Rucete ✏ AP Calculus In a Nutshell
1. Functions
This chapter reviews foundational precalculus concepts crucial for AP Calculus, including function properties, operations, transformations, and representations. It also introduces parametrically defined functions and polar coordinates for BC students.
Basic Definitions
• A function f assigns each input x in the domain to one and only one output f(x) in the range.
• Representations: equations, graphs, tables.
• Vertical Line Test: verifies whether a graph represents a function.
Domain and Range
• Domain = all permissible inputs (x-values).
• Range = all possible outputs (y-values).
• Exclude values that: – Make denominator zero – Result in even root of negative number – Are undefined in log or trig contexts
Function Operations
• Sum, difference, product, quotient: (f ± g)(x) = f(x) ± g(x), (fg)(x), (f/g)(x) (where g(x) ≠ 0)
Function Composition
• (f ∘ g)(x) = f(g(x)) • Domain: all x where g(x) is defined and f(g(x)) is defined
• In general, f(g(x)) ≠ g(f(x))
Even and Odd Functions
• Even: f(–x) = f(x) → symmetric about y-axis
• Odd: f(–x) = –f(x) → symmetric about origin
One-to-One and Inverses
• One-to-one: passes horizontal line test
• Inverse function f⁻¹(x): switch x and y, solve for y
• Graphs of f and f⁻¹ are reflections over y = x
Transformations of Functions
• |f(x)|: reflect negative parts over x-axis
• f(|x|): reflect right half over y-axis
• –f(x): reflection over x-axis
• f(–x): reflection over y-axis
Polynomial and Rational Functions
• Polynomial: f(x) = aₙxⁿ + ... + a₀ – Domain: all reals – Degree = highest power of x
• Rational: f(x) = P(x)/Q(x), where Q(x) ≠ 0
Trigonometric Functions
• Periodic functions repeat every p units
• sin x, cos x, sec x, csc x → period = 2π tan x, cot x → period = π
• Amplitude: height = |A| in A sin(bx)
• Inverse trig functions: – sin⁻¹x: domain [–1, 1], range [–π/2, π/2] – cos⁻¹x: domain [–1, 1], range [0, π] – tan⁻¹x: domain all reals, range (–π/2, π/2)
Exponential and Logarithmic Functions
• Exponential: f(x) = a^x (a > 0, a ≠ 1)
• Logarithmic: inverse of exponential – logₐ(x): domain x > 0 – ln(x): natural log (base e)
• Graphs of exp and log functions are reflections across y = x
Parametric Functions (BC)
• Functions defined in terms of a third variable (usually t): – x = f(t), y = g(t)
• Useful for motion problems, projectile paths, and curves that fail the vertical line test
• Derivatives: – dx/dt and dy/dt → slope = dy/dx = (dy/dt) / (dx/dt)
• Speed = √[(dx/dt)² + (dy/dt)²]
Example Graph: Circle
– x = cos(t), y = sin(t), 0 ≤ t ≤ 2π
Polar Functions (BC)
• Use polar coordinates (r, θ) instead of (x, y)
• Convert between forms: – x = r cos(θ), y = r sin(θ) – r = √(x² + y²), θ = tan⁻¹(y/x)
• Common graphs: – Circles: r = a – Roses: r = a sin(nθ) or a cos(nθ) – Limacons: r = a ± b sin(θ)
Example Graph: r = 1 + sin(θ)