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SparkChartsTM—created by Harvard students for students everywhere—serve as study companions and reference tools that cover a wide range of college and graduate school subjects, including Business, Computer Programming, Medicine, Law, Foreign Language, Humanities, and Science. Titles like How to Study, Microsoft Word for Windows, Microsoft Powerpoint for Windows, and HTML give you what it takes to find success in school and beyond. Outlines and summaries cover key points, while diagrams and tables make difficult concepts easier to digest. This two-page chart reviews derivatives and differentiation, integrals and integration. It also covers applications in: Geometry Motion Probability and statistics Microeconomics Finance
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CALCULUS REFERENCE 3/18/03 10:49 AM Page 1 SPARKCHARTSTM CALCULUS REFERENCE SPARK CHARTS TM THEORY SPARKCHARTS TM DERIVATIVES AND DIFFERENTIATION f (x+h)−f (x) h h→0 Definition: f � (x) = lim is continuous and differentiable on the interval and F � (x) = f (x). d f (x) ± g(x) = f � (x) ± g � (x) 1. Sum and Difference: dx � � d cf (x) = cf � (x) 2. Scalar Multiple: dx � � d f (x)g(x) = f � (x)g(x) + f (x)g � (x) 3. Product: dx � Mnemonic: If f is “hi” and g is “ho,” then the product rule is “ho d hi plus hi d ho.” � � � (x)g � (x) f (x) d = f (x)g(x)−f (g(x)) 2 g(x) dx 4. Quotient: 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) dy du dy = du • Second formulation: dx dx dy dx 9 781586 638962 $5.95 CAN $3.95 1. Left-hand rectangle approximation: n−1 � Ln = ∆x f (xk ) dx dx y) dy = 3 dx − 2y dx cos y + x d(cos , dx dx and then cos y − x(sin y)y � − 2yy � = 3. Finally, solve for y � = cos y−3 . x sin y+2y Mn = ∆x =0 2. Linear: d (mx dx + b) = m 3. Powers: d (xn ) dx = nx n−1 (true for all real n �= 0) 4. Polynomials: d (an xn dx + · · · + a2 x + a1 x + a0 ) = an nx 6. Logarithmic • Base e: d (ln x) dx d (sin dx x 1 x x) = cos x d (tan dx d (sec dx = x) = sec 2 x x) = sec x tan x n−1 • Arbitrary base: d (ax ) dx 5. Simpson’s Rule: Sn = f (x0 )+4f (x1 )+2f (x2 )+· · ·+2f (xn−2 )+4f (xn−1 )+f (xn ) � ∆x 3 � � � � • Arbitrary base: d (log a dx • Cosine: • Cotangent: • Cosecant: d (cos dx d (cot dx d (csc dx • Definite integrals: concatenation: = a ln a x) = x) = − sin x x) = − csc 2 x x) = − csc x cot x • Arccosine: d (cos −1 dx 1 1+x2 • Arccotangent: 1 x) = − √1−x 2 x) = d (cot −1 dx • Arcsecant: d (sec −1 dx x) = √1 x x2 −1 • Arccosecant: 1 x) = − 1+x 2 d (csc −1 dx x) = − x√x12 −1 INTEGRALS AND INTEGRATION � � b f (x) dx = − a p f (x) dx + a • Definite integrals: comparison: If f (x) ≤ g(x) on the interval [a, b], then 1 x ln a d (tan −1 dx √ 1 1−x2 • Definite integrals: reversing the limits: x d (sin −1 dx � � a f (x) dx b b f (x) dx = p b a � f (x) dx ≤ � � b f (x) dx a b g(x) dx. a � � � 2. Substitution Rule—a.k.a.