Partners Names:

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

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?

Answer:

P(x)>0 for the intervals:

P(x) <0 for the intervals:

5. Based on this information about P,

Answer:

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.

Answer:

8. For what intervals between 0 and 4 is P’(x) >0? When is P’(x) <0?

Answer:

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?

Answer:

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.