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Consider a firm with two inputs K and L that produce an output Q(K, L). The firm’s cost function is C(K, L) = K + 2L It is required to minimize C(K, L) subject to the constraint Q(K, L) = q where q is a positive real constant. You may assume that the optimization problem has a solution at an...

Hi, Can anyone help with the last part of this question, I just cannot seem to derive the scheme! \frac{u_{j}^{n+1}- u_{j}^{n-1}}{ 2 \Delta t} - 5 \frac{u_{j+1}^{n} - u_{j-1}^{n}}{ 2 \Delta x } = 0

When looking for a possible new pivot, why do we only search from row k to n in column k? In other words, what unwanted thing could happen if we used a row above row k as the new pivot row? am not really sure why, we choose the next pivot to be below the kth...

For any integer value of n, define F(n) as the combined operation count of forward and backsubstitutions and define G(n) as the operation count of gaussian elimination. When n gets larger and larger,why does F(n) grow much more slowly that G(n) does? I have no idea why? can anyone point me in...

1)A light bucket contains a ball of mass m. The bucket is attached to a rigid rod and swung in a vertical circle at a constant angular velocity. The distance from the pivot point to the bottom of the bucket is a. What is the minimum angular velocity of the bucket such that the ball does not...

Hello, Can someone please just clarify my notes for me, i really cant understand how my teacher gets j = - cos \phi \hat{\phi}-sin \phi \hat{r} I have attached the question and solution, but i have no idea how to work out j, the unit vector in terms of phi and r, please can someone show...

Let *^{n}(a) = {a*.......*a } n times thus *^{1} = a *^{2} = a*a *^{3} = a*a*a etc prove by induction that *^{n}a = (a+1)^{n} -1 for all n in N am really having problems understand how to do this, please can someone show a step by step solution guide? so far i...

a_{n} = \sqrt{n} + 2sin (n) let m >0 take N= (2+m)^{2} and let n >N then \geq \sqrt{n} -2 > 2+m-2 = m because n >N \sqrt{n} > \sqrt{N} = 2+m where does N come from ? we want a_{n} > m start with a_{n} = \sqrt{n} + 2sin (n) > \sqrt{n} -2 which exceeds m...

a_{n} = \dfrac{n^{3}}{1+ln(n)} a) for all n ∈ N. Say whether this sequence is divergent, convergent, and/or bounded. i think this sequence is divergent, just from looking at it? b) Give a proof of your answer to part (b). Standard properties...

Can someone please explain this proof to me, I dont understand how N = 4M^{2}

Please can someone explain the proof of this one for me, I just dont get it.. I get | n*e^{-n}-0 | < \epsilon

really struggling with this question... Determine the limit of this sequences, weather it converges or diverges and prove your limit is the correct one. a_{n} = sin(\frac{2\pi}{n}) the lim as n approaches infinity is 0 so | sin(\frac{2\pi}{n})-0 | <\epsilon sin(\frac{2\pi}{n})...

a_{n} = \frac{2n^{2}+n}{n^{2}+2n} then the limit as n approaches infinity is ? not sure what to do... i tired to factorise and \frac{2n+1}{1+2n} but this gives me 1, the correct answer is 2?

Find the supremum, infimum, maximum and minimum of the following sets, or indicate that they don’t exist. not sure about the other three, but is the first one, min = 0 max = 2 inf = does not exits sup = 2?

find a value of x between 0 and 77 such that 10x = 1(mod77) I did the algorithm and got 1 = -23 *10 + 3*77 so should the value of x be 54?

prove that 7 is the only prime p of the form n^{3}-1 for n in the set of natural numbers very lost with this question, I have factored it to however how do i 'prove' that 7 is the only prime number that can be written in this way ? (n^{2}+n+1)(n-1)

prove by induction that 11 | 5^{2n}-3^{n} let f(n) = 5^{2n}-3^{n} /tex] f(1) = 5^{2} - 3 = 22 which is divisible by 11 assume it is true for n =k f(k) = 5^{2k}-3^{k} induction step f(k+1) = 5^{2{k+1}}-3^{k+1} = 5^{2}(5^{2k})- 3(3^{k}) f(k+1) - f(k) =...

[A relation is defined on the set of all rational numbers as follows, a~b means that there is a rational number k >0 , such that a = b^{k} prove that ~ is an equivalence relation. on the set of all natural numbers N. Okay so I have to show it is reflective, symmetric and transitive...

Let f be a polynomial of the form f(x) = x^{n} +C_{1}x^{n-1} + .....C_{n-1}x+c_{n} where each coefficient c_{j} is an integer. Assume that n is greater than or equal to 1, and let \alpha be a rational number such that f(\alpha) = 0 prove that \alpha must in fact be an...

I have attempted part 'a', but have no idead how to do part b or c, any. av + bv = 0 a \lambda v + b \lambda v = 0 \lambda (au+bv) = \lambda 0 = 0 a \lambda u + b \mu v =0 a A u + bAv = A (au+bv)= A0 = 0 Is this making any sense...