Math Textbooks (487)

If \(y=f(u)\) and \(u=g(x),\) where \(f\) and \(g\) are twice differentiable functions, show that \(\frac{d^{2} y}{d x^{2}}=\frac{d^{2} y}{d u^{2}}\left(\frac{d u}{d x}\right)^{2}+\frac{d y}{d u} \frac{d^{2} u}{d x^{2}}\)

f(x)=2 x+\frac{1}{2 x}

Let \(V\) be a vector space over \(F\). Prove that \(0 v=0\) and \(r 0=0\) for all \(v \in V\) and \(r \in F\). Describe the different 0 's in these equations. Prove that if \(r v=0\), then \(r=0\) or \(v=0\). Prove that \(r v=v\) implies that \(v=0\) or \(r=1\).

A fund manager has a portfolio worth $$\$ 50$$ million with a beta of \(0.87\) . The manager is concerned about the performance of the market over the next two months and plans to use three-month futures contracts on the \(S\&P\) 500 to hedge the risk. The current level of the index is \(1250,\) one contract is on 250 times the index, the risk-free rate is \(6 \%\) per annum, and the dividend yield on the index is \(3 \%\) per annum. a. What is the theoretical futures price for the three-month futures contract? b. What position should the fund manager take to eliminate all exposure to the market over the next two months? c. Calculate the effect of your strategy on the fund manager's returns if the level of the market in two months is \(1.000,1,100,1,200,1,300,\) and \(1,400.\)

Describe two different ways to develop the concept of mean. How can each be developed? Which idea leads to the method of computing the mean?