Prove |a+b+c|\leq |a|+|b|+|c| for all a,b,c\in \mathbb{R}.Hint:Apply triangle inequality twice .(b) Use induction to prove |a_1+ a_2+...+a_n| \leq|a_1|+|a_2|+...+|a_n| for n numbers a_1, a_2,..., a_n. Use mathematical induction to prove that 5^n+...
Theorem 1 Every triangle can be folded along a straight line so that the area of the doubly covered part is the fractionof the area of the original triangle.This is best possible in the sense that, given any positive number, there exist triangles that cannot be folded to yield an area of...
How large should n be If you want the error bound to be below 0.005 for the \int_{0}^{2} \frac{3}{x^2}dx using Simpson's Rule? Prove that \pi \leq \int_0^\pi \sqrt {1 + \sin^4 x} dx \leq \pi \sqrt 2. Hint: Use a comparison theorem. ...
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Learn to calculate the area of an equilateral triangle. Understand the derivation of the formula for calculating the area and learn to prove it...
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To prove the instability of the solution (3.14), we make use of the Lyapunov instability theorem [11]. As an appropriate Lyapunov function we choose the difference FL = E − E0 (3.3). As was proved above, its derivative is a function of fixed sign F˙L < 0 everywhere outside the ...
To measure the difference between two intersecting families F, G2 [n] we introduce the quantity D(F, G) =|{(F, G):F ∈F,G ∈G,F ∩ G = }|. We prove that if F is k-uniform and G is l-uniform, then for large enough n and for any i = j Fi = {F [n]:i ∈ F, ...
(Hint: For the first inequality, use Bernoulli; for the second one, use the Binomial theorem) Show that \frac{2}{3} \leq \frac{x^2 + 1}{x^2 + x + 1} \leq 2 for any x. Let G=(\frac{x,y}{x^{4=y_4=e,xyxy^{-1}=e) Show that |G|\leq16. How to prove a set ...
In addition, the Jensen inequality is an important tool in Lp-spaces and in connection to modular and Orlicz spaces; see, e.g., [4, 5]. In particular, Orlicz spaces are often generated by convex φ-functions [6], and the convexity allows one to prove several important properties for ...