bound (Taylor remainder):
|R_n(x)| ≤ M*(|x - a|)^(n+1)/(n+1)! where M bounds the (n+1)th derivative