Functions

1. Function Properties

• Domain: the set of all valid inputs. Range: the set of all possible outputs.
• Monotonicity: a function is increasing on an interval if larger inputs give larger outputs there (decreasing if the opposite).
• Parity: a function is even if f(−x) = f(x) for all x (symmetric about the y-axis, e.g. f(x)=x²); odd if f(−x) = −f(x) (symmetric about the origin, e.g. f(x)=x³). Most functions are neither.

Common mistakes: confusing "the function is increasing" with "the function is positive" (these are unrelated properties); assuming every function is either even or odd — most are neither.

2. Elementary Functions

• Power functions: f(x) = xⁿ.
• Exponential functions: f(x) = aˣ (a > 0, a ≠ 1). Always positive, passes through (0, 1).
• Logarithmic functions: f(x) = log_a(x), the inverse of aˣ. Domain is x > 0 only.
  • Key rule: if log_a(x) = y, then x = aʸ.
• Trigonometric functions: sin(x), cos(x), tan(x) — periodic functions describing angles/rotation.

Common mistakes: taking log of a negative number or zero (undefined); confusing log_a(x) = y with x = a·y instead of x = aʸ (a very common slip).

3. Sequences

• Arithmetic sequence: each term differs from the previous by a constant common difference d.
  • nth term: aₙ = a₁ + (n−1)d
  • Sum of first n terms: Sₙ = n/2 · (a₁ + aₙ) = n/2 · [2a₁ + (n−1)d]
• Geometric sequence: each term is the previous term multiplied by a constant common ratio r.
  • nth term: aₙ = a₁ · r^(n−1)
  • Sum of first n terms (r ≠ 1): Sₙ = a₁(1 − rⁿ)/(1 − r)

Common mistakes: in arithmetic sequences, forgetting the (n−1) factor — e.g. computing the 10th term as a₁ + 10d instead of a₁ + 9d; in geometric sequences, using rⁿ instead of r^(n−1) for the nth term.

4. Basics of Derivatives and Calculus

• The derivative f′(x) represents the instantaneous rate of change of f at x — geometrically, the slope of the tangent line to the curve at that point.
• Formal definition (limit of average rate of change as the interval shrinks to zero): f′(x) = lim(h→0) [f(x+h) − f(x)] / h.
• Power rule: if f(x) = xⁿ, then f′(x) = n·x^(n−1).
• For a polynomial, differentiate term by term: if f(x) = axⁿ + bx + c, then f′(x) = a·n·x^(n−1) + b.
• Application: to find where a function has a maximum or minimum, solve f′(x) = 0.

Common mistakes: confusing f(a) (the function's value) with f′(a) (the derivative's value at a) — these are two completely different numbers; forgetting the derivative of a constant is 0, not the constant itself; sign errors when differentiating terms with negative coefficients.