Review of algebra and analysis
When solving an equation first write the domain: an expression under a logarithm is positive, a denominator is not zero, an expression under an even root is not negative; then check the roots you find against it. An exponential equation is brought to the same base (aˣ = aʸ ⟹ x = y); a logarithmic equation is solved with logₐ f = logₐ g ⟹ f = g or by the definition. In an arithmetic progression aₙ = a₁ + (n − 1)d and Sₙ = (a₁ + aₙ)n/2; in a geometric progression bₙ = b₁qⁿ⁻¹ and Sₙ = b₁(qⁿ − 1)/(q − 1). Derivatives: (xⁿ)′ = nxⁿ⁻¹, (sin x)′ = cos x, (eˣ)′ = eˣ, (ln x)′ = 1/x, and for a composite function we multiply by the derivative of the inner function. An extremum occurs where f′(x) = 0 and the sign changes; the tangent is y = f(a) + f′(a)(x − a). Integrals: ∫xⁿdx = xⁿ⁺¹/(n + 1) + C, ∫cos x dx = sin x + C, the definite integral ∫ₐᵇf dx = F(b) − F(a), and the area is ∫(upper − lower)dx. In an exam check each answer by substituting back or with a sketch.
“Mistake detective”: give a friend three “solved” problems, each with one error (a spurious root kept in a logarithmic equation, the inner derivative 2 forgotten in the derivative of (2x + 1)³, a wrong number of terms in a progression sum). The friend finds and fixes the errors, then you swap roles.