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The usual symbol for the maximum degree of any vertex in a graph is ?. Show that the chromatic number of a graph is no more than ? + 1. (In fact Brooks proved that if G is not complete or an odd cycle, then χ(G) ≤ ?. Though there are now many proofs of this fact, none are easy!)
Let f be a differentiable function on an interval of the form (a, +∞). Prove that if there is a number r not equal to 0 such that limx→∞(rf′(x) + f(x)) = L is finite, then limx→∞ f′(x) = 0 and limx→∞ f(x) = L. Hint: apply L'Hˆopital's Rule to [e..
Using the equation (W = FΔr cos θ = F→ ·Δr→) calculate the work done by the gravitational force in going from o to C along the following paths.
Rate related problems and Determine the general expression for the car's acceleration at any time t, and show that car is decelerating 2(2/3) seconds after starting from rest.
In a standard deck of cards, what is the smallest number of cards you must draw to guarantee that you have at least one pair?
Show that |I|= 1 for any the vector norm used. As always, I is the identity matrix.
Arjun's school is selling tickets to a fall musical.
A parking area that is to be resurfaced is shaped like a trapezoid
Suppose that fluid is leaking from an overhead pipe and spreading in the shape of a circle on the ground. If the radius of the circle is increasing at a rate of 0.5 feet per minute, how fast is the area of the circle increasing when the radius is ..
At its current short-run level of production, a firm's average variable costs equal $20,and its average fixed costs equal$70. It's total costs at its current production level equal $200,000. What is the firms current output level? What are its..
1. Find the direction angles of the vector: 2i-2j-42√k. 2. Find all numbers x for which the angle between c = 3xi + 3j + 3k and d = 3i + 3xj + 3k is π3.
Reflect on the content that you completed in ALEKS this week. Provide one (1) example of information you learned from your completed work that you believe is applicable to a real-world situation. Justify your response. Topic: Geometry, Measurement, ..
Let A be a finite set with subsets A1, ..., An, and let d1, Find a bipartite graph with a set of preferences such that no matching of maximum size is stable and no stable matching has maximum size.
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