Calculus I final exam

[ydoaPs]

I just got back from taking the final exam for my Calc 1 class. I did pretty well on it. The only question I wasn't sure about was the last one. It was a trapezoidal rule problem. I completely spaced how to do it. I drew the graph and divided it into sections. I drew the trapezoids. I then broke the trapezoids into triangles and rectangles. I found the areas of all triangles and rectangles then added them togather. It was y=x² from 0 to 2 with 4 trapezoids and I got 2.688(i think).

 

I think I ended up doing it the right way, but in a longer form. When i got home, I derived the formula for what I did.

[math]\frac{b-a}{n}(f(x_1)+f(x_2)+f(x_3)+{\frac{1}{2}}f(b))[/math] where n is the number of trapezoids. Did I even come close?

[matt grime]

Slightly puzzled: you use b twice for different things, a generic 'n' for the number of trapezoids but you only use 4 values of f, and one of them is divided by 2 and none of the others is, and surely f(a) and f(b) ought to appear somewhere too.

[ydoaPs]

no, b meant the same thing. i just wasn't thinking when i did the LaTeX. the one value that is divided by two is due to the triangles.

 

 

edit: i didn't have f(a) in there because in my problem, f(a)=0. i should have factored that in.

 

edit#2:

[math]\frac{b-a}{n}(f(a)+f(x_1)+f(x_2)+f(x_3)+...+{\frac{1}{2}}f(b))[/math]

[matt grime]

It is still not clear if you're doing this in general of for some specific function and interval: you are mixing and matching, so why not put a=0, b=2, use 4 trapezoids (n=4) instead? b can't have meant 'the same thing', since it was the upper limit of the integral (b-a) and a subscript (x_b). Since b was in this example 2 you'd have x_2 appearing twice which presumably you don't want. Since you're doing equal sized subintervals why not use that fact too?

[ydoaPs]

It is still not clear if you're doing this in general of for some specific function and interval: you are mixing and matching, so why not put a=0, b=2, use 4 trapezoids (n=4) instead?

the formula was in general

 

b can't have meant 'the same thing', since it was the upper limit of the integral (b-a) and a subscript (x_b). Since b was in this example 2 you'd have x_2 appearing twice which presumably you don't want.

that's why i fixed it after your first post.

 

Since you're doing equal sized subintervals why not use that fact too?

hence the interval being factored out. that's what the [math]\frac{b-a}{n}[/math] is.

[matt grime]

the formula was in general

 

what generality? You assumed a=0 at one point (and not at another) and that f(a)=0, so it was not very general, you also assumed that there are 3 points defining the subdivision into trapezia and that they are equally spaced, again that is not very 'general', and n=4 so why not put that in?

 

 

You also need to divide f(a) by 2.

[ydoaPs]

what generality? You assumed a=0 at one point (and not at another) and that f(a)=0, so it was not very general

hence the fixing.

 

you also assumed that there are 3 points defining the subdivision into trapezia

i'll fix that.

 

and that they are equally spaced, again that is not very 'general'

don't you usually use equally spaced intervals?

 

 

You also need to divide f(a) by 2.

why?

[matt grime]

Presumably for the same reason you're dividing f(b) by two: the area of a trapesium is one half the sum of the two sides times the base. All 'sides' are used twice except the first and the last in the approximation.

 

And no there's no reason to suppose that trapezia should have evenly sized bases, especially if you're going to claim a general formula.

[ydoaPs]

Presumably for the same reason you're dividing f(b) by two: the area of a trapesium is one half the sum of the two sides times the base. All 'sides' are used twice except the first and the last in the approximation.

when i was writing out the triangles, everything but the f(b) cancelled out.

 

can you tell me if i have the right answer?

[matt grime]

when i was writing out the triangles' date=' everything but the f(b) cancelled out.

[/quote']

 

 

but that was because you were using a certain type of function that satisfied f(a)=0, weren't you? I have no intention of working out that sum. There are general forumlae for doing integration by approximation all over the place (in your textbook for instance, or here http://en.wikipedia.org/wiki/Trapezoidal_rule).