Math 170 Final Formula Sheet, Study notes of Differential Equations

Download Math 170 Final Formula Sheet and more Study notes Differential Equations in PDF only on Docsity! 1 Math 170 Final Formula Sheet Probability Formula 1 Suppose we have an Events A such that • The Probability that Event A has Outcome X1 is P1 • The Probability that Event A has Outcome X2 is P2 • The Probability that (Event A has Outcome X1) AND (Event A has Outcome X2) is P3 then • The Probability that (Event A1 has Outcome X1) OR (Event A2 has Outcome X2) = P1 + P2 − P3 Independent Events Suppose we have two Events B1, B2 • Probability that Event B1 has Outcome Y1 is Q1 • Probability that Event B2 has Outcome Y2 is Q2 If Event B1 and Event B2 are independent (i.e. the out come of Event B1 does not effect the outcome of Event B2) then • The Probability that (Event B1 has Outcome Y1) AND (Event B2 has Outcome Y2) = Q1 ×Q2. 2 Dynamical Systems • Let Pn+1 = f(Pn) be a discrete dynamical system. Then the equilib- rium of the system occur at values of P such that P = f(P ). • An equilibrium P is – Stable from the left if and only if for all small enough ε, when P0 = P − ε then limn→∞ Pn = P – Stable from the right if and only if for all small enough ε, when P0 = P + ε then limn→∞ Pn = P Complex Numbers and Polynomials • Let f(x) be a polynomial in x with real coefficients. Then f(x) = f(x). So, if f(x) = 0 then f(x) = 0 • If f(x) = ax2 + bx + c where a, b, c are real numbers then f(x) = 0 if and only if x = −b±√b2 − 4ac 2a • If f(x) is a polynomial of degree n then f factors into n linear factors over the complex numbers. • If a, b are real numbers then – a + bi = a− bi – Norm(a + bi) = √ a2 + b2