Recent content by Born

I get what you're saying. The only problem is that Apostol still hasn't mentioned sequences. The only relevant thing I can think of that has been covered in the book till now is the Least Upper Bound Property of Numbers. Which would simply state that ##f(x)## is the supremum for ##Q'##

RUber, I see what you're saying, however the new step regions (S' and T') would produce functions bearing the following relationship with the function##f(x)##: ##s'(x) < f(x) \le t'(x)##. Which doesn't help since the definition of the integral requires "##\le##" for both step function...

In section 1.18 ("The area of an ordinate set expressed as an integral"), Apostol proves two theorems. the first, theorem 1.10, deals with the area of a function's ordinate set; the second, theorem 1.11, deals with the area of the graph of the function of theorem 1.10. (I have attached two...

Yes, that makes sense. However, what I want to pove is that if the base and height of a right triangle are inside another then the hypotenuses cannot be congruent. If I can prove that then the isosceles triangles can be argumented to be case of two pairs of juxtaposed right triangles

I just uploaded a picture to make it clearer.

Homework Statement Problem 99 from "Kiselev's Geometry Book I - Planimetry": Two isosceles triangles with a common vertex and congruent lateral sides cannot fit one inside the other. Homework EquationsThe Attempt at a Solution The statement is obviously true. If we visualize each isosceles...

##\mathrm{Follow\ up:}## Concerning the pentagon: labeling the five vertex points A, B, C, D, E; and forming the sides AB, BC, CD, DE, and EA. A diagonal is made from point A to point D crossing the side BC. This is possible since the diagonal would only share one point with the side BC- (This...

Simon, MidgetDwarf, and lurflurf, I think you'll like what I've come up with. I'm sorry to not be able to show some pictures but I believe the written proof will suffice. Hope it's clear enough. Thank you for your help. The three properties of straight lines in the proof are the following: (1)...

Homework Statement Problem 55 from Kiselevś Geometry - Book I. Planimetry: "Prove that each diagonal of a quadrilateral either lies entirely in its interior, or entirely in its exterior. Give an example of a pentagon for which this is false." Homework EquationsThe Attempt at a Solution The...

Got it. Thanks! :)

Homework Statement Kleppner and Kolenkow "An Introduction to Mechanics (2nd ed.)" prob. 3.2: Mass MA = 4 kg rests on top of mass MB = 5 kg that rests on a frictionless table. The coefficient of friction between the two blocks is such that the blocks just start to slip when the horizontal...

Wow total brain fart. ##\ddot{y}=g\cos^2(\theta)## NOT: ##\ddot{y}= g \cos(\theta)## Therefore ##\ddot{x}=g\sin(\theta)\cos(\theta)=\frac{g\sin(2\theta)}{2}## So, I'm back were I started. The way the equation behaves is pretty interesting though, with a max value of ##\frac{1}{2}##, it starts...

Sorry, edited the mistake. The question would be; is there a simpler (and more elegant) way to count the number of elements in the set?

Let ##T = \{ \frac{n}{m}\in \mathbb{Q} \vert n, m \in \{ 1, 2, ..., 9 \} \}## No values can repeat (e.g. ##\frac{2}{2},\frac{3}{3},...##) How many elements does the set have. I could just go ahead and count the elements and eliminate the repeats, but I'm wondering if there is a simpler (and...

So I think this post is dead but I'll give it a shot and try to revive it. I'm sorry if it seems inappropriate but I must also check my answer to this problem Here is my my analysis: ##2T+N_{scaffold}-M_{parinter}g=M_{painter}a_{painter}##...