Determining the input of a function given an output (Calculus involved)

Question

My Calculus teacher gave us a program on to calculate the definite integrals of a given interval using the trapezoidal rule. I know that programmed functions take an input and produce an output as arithmetic functions would but I don't know how to do the inverse: find the input given the output.

The problem states:

"Use the trapezoidal rule with varying numbers, n, of increments to estimate the distance traveled from t=0 to t=9. Find a number D for which the trapezoidal sum is within 0.01 unit of this limit (468) when n > D."

I've estimated the limit through "plug and chug" with the calculator and I know that with a regular algebraic function, I could easily do:

limit (468) = algebraic expression with variable x (then solve for x)

However, how would I do this for a programmed function? How would I determine the input of a programmed function given output?

I am calculating the definite integral for the polynomial, (x^2+11x+28)/(x+4), between the interval 0 and 9. The trapezoidal rule function in my calculator calculates the definite integral between the interval 0 and 9 using a given number of trapezoids, n.

Overall, I want to know how to do this:

Solve for n: 468 = trapezoidal_rule(a = 0, b = 9, n);

The code for trapezoidal_rule(a, b, n) on my TI-83:

Prompt A
Prompt B
Prompt N
(B-A)/N->D
0->S
A->X
Y1/2->S
For(K,1,N-1,1)
X+D->X
Y1+S->S
End
B->X
Y1/2+S->S
SD->I
Disp "INTEGRAL"
Disp I

Because I'm not familiar with this syntax nor am I familiar with computer algorithms, I was hoping someone could help me turn this code into an algebraic equation or point me in the direction to do so.

Edit: This is not part of my homework—just intellectual curiosity

Answer 1

the polynomial, (x^2+11x+28)/(x+4)

This is equal to x+7. The trapezoidal rule should give exactly correct results for this function! I'm guessing that this isn't actually the function you're working with...

There is no general way to determine, given the output of a function, what its input was. (For one thing, many functions can map multiple different inputs to the same output.)

So, there is a formula for the error when you apply the trapezoidal rule with a given number of steps to a given function, and you could use that here to work out the value of n you need ... but (1) it's not terribly beautiful, and (2) it doesn't seem like a very reasonable thing to expect you to do when you're just starting to look at the trapezoidal rule. I'd guess that your teacher actually just wanted you to "plug and chug".

I don't know (see above) what function you're actually integrating, but let's pretend it's just x^2+11x+28. I'll call this f(x) below. The integral of this from 0 to 9 is actually 940.5. Suppose you divide the interval [0,9] into n pieces. Then the trapezoidal rule gives you: [f(0)/2 + f(1*9/n) + f(2*9/n) + ... + f((n-1)*9/n) + f(9)/2] * 9/n.

Let's separate this out into the contributions from x^2, from 11x, and from 28. It turns out that the trapezoidal approximation gives exactly the right result for the latter two. (Exercise: Work out why.) So the error you get from the trapezoidal rule is exactly the same as the error you'd have got from f(x) = x^2.

The actual integral of x^2 from 0 to 9 is (9^3-0^3)/3 = 243. The trapezoidal approximation is [0/2 + 1^2+2^2+...+(n-1)^2 + n^2/2] * (9/n)^2 * (9/n). (Exercise: work out why.) There's a standard formula for sums of consecutive squares: 1^2 + ... + n^2 = n(n+1/2)(n+1)/3. So our trapezoidal approximation to the integral of x^2 is (9/n)^3 times [(n-1)(n-1/2)n/3 + n^2/2] = (9/n)^3 times [n^3/3+1/6] = 243 + (9/n)^3/6.

In other words, the error in this case is exactly (9/n)^3/6 = (243/2) / n^3.

So, for instance, the error will be less than 0.01 when (243/2) / n^3 < 0.01, which is the same as n^3 > 100*243/2 = 12150, which is true when n >= 23.

[EDITED to add: I haven't checked any of my algebra or arithmetic carefully; there may be small errors. I take it what you're interested is the ideas rather than the specific numbers.]