CSSS 505
                                               Calculus Summary Formulas
                            
                            
                                                                            Differentiation Formulas 
                            
                           1.     d (xn) = nxnŠ1                                                 17.  dy = dy × du    Chain Rule 
                                  dx                                                                    dx      dx     dx
                           2.     d (fg) = fg′+ gf ′ 
                                  dx
                           3.     d ( f ) = gf ′Š fg′  
                                  dx g              g2
                                  d                     ′           ′
                           4.     dx f (g(x)) = f (g(x))g (x)  
                           5.     d (sin x) = cosx 
                                  dx
                           6.     d (cosx) = Šsin x 
                                  dx
                           7.     d (tanx) = sec2 x 
                                  dx
                           8.     d (cotx) = Šcsc2 x  
                                  dx
                           9.     d (secx) = secxtanx 
                                  dx
                           10.    d (cscx) = Šcscxcotx 
                                  dx
                           11.    d (ex) = ex 
                                  dx
                           12.    d (ax) = ax lna 
                                  dx
                           13.    d (lnx) = 1  
                                  dx              x
                           14.    d (Arcsinx) =              1        
                                  dx                       1Šx2
                           15.    d (Arctanx) =             1      
                                  dx                     1+x2
                           16.    d (Arcsecx) =                 1          
                                  dx                     | x |   x2 Š1
                            
                                                                            Trigonometric Formulas 
                            
                           1.    sin2θ +cos2θ =1                                                 13.  tanθ = sinθ =              1  
                           2.    1+tan2θ =sec2θ                                                                   cosθ        cotθ
                           3.    1+cot2θ =csc2θ                                                                   cosθ           1
                           4.    sin(Šθ) = Šsinθ                                                 14.  cotθ = sinθ = tanθ  
                           5.    cos(Šθ) = cosθ                                                  15.  secθ =         1     
                           6.    tan(Šθ) = Štanθ                                                                  cosθ
                           7.    sin(A+ B) = sin AcosB+sinBcosA                                  16.  cscθ =         1  
                           8.    sin(AŠB) =sin AcosBŠsinBcosA                                                     sinθ
                           9.    cos(A+B)=cosAcosBŠsinAsinB                                      17.  cos(π Šθ) = sinθ  
                                                                                                              2
                           10.  cos(AŠ B) = cos AcosB+sin AsinB                                  18.  sin(π Šθ) = cosθ  
                                                                                                             2
                           11.  sin 2θ = 2sinθ cosθ  
                           12.  cos2θ = cos2θ Šsin2θ = 2cos2θ Š1=1Š2sin2θ
                                  
                            
                            
                            
                             
                                                                               Integration Formulas 
                             
                            Definition of a Improper Integral 
                                       b
                                       ∫ f (x) dx is an improper integral if 
                                       a
                                      1.     f  becomes infinite at one or more points of the interval of integration, or 
                                      2.    one or both of the limits of integration is infinite, or 
                                      3.    both (1) and (2) hold. 
                                       
                             
                             
                             
                            1.    ∫a dx = ax +C                                                   12.  ∫cscx dx = lncscx Šcot x +C  
                                                xn+1                                              13.     sec2 x dx = tan x +C  
                            2.    ∫ xn dx =            +C,   n ≠ Š1                                     ∫
                                                n+1                                               14.  ∫secxtan x dx = secx +C  
                            3.      1dx=ln x + C 
                                  ∫ x                                                             15.  ∫csc2 x dx = Šcot x +C  
                            4.    ∫ex dx = ex +C                                                  16.  ∫cscxcot x dx = Šcscx +C 
                                                ax                                                             2
                            5.    ∫axdx =            +C                                           17.  ∫ tan     x dx = tan x Š x +C  
                                               lna                                                            dx          1             x 
                            6.    ∫lnx dx = xlnxŠ x+C                                             18.  ∫ a2 + x2 = a Arctana+C  
                                                                                                                                          
                            7.    ∫sin x dx = Šcosx +C                                            19.          dx        = Arcsin x+C  
                                                                                                        ∫      2      2                a 
                            8.    ∫cosx dx = sin x +C                                                        a Šx                         
                            9.    ∫tanx dx = lnsecx +C  or   Šlncosx +C                           20.  ∫        dx         = 1 Arcsec x +C = 1 Arccos a +C
                                                                                                          x x2 Ša2            a              a           a              x
                            10.  ∫cot x dx = lnsin x +C                                                 
                            11.  ∫secx dx = lnsecx + tan x +C 
                             
                                                                   Formulas and Theorems 
                        1a. Definition of Limit:  Let  f  be a function defined on an open interval containing  c  (except 
                        possibly at  c ) and let L be a real number.  Then   lim     f (x) = L   means that for each ε  > 0  there 
                                                                            x →a
                        exists a δ > 0  such that   f (x) Š L < ε   whenever  0 <  x Š c < δ . 
                        1b. A function y = f (x) is continuous at   x = a  if   
                                           i).      f(a) exists 
                                           ii).        lim    f (x) exists 
                                                     x →a
                                           iii).       lim = f(a)                
                                                     x →a
                        4. Intermediate-Value Theorem                                              []
                                 A function   y = f (x) that is continuous on a closed interval  a,b  takes on every value 
                                 between  f (a)  and  f (b) . 
                                                                 []
                                 Note:  If   f is continuous on  a,b  and  f (a)  and   f (b) differ in sign, then the equation 
                                             f (x) = 0 has at least one solution in the open interval (a,b). 
                        5.       Limits of Rational Functions as  x → ±∞ 
                          i). lim f(x)=0 if the degree of   f(x)< the degree of g(x) 
                                                     x →±∞g(x)
                                                                                  x2 Š2x
                            Example:  lim 3                                                  =0 
                                                                         x →∞ x +3
                          ii). lim f(x)is infinite if the degrees of  f(x) > the degree of g(x) 
                                                     x→±∞ g(x)
                                                                                   x3 +2x
                            Example
                                                                       :   lim       2       =∞ 
                                                                         x →∞ x Š8
                          iii). lim f(x)is finite if the degree of   f(x) = the degree of g(x) 
                                                     x→±∞ g(x)
                            
                                                                                  2x2 Š3x+2             2
                            Example:   lim                                                     2 =Š5 
                                                                         x →∞ 10xŠ5x
                         
                        6.       Average and Instantaneous Rate of Change 
                                                                          (        )       (       )
                                 i).       Average Rate of Change:  If   x , y        and   x , y     are points on the graph of 
                                                                            0    0           1 1
                                            y = f (x), then the average rate of change of  y  with respect to  x  over the interval 
                                                         f (x ) Š f (x )      y Š y        ∆y
                                           []x , x   is      1         0   = 1       0 =       . 
                                              0   1          x Šx             x Šx         ∆x
                                                              1     0          1     0
                                                                                ()
                                 ii).      Instantaneous Rate of Change:  If   x0, y0  is a point on the graph of   y = f (x), then 
                                                                                                                     ′
                                           the instantaneous rate of change of   y  with respect to   x  at   x0  is   f (x0) . 
                                    ′                f (x + h) Š f (x)
                        7.        f (x) =     lim            h            
                                            h→0