answer:The gradient is: {eq}nabla f(x,y) = left< 2x, 2y right>{/eq} At the point (1,2), we have: {eq}nabla f(1,2) = left< 2, 4 right>{/eq} Normalizing the direction vector: {eq}displaystyle hat u = frac{left< 1,-1right>}{sqrt{1^2 + (-1)^2}} = frac{left< 1,-1right>}{sqrt2}{/eq} The directional derivative is: {eq}begin{align*} D_{hat u} &= left< 2, 4 right> cdot frac{left< 1,-1right>}{sqrt2} = frac{2 - 4}{sqrt2} = -sqrt2 end{align*}{/eq}

answer:To find the values of N that satisfy the inequality 10N log N > 2N^2, we can use the following steps: 1. Rewrite the inequality in terms of the function f(x) = 5log(x) - x^2. 10N log N > 2N^2 5log(N) > N f(N) > 0 2. Find the critical points of f(x) by taking the first derivative and setting it equal to zero. f'(x) = frac{5}{x} - 2x = 0 x = frac{5}{2} 3. Determine the intervals where f(x) is increasing or decreasing by examining the sign of f'(x). f'(x) > 0 text{ for } 0 < x < frac{5}{2} f'(x) < 0 text{ for } x > frac{5}{2} 4. Find the maximum value of f(x) by evaluating f(x) at the critical point. fleft(frac{5}{2}right) = frac{25}{2} log left(frac{5}{2}right) - frac{25}{4} = frac{25}{4} left(log 5 - 1right) 5. Determine the values of N that satisfy f(N) > 0. f(N) > 0 text{ for } 0 < N < frac{5}{2} 6. Since N must be an integer, the only values of N that satisfy the inequality are N = 1, 2, 3, 4. Therefore, the values of N that satisfy the inequality 10N log N > 2N^2 are N = 1, 2, 3, 4.

answer:a. Net Income: ``` Net Income = (EBIT - Interest Expenses) x (1 - Tax Rate) Net Income = (250 million - 100 million) x (1 - 0.30) Net Income = 105 million ``` b. Interest Tax Shield: ``` Interest Tax Shield = Interest Expenses x Tax Rate Interest Tax Shield = 100 million x 0.30 Interest Tax Shield = 30 million ```

answer:The balanced chemical equation for the reaction is: {eq}2S + 3O_2 --> 2SO_3 {/eq} Moles of S = mass/Atomic weight = 10.5 g/32.065 g/mol = 0.327 mol. From the balanced equation, 2 moles of S produce 2 moles of SO{eq}_{3} {/eq}. Hence, 0.327 moles of S produce 0.327 moles of SO{eq}_{3} {/eq}. 1 mole of SO{eq}_{3} {/eq} molecules = {eq}6.023times10^{23} {/eq} number of SO{eq}_{3} {/eq} molecules. Hence, 0.327 moles of SO{eq}_{3} {/eq} molecules = {eq}(0.327)times(6.023times10^{23}) = 1.97times10^{23} {/eq} number of SO{eq}_{3} {/eq} molecules.