answer:The directional derivative of the function {eq}f(x, y) = x^3 - y^3 {/eq} at the point {eq}P(8, 5) {/eq} in the direction of the vector {eq}mathrm{v} = frac{sqrt{2}}{2} (mathrm{i} + mathrm{j}) {/eq} is given by the dot product of the gradient of {eq}f(x, y) {/eq} at {eq}P(8, 5) {/eq} and the unit vector in the direction of {eq}mathrm{v} {/eq}. The gradient of {eq}f(x, y) {/eq} is {eq}nabla f(x, y) = frac{partial f}{partial x} mathbf{i} + frac{partial f}{partial y} mathbf{j} = 3x^2 mathbf{i} - 3y^2 mathbf{j} {/eq} So, the gradient of {eq}f(x, y) {/eq} at {eq}P(8, 5) {/eq} is {eq}nabla f(8, 5) = 3(8)^2 mathbf{i} - 3(5)^2 mathbf{j} = 192 mathbf{i} - 75 mathbf{j} {/eq} The unit vector in the direction of {eq}mathrm{v} {/eq} is {eq}frac{mathrm{v}}{|mathrm{v}|} = frac{frac{sqrt{2}}{2} (mathrm{i} + mathrm{j})}{sqrt{left(frac{sqrt{2}}{2}right)^2 + left(frac{sqrt{2}}{2}right)^2}} = frac{sqrt{2}}{2} (mathrm{i} + mathrm{j}) {/eq} Therefore, the directional derivative of {eq}f(x, y) {/eq} at {eq}P(8, 5) {/eq} in the direction of {eq}mathrm{v} {/eq} is {eq}nabla f(8, 5) cdot frac{mathrm{v}}{|mathrm{v}|} = (192 mathbf{i} - 75 mathbf{j}) cdot left(frac{sqrt{2}}{2} (mathrm{i} + mathrm{j})right) = frac{sqrt{2}}{2} (192 + (-75)) = frac{117sqrt{2}}{2} approx 82.73 {/eq}

answer:The right to be represented by counsel is a fundamental right guaranteed by the Sixth Amendment to the United States Constitution. This right ensures that all persons accused of a crime have the right to be represented by an attorney during all stages of the criminal justice process, including trial, sentencing, and appeals. The right to an attorney is essential to ensure that all persons are treated fairly and equally under the law, regardless of their financial resources or ability to hire an attorney. The right to be represented by counsel is also known as the right to an attorney.

answer:Yes, 12345 is divisible by 5 because the last digit of the number is 5, which is a multiple of 5.

answer:A. Here is the graph of the given curves: [Insert graph image] B. To find the area of the region: Step 1: We find the points of intersection: [ begin{align*} y &= x^2 - 4 y &= 0 end{align*} ] Setting the equations equal to each other: [ x^2 - 4 = 0 ] Solving for ( x ) gives: [ x^2 = 4 ] [ x = -2, 2 ] Step 2: From the graph, we see that ( y = 0 ) forms the upper curve, while ( y = x^2 - 4 ) forms the lower curve. The area A between these curves is given by the integral: [ A = int_{a}^{b} (y_{upper} - y_{lower}) , dx ] From our points of intersection, ( a = -2 ) and ( b = 2 ), so we have: [ A = int_{-2}^{2} (0 - (x^2 - 4)) , dx ] [ A = int_{-2}^{2} (4 - x^2) , dx ] [ A = left[4x - frac{x^3}{3}right]_{-2}^{2} ] [ A = left[4(2) - frac{2^3}{3} - 4(-2) + frac{(-2)^3}{3}right] ] [ A = left[8 - frac{8}{3} + 8 - frac{8}{3}right] ] [ A = frac{32}{3} text{square units} ] The area of the region enclosed by the curves is ( frac{32}{3} ) square units.