{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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! ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online