Limits, Derivatives, Integrals
                                                   PeyamRyanTabrizian
                                                 Monday,August8th, 2011
                        1    Limits
                        Evaluate the following limits. You may use l’Hopital’s rule!
                          (a) lim       1
                                 x→∞ 2x+3
                          (b) limx→6 x−6
                                      |x−6|
                                        4  2
                          (c) limx→∞ x −x
                                         3
                                       2x −1
                                              1
                          (d) lim    + cos(x)x
                                 x→0
                          (e) limx→∞e−xln(x)
                                       2    1

                          (f) limx→0x sin x
                          (g) limx→∞√x2 +1−x
                          (h) lim     √x+3−2
                                 x→1    x−1
                                       x      1 2
                          (i) limx→0 e −1−x−2x
                                          x3
                                        √ 2
                          (j) lim         x +1
                                 x→−∞      x
                                                                1
                         2     Derivatives
                         Find the derivatives of the following functions
                                           x
                           (a) f(x) = eee
                           (b) f(x) = sin(x)+x
                                          ln(x)
                           (c) f(x) = xtan(x)
                           (d) f(x) = tan(sin(cos(2x)))
                           (e) y′, where xy + xy2 + x2y = 1
                           (f) y′ at (1,2), where x2 + 2xy − y2 + x = 2
                           (g) f′(x) = ln(x)ln(x)
                           (h) f′′′(x), where f(x) = xex
                         3     Integrals
                         Evaluate the following integrals.
                           (a) R3 √9−x2dx
                               R−3
                           (b)   2 |x − 1|dx
                                 0
                               R 4 2
                           (c)    x +x dx
                                    x
                               R x√         x
                           (d)    e   1+e dx
                               R√
                           (e)     π xsin(x2)dx
                                 0
                           (f) Rπ xcos(x)dx
                                         2
                                 −π 1+x
                           (g) The average value of f(x) = sin(x)cos(x)4 on [0,π]
                           (h) Re2   dx
                                 e  xln(x)
                            (i) The derivative of g(x) = Rcos(x) ln(1 + x)dx
                                                            x
                                                            e
                               R     −1         R
                            (j)   tan  (x)dx +    √ 1   dx
                                       2               2
                                   1+x              1−x
                                                                   2