Eqn Sheet - Math 122 Equation Sheet FORMULAE SHEET • •...

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Unformatted text preview: Math 122 Equation Sheet FORMULAE SHEET • • A − B = ( A − B )( A + B ) • A3 − B 3 = ( A − B)( A 2 + AB + B 2 ) • A + B = ( A + B)( A − AB + B ) A C A D AD ÷=⋅= B D B C BC 2 2 3 • • • 3 2 2 Two parallel lines have the same slope Two perpendicular lines have slopes such that m1 = -1/m2 The slope of the line passing by two points A(x1, y1) and B(x2, y2) is given by • y 2 − y1 m= x 2 − x1 • Given the slope m of a line and a point A(x0, y0) on the line we write the equation of the line using the point slope formula y - y0 = m (x-x0) The properties of logarithms are the following o log A + log B = log (AB) o log A – log B = log (A/B) B o log A = B log A • A = P (1 + rt ) The derivative as a limit: • h →0 f ′(a) = lim x →a Here are the most common derivatives: (in all the formulas below k denotes a constant) • • • • dx b = b x ln b (b > 0) dx d 1 ln x = (x > 0) • n ⎠ • ⎞ ⎟ ⎟ ⎠ It can be shown that if f ( x) and g ( x) are both differentiable and if both functions ∞ 0 approach infinity (i.e. ), or zero (i.e. ) ∞ 0 when x → a then f ′( x) lim g ( x) = lim g ′( x) x→a • d cos x = − sin x dx • L’HOPITAL’S RULE f ( x) d tan x = sec 2 x dx d sec x = sec x ⋅ tan x dx • • • • x→a • CONTINUITY A function x dx 1 (b, x > 0) d log b x = x ln b dx d sin x = cos x dx d csc x = − csc x ⋅ cot x dx d cot x = − csc 2 x dx 1 d arcsin x = dx 1− x2 d −1 arccos x = dx 1− x2 d 1 arctan x = dx 1+ x2 d sinh x = cosh x dx d cosh x = sinh x dx CHAIN RULE d • ( f ( x) )n = n ⋅ ( f ( x) )n−1 f ′( x) dx d • f ( g ( x)) = f ′( g ( x)) ⋅ g ′( x) dx LINEAR APPROXIMATION The linear approximation of f ( x) near x = a is given by the formula L ( x ) = f ′(a )( x − a ) + f ( a ) SIMPLE INTEGRATION FORMULAS • ∫ Kdx = Kx + C ( K and C are constants) • • • dx e = ex dx • Future value S = P⎛ (1 + i ) − 1 ⎞ ⎟ ⎜ ⎟ ⎜ i Present value d k =0 dx dn x = n ⋅ x n −1 dx d (k ⋅ f ( x) ) = k ⋅ f ′( x) dx d ( f ⋅ g ) = f ′ ⋅ g + f ⋅ g ′ (product rule) dx df f ′ ⋅ g − f ⋅ g ′ (quotient rule) = dx g g2 • Effective rate r = ⎛1 + r ⎞ − 1 ⎜ ⎟ eff ⎝ m⎠ ⎛ 1 − (1 + i) − n A = P⎜ ⎜ i ⎝ f ( x + h) − f ( x) or h f ( x) − f (a ) x−a RULES OF DIFFERENTIATION m ⎝ x →a + f ′( x) = lim mt Continuous Compound Interest A = Pe rt • • x →a − • r⎞ ⎛ A = P ⎜1 + ⎟ ⎝ m⎠ Compound Interest is defined at x = a. lim f ( x) = lim f ( x) = f (a) • Simple Interest f ( x) x ∫ x dx = n + 1 + C ( n ≠ −1 ) ∫ e dx = e + C n +1 n x x 1 ∫ x dx = ln x + C SUBSTITUTION ∫ f ′(u ( x)) ⋅ u ′( x)dx = f (u ( x)) + C INTEGRATION BY PARTS ∫ udv = uv − ∫ vdu FUNDAMENTAL PART I d dx THEOREM OF CALCULUS u( x) ∫ f (t )dt = f (u ( x)) ⋅ u ′( x) a FUNDAMENTAL PART II THEOREM OF CALCULUS b ∫ f ( x)dx = F (b) − F (a) a where F ( x) is an antiderivative of f ( x) f ( x) is continuous at x = a if We’ve helped over 50,000 students get better grades since 1999! ...
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