f ( x ) dx using the Fundamental Theorem by t0231232

VIEWS: 0 PAGES: 11

									MTH 252 – Calculus II – Fall 2006                                                          Final – 12.13.2006


                                              y




                                                                  f
                                          1


                                                                             x
                                                  1




  1. The graph of f (x) = −x2 + 4x + 2 is shown. Answer the following:                            (7 points each)

                        4
     (a) Estimate           f (x) dx with 4 subintervals using left endpoints.
                    0
                     4
     (b) Evaluate           f (x) dx using the Fundamental Theorem of Calculus.
                    0



  2. (Refer to your notes on the “Lanchester’s Laws” project.)

    In Project #2, Admiral Nelson devised a strategy by which a British fleet was able to defeat a combined
    French-Spanish fleet with superior numbers. Nelson’s strategy was as follows:

                            Skirmish I:   send 32 British ships against 23 opposing ships
                                          (the British fleet prevails with 22 ships remaining)
                        Skirmish II:      send 8 British ships against 23 opposing ships
                                          (the opposing fleet prevails with 21 ships remaining)

    In the project, we saw that after these two skirmishes, Nelson sent the remaining British fleet against
    the remaining French-Spanish fleet, and the British fleet was victorious with 6 ships remaining.
    Let’s play out a different scenario. After the two skirmishes, with 22 British ships and 21 opposing
    ships remaining, suppose that the commandants of the opposing fleet decide to use Admiral Nelson’s
    strategy against him. They propose the following:

                    Skirmish “A”:         send 12 French-Spanish ships against 11 British ships
                    Skirmish “B”:         send 9 French-Spanish ships against 11 British ships

    For each of these new skirmishes, use the general solution y 2 − x2 = C to determine the victor. You
    won’t need to change the differential equations, just the initial conditions.
       • Which fleet will prevail in Skirmish “A”? How many ships will remain in that fleet?             (4 points)

       • Which fleet will prevail in Skirmish “B”? How many ships will remain in that fleet?             (4 points)

       • Did the French-Spanish commandants use Admiral Nelson’s strategy successfully? You shouldn’t
         have to do any more calculations for this, just use the figures from parts (a) and (b). (Remember
         that partial ships can’t float and won’t be able to fight.)                                 (2 points)
                               1
3. Evaluate the integral           x3   1 − x4 dx.                                            (10 points)
                           0




                                              n2
4. Determine if the sequence an =                is convergent or divergent.                  (10 points)
                                             1+n




5. Evaluate the integral       x cos x dx.                                                    (10 points)




                                         4




                                         3




                                        y2




                                         1




                                         0
                                             0   0.5    1    1.5   2     2.5   3
                                                             x
                                    Exercises #6 and #7 refer to the graph above.


6. Find the area of the region bounded by the curves y = e−x + 2, y = x − 1 , x = 1, and x = 2.
                                                                          2
                                                                                              (12 points)




7. Find the volume of the solid obtained by rotating the region bounded by the curves y = e−x + 2,
   y = x − 1 , x = 1, and x = 2 about the x-axis.
           2                                                                             (12 points)
                    x
8. Let g(x) =           f (t) dt, where f is the function whose graph is shown. Answer the following:
                0
                                                                               (3 points for each part, 12 points total)


                                                                               y

    (a) Evaluate g(0), g(1), and g(2).


                                                                                           f
   (b) Estimate g(3), g(4), and g(5).
                                                                           1


    (c) On what interval is g increasing?                                              1
                                                                                                                           t


   (d) Sketch a rough graph of g.




                                                       ∞
                                                          xn
9. Find the interval of convergence of the series             .                                             (10 points)
                                                      n=1
                                                          n2n
> with(student) :
> # Exercise 1
  f := x -> -1*x^2 + 4*x - 2 ;
                                             2
                              f := x       -x + 4 x - 2

> leftbox( f(x) , x = 0..4 , 4 ) ;




> Int( -1*x^2 + 4*x - 2 , x = 0..4 ) = int( -1*x^2 + 4*x - 2 , x = 0..4 ) ;

                                  4

                                       2                  8
                                      -x + 4 x - 2 dx =
                                                          3
                              0

> Int( -1*x^2 + 4*x - 2 , x ) = int( -1*x^2 + 4*x - 2 , x ) + C ;

                         2                1 3     2
                       -x + 4 x - 2 dx = - x + 2 x - 2 x + C
                                          3

