Math 109 Lab on Using Winplot To Investigate Function Properties with Calculus.
M. Flashman  Spring, 2003
Partners Names: 1                                                          2                                                     .
Submit responses to the following lab activities (done with a partner) by 5 pm Friday , April 11th.

The function P(x)= sin(x^2) has many properties that can be investigated easily with Winplot.
We are interested in this function because the differential equation f (x) = sin(x^2) has no solution that can be expressed in the form of an elementary function.

By investigating P(x) we can say much about the function f(x) where f (x) = P(x) and f(0)=0.

1. Use WINPLOT to graph P(X).
2. Is P(X) symmetric?  With respect to the Y-axis? Can you explain why?

3. For how many xs in the interval [0 , 4 ] is P(x) = 0? Give estimates for these xs.
Describe the smallest positive of these xs in terms of pi.

4. For what intervals between 0 and 4  is P(x) >0?   When is P(x) <0?
P(x)>0 for the intervals:

P(x) <0 for the intervals:

5. Based on this information about P, for what intervals between 4 and 4 is f increasing?  When is f decreasing? When does f have its local maxs and mins for this interval?
f is decreasing for the intervals:
f is increasing for the intervals:
f has  local maxima at x =
f has  local minima at x =

6. Use Winplot to find the graph of P'(x). Compare this with the graph of P'(x) that you find by using the derivative calculus.

The derivative of P , P'(x) =

7. For how many xs  in the interval [0 , 4 ] is P(x) = 0? Give estimates for these xs.
Describe the smallest of these xs in terms of pi.

8. For what intervals between 0 and 4 is P(x) >0?   When is P(x) <0?
P'(x)>0 for x in the intervals:

P'(x)<0 for x in the intervals:

9. Based on this information about P(x) = f(x), for what intervals between 4 and 4 is f concave up?  When is f concave down? When does f have its inflection points for this interval?
f is concave down for the intervals:

f is concave up for the intervals:

f has inflection points when x =

10. Based on your results so far, draw a sketch of the graph of f(x).

11. 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. Compare your graph with the information developed in the previous steps with regard to the usual calculus features.