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.”
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                                                           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
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                          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.
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