CalculusReference - CALCULUS REFERENCE 3/18/03 10:49 AM...

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SPARK CHARTS Calculus Reference page 1 of 2 This downloadable PDF copyright © 2004 by SparkNotes LLC. SPARK ±²³´µ¶ µM SPARK CHARTS TM Copyright © 2003 by SparkNotes LLC. All rights reserved. SparkCharts is a registered trademark of SparkNotes LLC. 10 9 8 7 6 5 4 3 2 1 Printed the USA $3.95 $5.95 CAN CALCULUS REFERENCE 9 781586 638962 5 0 3 9 5 ISBN 1-58663-896-3 DERIVATIVES AND DIFFERENTIATION Definition: f ( x ) = lim h 0 f ( x + h ) f ( x ) h DERIVATIVE RULES 1. Sum and Difference: d dx f ( x ) ± g ( x ) ± = f ( x ) ± g ( x ) 2. Scalar Multiple: d dx cf ( x ) ± = cf ( x ) 3. Product: d dx f ( x ) g ( x ) ± = f ( x ) g ( x ) + f ( x ) g ( x ) Mnemonic: If f is “hi” and g is “ho,” then the product rule is “ho d hi plus hi d ho.” 4. Quotient: d dx ² f ( x ) g ( x ) ³ = f ( x ) g ( x ) f ( x ) g ( x ) ( g ( x )) 2 Mnemonic: “Ho d hi minus hi d ho over ho ho.” 5. The Chain Rule First formulation: ( f g ) ( x ) = f ( g ( x )) g ( x ) Second formulation: dy dx = dy du du dx 6. Implicit differentiation: Used for curves when it is difficult to express y as a function of x. Differentiate both sides of the equation with respect to x . Use the chain rule carefully whenever y appears. Then, rewrite dy dx = y and solve for y . Ex: x cos y y 2 = 3 x . Differentiate to first obtain dx dx cos y + x d (cos y ) dx 2 y dy dx = 3 dx dx , and then cos y x (sin y ) y 2 yy = 3 . Finally, solve for y = cos y 3 x sin y +2 y . COMMON DERIVATIVES 1. Constants: d dx ( c ) = 0 2 . Linear: d dx ( mx + b ) = m 3 . Powers: d dx ( x n ) = nx n 1 (true for all real n = 0 ) 4 . Polynomials: d dx ( a n x n + · · · + a 2 x 2 + a 1 x + a 0 ) = a n nx n 1 + · · · + 2 a 2 x + a 1 5. Exponential Base e : d dx ( e x ) = e x Arbitrary base: d dx ( a x ) = a x ln a 6. Logarithmic Base e : d dx (ln x ) = 1 x Arbitrary base: d dx (log a x ) = 1 x ln a 7. Trigonometric Sine: d dx (sin x ) = cos x Cosine: d dx (cos x ) = sin x Tangent: d dx (tan x ) = sec 2 x Cotangent: d dx (cot x ) = csc 2 x Secant: d dx (sec x ) = sec x tan x Cosecant: d dx (csc x ) = csc x cot x 8. Inverse Trigonometric Arcsine: d dx (sin 1 x ) = 1 1 x 2 Arccosine: d dx (cos 1 x ) = 1 1 x 2 Arctangent: d dx (tan 1 x ) = 1 1+ x 2 Arccotangent: d dx (cot 1 x ) = 1 1+ x 2 Arcsecant: d dx (sec 1 x ) = 1 x x 2 1 Arccosecant: d dx (csc 1 x ) = 1 x x 2 1 INTEGRALS AND INTEGRATION DEFINITE INTEGRAL The definite integral ´ b a f ( x ) dx is the signed area between the function y = f ( x ) and the x -axis from x = a to x = b . Formal definition:
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CalculusReference - CALCULUS REFERENCE 3/18/03 10:49 AM...

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