>
> # Exercise 2, part (a)
  x0 := 11 ; y0 := 12 ; C := y0^2 - x0^2 ; y^2 - x^2 = C ; y^2 - 0^2 = C ;
  y = C^(1/2.) ;
                                x0 := 11
                                   y0 := 12
                                   C := 23
                                2       2
                               y - x = 23
                                    2
                                    y = 23
                             y = 4.795831523
> # Exercise 2, part (b)
  x0 := 11 ; y0 := 9 ; C := y0^2 - x0^2 ; y^2 - x^2 = C ; 0^2 - x^2 = C ; x
  = (-C)^(1/2.) ;
                                x0 := 11
                                    y0 := 9
                                C := -40
                               2        2
                               y - x = -40
                                    2
                                -x = -40
                             x = 6.324555320
> # Exercise 2, part (c)
  x0 := 6 ; y0 := 4 ; C := y0^2 - x0^2 ; y^2 - x^2 = C ; 0^2 - x^2 = C ; x
  = (-C)^(1/2.) ;
                                 x0 := 6
                                    y0 := 4
                                C := -20
                               2        2
                               y - x = -20
                                    2
                                -x = -20
                             x = 4.472135955
>
> with(DEtools) :
> # Exercise 2, part (a)
  phaseportrait( [diff(x(t),t)=-y(t), diff(y(t),t)=-x(t)], [x(t),y(t)],
  t = 0..10, [[x(0)=11,y(0)=12]], stepsize=1, x = 0..30, y = 0..30 ) ;




>
> phaseportrait( [diff(x(t),t)=-y(t), diff(y(t),t)=-x(t)], [x(t),y(t)],
  t = 0..10, [[x(0)=11,y(0)=9]], stepsize=1, x = 0..30, y = 0..30 ) ;




> restart ;   # this clears out the value of 'C' which we use again below
> # Exercise 3
  Int( x^3*(1-x^4)^(1/2) , x = 0..1 ) = int( x^3*(1-x^4)^(1/2) , x = 0..1 )
  ;
                                     1

                                             3            4       1
                                         x       1 - x dx =
                                                                  6
                                 0

> Int( x^3*(1-x^4)^(1/2) , x ) = int( x^3*(1-x^4)^(1/2) , x ) + C ;
                                                                      (3/ 2)
                         3               4            1           4
                        x    1 - x dx = - (1 - x )                             +C
                                         6

> # Exercise 4
  Limit( n^2/(1+n) , n = infinity ) = limit( n^2/(1+n) , n = infinity ) ;
                                                      2
                                                  n
                                 lim                          =
                             n                   1+n

> # Exercise 5
  Int( x*cos(x) , x ) = int( x*cos(x) , x ) + C ;

                        x cos(x) dx = cos(x) + x sin(x) + C


>
> # the following code is how we created the figure for Exercises 6 and 7
> with(plots) :
> p1 := plot( exp(-x) + 2 , x = 0..3 , y = 0..4 , thickness = 2 , color =
  blue ) :
> p2 := plot( x - 1/2, x = 0..3 , y = 0..4 , thickness = 2 , color = green
  ) :
> p3 := implicitplot( x = 1 , x = 0..3 , y = 0..4 , thickness = 2 , color =
  red ) :
> p4 := implicitplot( x = 2 , x = 0..3 , y = 0..4 , thickness = 2 , color =
  red ) :
> plots[display]({p1,p2,p3,p4}) ;




>
> # Exercise 6
  Int( exp(-x)+2 - (x - 1/2) , x = 1..2 ) = int( exp(-x)+2 - (x - 1/2) , x
  = 1..2 ) ;
                                             2
                                                     (-x)       5                 (-1)        (-2)
                                                 e          +       - x dx = e           -e          +1
                                                                2
                                         1

> int( exp(-x)+2 - (x - 1/2) , x = 1..2.0 ) ;
                               1.232544158
> # Exercise 7
  Int( pi*( (exp(-x)+2)^2 - (x - 1/2)^2 ) , x = 1..2 ) = int( pi*(
  (exp(-x)+2)^2 - (x - 1/2)^2 ) , x = 1..2 ) ;
             2                   2
                                                            2
                     (-x)                              1                                 (-2)             (-1)                        (-4)
                                                                              7                                      35       1
                 e          +2       -           x-                  dx = -         e           +4    e          +        -       e
                                                       2                      2                                      12       2
         1

> int( pi*( (exp(-x)+2)^2 - (x - 1/2)^2 ) , x = 1..2.0 ) ;
                              3.905353121

>
> # Exercise 9
  Sum( x^n/(n*2^n) , n = 1..infinity ) = sum( x^n/(n*2^n) , n = 1..infinity
  ) ;
                                    n
                                   x                     1
                                                = -ln 1 - x
                         n= 1           n                2
                                   n2

> # think about what happens when 'x < 2', 'x > 2' , or 'x = 2'
> # Ratio Test
  Limit( abs((x^(n+1)/((n+1)*2^(n+1)))/(x^n/(n*2^n))) , n = infinity ) ;
                                            (n + 1)        n
                                            x         n2
                             lim
                         n                          (n + 1) n
                                       ( n + 1) 2              x

> # can you see how things cancel out in the expression above?
  limit( (x^(n+1)/((n+1)*2^(n+1)))/(x^n/(n*2^n)) , n = infinity ) ;
  # be careful with this simplified version -- I dropped the absolute value
  here!
                                  1
                                    x
                                  2

>

								
To top