In the differential equations slides, the equation f ( x ) + x f ′ ( x ) = h ( x ) is solved using the product rule. Which expression below correctly represents the solution shown in the slides? a) f ( x ) = ∫ h ( x ) d x f(x)=\int h(x)\,dx b) f ( x ) = ∫ h ( x ) d x x f(x)=\dfrac{\int h(x)\,dx}{x} c) f ( x ) = x ∫ h ( x ) d x f(x)=x\int h(x)\,dx d) f ( x ) = h ′ ( x ) + x f(x)=h'(x)+x e) f ( x ) = c e x f(x)=ce^x
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In the calculus slides, it is shown that the rule d d x x n = n x n − 1 also works for fractional, negative, and even zero exponents. However, one function is still “missing” from the table of power derivatives. Which function is it? a) x − 2 x^{-2} b) x − 1 x^{-1} c) ln x \ln x d) e x e^x e) x \sqrt{x}
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A startup has 6 founders at a strategy retreat. Each pair of founders independently has a one-on-one strategy conversation with probability 1/2 . Question: What is the probability that the final conversation network has exactly 4 links ? A. 1001 32768 B. 1365 32768 C. 1820 32768 D. 3003 32768 E. 5005 32768 \frac{5005}{32768}
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A data center has 10 backup servers . Each pair of servers can establish a direct synchronization link independently with probability 1/9 . Question: Which statement is correct about this network? A. The expected degree is 8/9 , so the network is below the critical point. B. The expected degree is 1 1 , so the network is exactly at the critical point. C. The expected degree is 10/9 10/9 , so the network is above the critical point. D. The expected number of links is 10 , so the network is above the critical point. E. The expected degree is 2, so the network is far above the critical point.