?

Since, the marble will move the opposite way after collision, will the wall as well?

Does anyway know of any of person in history of science who made an impact during their time, like change his/her society's thought patterns? It can't be any of these people:

al-Kindi (Alkindus)

al-Razi (Rhazes)

Albert the Great

Antoni van Leeuwenhoek

Archimedes

Aristotle

Buridan

Christian Huygens

Claudius Ptolemy

Copernicus

Descartes

Edmund Halley

Euclid

Eudoxus

Francis Bacon

Galen

Galileo

Gottfried Leibniz

Henry Oldenburg

Hippocrates

ibn al-Haytham (Alhazen)

ibn Rushd (Averroës)

ibn Sina (Avicenna)

Isaac Newton

Jabir ibn Hayyan (Geber)

Kepler

Konrad Gesner

Leonardo da Vinci

Marcello Malpighi

Nicole Oresme

Paracelsus

Pierre Gassendi

Plato

Ptolemy I

Pythagoras

Robert Boyle

Robert Grosseteste

Robert Hooke

Roger Bacon

Siger of Brabant

Thomas Aquinas

Thomas Hobbes

Tycho Brahe

Vesalius

William Harvey

Prove the following: If the top 26 cards in a shuffled deck of cards have more red cards

than the bottom 26 have black cards, then there must be three cards in a row of the same color somewhere in the deck.

I dont need a proof but it would be nice to have the above translated symbolically. Thanks

Give the formal definition of the following statement:

lim x-->∞ f(x) = -∞ (negative infinity)

Prove the following using the definition:

lim x-->∞ x^(0.5) - x = -∞ (negative infinity)

What is the maximum possible area for a triangle inscribed in a circle of radius r? How do I go about doing this?

Let y = f(x) be differentiable and suppose that the graph of f does not pass through the

origin. Then the distance D from the origin to a point P(x, f(x)) on the graph is given by

D = √(x^2 + [f(x)]^2):

Show that if D has a local extreme value at c, then the line through (0, 0) and (c, f(c)) is

perpendicular to the line tangent the graph of f at (c, f(c)).

I would love to have a little more clarification on this question.

Suppose that f and g are differentiable functions and f(x)g'(x) - g(x)f'(x) has no zeros on some interval I. Assume that there are numbers a, b in I with a < b for which f(a) = f(b) = 0, and that f has no zeros in (a,b). Prove that if g(a) ≠ 0 and g(b) ≠ 0, then g has exactly one zero in (a, b). HINT: Suppose that g has no zeros in (a, b) and consider h = f / g. Then consider k = g / f.

I am trying to use Rolle's theorem, but I don't where to apply it. Please help.

A horizontal trough is 8 m long and has a vertical cross section in the form of a trapezoid. The bottom of the trough is 65 cm wide and the sides are inclined to the vertical at an angle of 30. Water is poured into the trough at a rate of 1500 cm^3/minute. How fast is the water level rising when the water is exactly 45 cm deep?

A circle of radius 1 with center on the y-axis is inscribed on the parabola y = 2x^2. Find the points of contact.

I've tried a method that got me 19/16, although I don't really understand the logic behind the steps, so I thought someone may explain more clearly how to do this, Thanks.

1. Set

f(x) = { x, x rational | 0, x irrational }, its a piecewise function separate by this |

Show that f is not differentiable at 0.

2. The dimensions of a rectangle are changing is such a way that the area of the rectangle remains constant. Find the rate of change of the height h with respect to the base b.

a) Show that S is bounded

b) Show that lub S and glb S are elements of S

The part after a is the subscript, like 1 is the subscript of a, just in case you get confused.

1. Let S be a bounded set of real numbers and suppose that the least upper bound of S = greatest lower bound of S. What can you conclude about S?

2.

a) Show that the least upper bound of a set of negative numbers cannot be positive.

b) Show that the greatest lower bound of a set of positive numbers cannot be negative.

3. Challenging one:

Let S be a nonempty bounded subset of R. (glb = greatest lower bound, lub = least upper bound)

(a) Let a > 0 and let aS = {as : s ∈ S}. Prove that glb(aS) = a glb(S) and lub(aS) = a lub(S).

(b) Let b < 0 and let bS = {bs : s ∈ S}. Prove that glb(bS) = b lub(S) and lub(bS) = b glb(S).

Fix a positive number P. Let R denote the set of all rectangles with perimeter P. Prove that there is a member of R that has maximum area. What are the dimensions of the rectangle of maximum area? Hint: Express the area of an arbitrary element of R as a function of the length of one of the sides.

given that f(x) and g(x) are continuous on [a,b], that f(a) < g(a), and g(b) < f(b), show that there exists at least one number c in (a,b) such that f(c) = g(c). Hint: consider f(x) = g(x).

Where can i get a free software that can tell me how much bandwidth i'v used up after i play online games? I used a couple of softwares already and they cannot even recognize that i'm playing a multiplayer game, so that cant tell me how much i've used up. is there one that can, cause i'm playing halo multiplayer online on pc, and i need to know the bandwidth cause my ISP gave us a limit of 40gb.

Does anybody know a good site to help me understand polymers for grade 12 chemistry. I need to know how to draw 2 repeating polymer units of lactic acid, and the book is really of no help.

rank these compounds' solubility in water as very low, low, medium, high:

1. C3H7CL

2. C3H7OH

3. (CH3)2CO

4. C3H8

I know that the answer is 4,1,3,2, but I don't why 1 and 3 are ranked in that order and not the other way around. PLEASE Explain.

A rectangular box is to be made from a 40cm by 36cm of metal. What size squares should be cut from the corners in order for the box to have a volume of AT LEAST 456cm^3.

My essay is based around the statement: "the domino effect of death" (describing how one death led to another).

First of all, what type of essay would this go under, cause I need to know what i would write and how i would write it in the 3 body paragraphs.

Also, my intro paragraph is preety much looking dry, all i can think about writing is stating all the deaths, and describing the domino effect, and a thesis statement, which is like 3-4 sentences. Any help there would be nice also.

Is "domino effect of death" my thesis statement, and if so how do i re-word so it is a thesis sentence?

Also, where do I place the thesis sentence in the introduction paragraph? --> this question is the most important!!!