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Final Review 3: Other Gadgets and Properties • Sigma Notation n X ai i=m – Definition: n X ai = am + am+1 + · · · + an−1 + an i=m – Constant multiple: n X cai = c i=m Sum and difference rule: n X n X ai i=m (ai + bi ) = i=m n X ai + i=m n X i=m bi , n X (ai − bi ) = i=m n X i=m ai − n X bi i=m – No product or quotient rule. n n n X X X – Special sums: i, i2 and i3 . The formulaes will be given in the final. Notice that the i=1 i=1 i=1 sums in the formulaes are taken from the first term to the nth term. Z b Z c Z b • Interval Additivity: f (x) dx = f (x) dx + f (x) dx regardless the order of a, b and c on the real number line. a a c • Intermediate Value Theorem: Suppose f (x) is a continuous function on the closed interval [a, b] and A is any real number between f (a) and f (b). Then there is at least one real number c between a and b such that f (c) = A. • Mean Value Theorem – of Derivatives: Suppose f (x) is a differentiable function on the close interval [a, b] (i.e. f 0 (x) is defined everywhere between [a, b]). Then there is at least one real number c between a and b such that f (b) − f (a) f 0 (c) = b−a Average slope on the interval [a, b]. – of Integrals: Suppose f (x) is a continuous function on the closed interval [a, b], then there is at least one real number c between a and b such that Z b 1 f (c) = f (t) dt b−a a Average value on the interval [a, b].