comparing geometric area with derivative of area function

[nikchic5]

Here is the question:

(a) Draw the line y = 2t + 1, and use geometry to find the area under this line, above the t-axis, and between the vertical lines t = 1 and t = 3.

(b) If x > 1, let A(x) be the area of the region that lies under the y = 2t + 1 between t = 1 and t = x. Sketch this region and use geometry to find an expression for the area A(x).

(c) Differentiate the area function A(x). What do you notice?

Any help would be really appreciated!! I really do not know how to do it and I am trying to study for a test coming up and this is one main problem we need to know how to do...Thanks so much for the help!!

 

[stapel]

Any help would be really appreciated!

"Help" with which part?

a) This involves nothing more than drawing some lines and applying the formulas for the areas of rectangles and triangles.

b) Draw the same figure as in (a), but with the right-hand limit not fixed. (That is, use "x" instead of "3", but otherwise use the same formulas.)

c) Differentiate what you got in (b), and compare with (a) and (b).

Please reply showing (or describing) everything you have tried so far. Thank you.

Eliz.

 

[nikchic5]

Hello...Thanks!

I really need to know how to do it...I didn't do any of it yet because I like to see exaclty how it is done and go from there...I have a few more questions exaclty like this one to figure out so I wanted to be able to see how you work out one of them and go from there....sorry if it is too much to ask...

 

[stapel]

I really need to know how to do it.

Did you not read my reply? (I'd said "how to do it", but you seem unaware of this, is why I ask.)

Do you not know how to draw straight lines? Do you not know how to find the area of a rectangle or a triangle? Do you not know how to work with variables?

Except for the differentiation in part (c), this is just simple stuff you did back in algebra. If you really can't get started, not even with the drawing, then you might want to conference with your advisor regarding your course placement.

If, on the other hand, you are familiar with algebra, then you should be able to make good progress on at least the first two parts.

I look forward to seeing what you've been able to come up with.

Eliz.

 

[skeeter]

Re: comparing geometric area with derivative of area functio

Here is the question:

(a) Draw the line y = 2t + 1, and use geometry to find the area under this line, above the t-axis, and between the vertical lines t = 1 and t = 3.

look at your sketch ... you have a trapezoid sitting with its end (the height of the trapezoid lies on the "t" axis) ...
b₁ = 2(1) + 1 = 3, b₂ = 2(3) + 1 = 7, and h = 3 - 1 = 2. Look up and use the formula for the area of a trapezoid to find the area of the region.

(b) If x > 1, let A(x) be the area of the region that lies under the y = 2t + 1 between t = 1 and t = x. Sketch this region and use geometry to find an expression for the area A(x).

same shape ... a trapezoid, correct? the first base hasn't changed, b₁ = 2(1) + 1 = 3. the length of the second base depends on the value of x, doesn't it? b₂ = 2(x) + 1 = 2x + 1. the height of the trapezoid also depends on x ... h = x - 1. Now use the formula for the area of a trapezoid to get the area A as a function of x, A(x).

(c) Differentiate the area function A(x). What do you notice?

do it

Any help would be really appreciated!! I really do not know how to do it and I am trying to study for a test coming up and this is one main problem we need to know how to do...Thanks so much for the help!!

 

[nikchic5]

Here is what I got...

(a)1/2(2)(3+7)=10
(b)1/2(x-1)(3+2x+4)=(x-1)(x+2)=x^2 +x-2
(c) a'(x)=2x+1 which is the exact same equation that is in part a just with t.

Is that correct? Thanks so much for all of your help!!