1

 y'=     -----------

           sqrt(1-x²)

         - 1

 y'=     -----------

           sqrt(1-x²)

         1

 y'=     --------

           1+x²

        - 1

 y'=     -----------

           1+x²

         1

 y'=     ---------------

           x * sqrt(x²-1)

        - 1

 y' =     ---------------

           x * sqrt(x²-1)

e^x - e^-x

2

e^x + e^-x

2

     1

y'= --

      x

lim_x->0 (1+x)^1/x

lim_x->infinty (1+1/x)^x

                   sinx

        lim_x->0  -----

                    x

                   cosx-1

        lim_x->0  -------

                    x

      1

y'=  -----

       x lna

T(x)=T_a+(T_o-T_a)e^kx

x= time

T_a=Ambient Temperature

T_o=Initial Temperature

         1

 y'=     -----------

           sqrt(1+x²)

              1

 y'=     -----------

           sqrt(x²-1)

y=tanh^-1(x)

          1

 y'=     -----------

           1 - x²

y=csch^-1(x)

         1

 y'=     -----------

           |x|sqrt(x²+1)

y=sech^-1(x)

       -  1

 y'=     -----------

           |x|sqrt(1-x²)

y=coth^-1(x)

         1

 y'=     -----------

           1 - x²

suppose f satifies

1. f is continunous on [a,b]

2. f is differentiable on (a,b)

then there exists

f'(c)=f(b)-f(a)/(b-a)

* the average change of f on [a,b]

*linear approximation

1. must be a function

2. f is continuous on [a,b]

3. f(a)#f(b)

then there exists a number N between f(a) and f(b) and a number c between a and b such that

f(c)=N

** use this to check if there are roots

Precise Definition of a Limit

Epsilon Delta Proofs

Given ε > 0 there exists δ > 0 such that

1. limit exists

2. f(a) is defined

3. f(a)=lim a

f'(x)=lim_h->0   f(x+h) - f(x)

                  ------------

                h

slope

instantaneous rate of change

velocity