Math 109                            Lab: Monday  April 14, 2003                    M. Flashman
Solutions to Differential Equations and Finding Areas with the Definite Integral

Partners Names: 1                                                          2                                                     .


Submit responses to the following lab activities (done with a partner) by 5 pm, Tuesday , April 22nd.

Let P(x)= sin(x^2).
We will be interested in this function because the differential equation dy/dx = sin(x^2) has no solution that can be expressed in the form of an elementary function.
 

  1. Use WINPLOT to graph P(x).

    2.  

     
     
     
     
     

  2. Use the differential equations item in the EQUAtions menu to draw the slope (tangent) field for dy/dx = sin(x^2).

    3.  

     
     
     

    Discuss briefly the relation between the graph of P(x)  and the slopes of the segments in the slope field.
     

    The slopes are positive for x in the intervals:
     

    The slopes are negative for x in the intervals:
     
     
     
     

  3. Use the IVPS solver under the One menu item for dy/dx trajectory to estimate the value of y(1) based on the initial condition that y(0) = 0. First by drawing, then from the Euler table:

    4. a. With  dx =1/10. Ans.  y(1) is approximately
     

    b. With  dx = 1/100. Ans.  y(1) is approximately
     

  4. Using Winplot- measurement- integration- indefinite with the “lower limit” set to 0, graph the solution to the differential equation f’(x) = sin(x^2) with the initial condition that f(0)=0. Based on this graph, estimate the value of y(1).

    5.  

     
     
     
     
     
     
     
     
     
     
     

  5. Using Winplot- measurement- integration- definite with the “lower limit” set to 0 and the “upper limit” set to 1 and the “left endpoint” method to draw the Euler rectangles used to estimate the area under the graph of P(x) from 0 to 1.

    6.  

     
     
     

    a. With n = 10. [so dx = .1]  .  Ans. The estimate for the area is                                         .

    b. With n = 100. [ so dx = .01]  Ans. The estimate for the area is                                         .
     
     

  6. Compare your results in problem 3 and 5. Explain your findings.

    7.  

     
     
     
     
     
     
     
     
     

  7. Use the IVPS and/or the measurement integration to find an estimate for the definite integral of P(x) over the interval [0,1] that has an error less than 1/10000=.0001. Discuss briefly the justification for the quality of your estimate and whether you believe it is an underestimate or an overestimate.

    8.