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.

Use WINPLOT to graph P(x).

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

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:
a. With dx =1/10. Ans. y(1) is approximately
b. With dx = 1/100. Ans. y(1) is approximately

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).

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.
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
.

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

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.