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SIMS Program 2010, Math 3B/3C Worksheet 1 Grace Kennedy SIMS Website: http://www.epsem.ucsb.edu/summer programs/sims.html CourseWebsite: http://math.ucsb.edu/∼kgracekennedy/SIMSsummer2010.html Monday August 16, 2010, Review from 3B and 3C 1 Derivative Formulas Thefollowing two properties are known as the linearity of the deriva- tive: d (f(x)+g(x)) = f′(x)+g′(x) dx “The derivative of the sum is the sum of the derivatives.” or “You can take the derivative term by term.” d [cf(x)] = cf′(x) dx “You can pull out constants.” Other Rules: d n n−1 • Power Rule: dx(x ) = nx What does this say about the derivative of a constant? • Product Rule: d [f(x)g(x)] = f(x)g′(x)+g(x)f′(x) dx • Quotient Rule: [f(x)] = g(x)f′(x)−f(x)g′(x) “LodiHi-HidiLo all over LoLo.” 2 g(x) [g(x)] • Chain Rule: d f(g(x)) = f′(g(x))g′(x) dx d x x • dx(e ) = e • d ln(|x|) = 1 dx x d d d 2 dx(sin(x)) = cos(x) dx(cos(x) = sin(x) dx(tan(x)) = sec (x) d (csc(x)) = −csc(x)cot(x) d (sec(x)) = sec(x)tan(x) d (cot(x)) − csc2(x) dx dx dx d (sin−1(x)) = √ 1 2 d (cos−1(x) = −√ 1 2 d (tan−1(x)) = 1 2 dx 1−x dx 1−x dx 1+x d −1 √1 d −1 √1 d −1 1 dx(csc (x)) = −x x2−1 dx(sec (x)) = x x2−1 dx(cot (x)) = −1+x2 1 2 Integration Formulas The Fundamental Theorem of Calculus: If f is continuous on [a,b], then the function g defined by g(x) = Z xf(t)dt, a ≤ x ≤ b a is the antiderivative of f, that is g′(x) = f(x) for a < x < b. “The integral and derivative undo each other.” The following two properties are known as the linearity of the integral: Z Z Z f(x)+g(x)dx= f(x)dx+ g(x)dx “The integral of the sum is the sum of the integrals.” or “You can integrate term by term.” Z Z cf(x)dx = c f(x)dx “You can pull out constants.” • Power Rule: R xndx = xn+1 +C (n 6= 1) n+1 • R dx = ln(|x|) +C R x x x R • e dx = e +C R R sin(x)dx = −cos(x)+C cos(x)dx = sin(x)+C sec2(x)dx = tan(x)+C R tan(x) =? R ln(x)dx =? etc... Everything on the first page gives you an integration formula. But some seemingly commonfunctionsdon’thaveanobviousderivative(liketan(x))while some seemingly obscure functions do (like sec2(x))! 3 Integration Methods You Should Know... if not now, by Tuesday at 9:50. • u-substitution • integration by parts • partial fractions • Trigonometric substitution is also covered in 3B, but we will not have time to cover it in this summer session. Just be aware of it’s existance. You might have to go learn it at one point. If you are interested, come see me in office hours. Homework : You have a short webwork assignment for Tuesday night and a written assignment for Wednesday in class. 2
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