Programming and Computer Science in Java:

See Course Home Page.

**Due
Date: Mon, Mar 17, 2003**

For each of the following questions, anything which your
programs *compute* must be computed within a new method, and not directly
inside your main method.

In class on Friday, we covered in detail how to approximate the derivative and the integral of a function f(x). Your homework is to duplicate these programs.

** Question 1:
Approximating a Derivative. **Write a Java program which reads in an
integer, a, and computes (as a double) the derivative of a function, f(x), at
the point x=a -- that is, f'(a). The function f(x) should be built-in to
your program, as in:

public static double f(double x) { return x * x; } // f(x) = x^2

Very briefly, how do you do
this? Because we covered this in detail in class, we will provide very
brief notes here. First, you write a function, approxSlope:

public static double
approxSlope(double a, double h)

which takes two parameters -- a and h -- and returns the slope of the secant
line through the points (a,f(a)) and (a+h,f(a+h)). Note that for very
small h, this value approximates the derivative f'(a), and for infinitely small
h (if there were such a thing), it would in fact *be* that derivative.

So now you can write your
derivative function:

public static double
derivative(double a)

which takes one parameter, a, and returns the derivative of f at x=a, that is,
f'(a). It does so by starting with a small value for h, say 0.01, and
repeatedly calling approxSlope(a,h), each time halving the value of h, until the
difference between successive calls is smaller than some rather small epsilon,
say 0.000000001. Now, due to roundoff error in doubles, there is a chance
this process will never terminate. Thus, you should also terminate the
loop if it has run more than some maximum number of iterations.

Try your program out for different functions f(x), and verify your program's operation both by hand (for easily differentiable functions) and, as needed, by calculator.

** Question 2:
Approximating an Integral. **This problem is very similar to the
preceding problem, curiously enough. Write a Java program which reads in
two integers, a and b, where a <= b, and computes (as a double) the integral
of a function, f(x), from the point x=a to the point x=b. The function
f(x) should be built-in to your program, as in:

public static double f(double x) { return x * x; } // f(x) = x^2

Very briefly, how do you do
this? Because we covered this in detail in class, we will provide very
brief notes here. First, you write a function, approxArea:

public static double
approxArea(double a, double b, int rectCount)

which takes three parameters -- a, b, and rectCount -- and returns the area
under the curve f(x) from x=a to x=b as approximated by rectCount
rectangles. First, note the obvious: each rectangle has a
fixed-width of (b-a)/rectCount. Now, for each rectangle, use function's
value at the left x as the height, so the first rectangle's height will just be
f(a) and its area will be f(a)*width, where width = (b-a)/rectCount. Note
that for a large rectCount, this value approximates the integral of f(x) from
x=a to x=b, and for infinitely large values of rectCount (if there were such a
thing), it would in fact *be* that integral.

So now you can write your
derivative function:

public static double
integral(double a, double b)

which takes two parameters, a and b, and returns the integral of f(x) from x=a
to x=b. It does so by starting with a moderate value for rectCount, say
1000, and repeatedly calling approxArea(a,b,rectCount), each time doubling the
value of rectCount, until the difference between successive calls is smaller
than some rather small epsilon, say 0.000000001. Now, due to roundoff
error in doubles, there is a chance this process will never terminate.
Thus, you should also terminate the loop if it has run more than some maximum
number of iterations.

Try your program out for different functions f(x), and verify your program's operation both by hand (for easily integrable functions) and, as needed, by calculator.

Good luck!

DK

See Course Home Page.