5.3 – The Fundamental Theorem of Calculus

							5.3 – The Fundamental Theorem of
      Calculus
Examples
Use the TI-89 to determine g.
                                  2. g  x    t 3 dt
                                                x
1. g  x    cos t dt
              x

             1                                  3

                       1
3. g  x   
                  x
                             dt
                 1    1 t 2


Question So what does g represent here?
Answer A specific antiderivative of the function f.
 First Fundamental Theorem of
 Calculus – Part 1
If f is continuous on [a, b], then the function g is
defined by

                g  x    f  t  dt
                           x

                          a


is continuous on [a, b] and differentiable on
(a, b), and g'(x) = f (x). That is, g is the
antiderivative of f in terms of x.
 First Fundamental Theorem of Calculus
 – Part 1 (Alternate Definition)

If f is continuous on [a, b], then

             d  x
                     f t  dt   f x 
             dx  a
                              
                               
Examples
Determine the derivative of g in your head.

                                                     3
1. g  x    ln t dt       2. g  x   
              x                               x
                                                         dt
             1                            3       t  5t
                                                   2



3. g  x    cos  t  dt 4. g  x   
              u                               2
                         2
                                                    9csc t dt
             1                                x
Examples
Use the TI-89 to determine the derivative of g.
Establish a property for what you observe.
     d x2
     dx 1
  5.       1  r 3 dr

     d cos x
  6.
     dx 3 ln t dt
 First Fundamental Theorem of Calculus
 – Part 1 (Alternate Definition)

If f is continuous on [a, b] and u is a function of
x, then

            d  u               f u  du
            dx 
                a f t  dt 
                                        dx
Examples

Determine the derivative of g by hand.
                 e2 x
7. g  x               t  sin t  dt
                 2



8. g  x    sin t dt
                     2
                              5
                     ln x
 First Fundamental Theorem of
 Calculus – Part 2
If f is continuous on [a, b], then the function g is
defined by

             f  x  dx  F  b   F  a 
              b

             a


where F is the general antiderivative of f,
that is, a function such that F ' = f.