Science:Math Exam Resources/Courses/MATH105/April 2011/Question 08 (c)

MATH105 April 2011

• Q1 (a) • Q1 (b) • Q2 • Q3 • Q4 • Q5 (a) • Q5 (b) • Q5 (c) • Q6 • Q7 • Q8 (a) • Q8 (b) • Q8 (c) • Q9 (a) • Q9 (b) • Q9 (c) • Q9 (d) • Q9 (e) • Q9 (f) • Q9 (g) •

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[hide] Question 08 (c)
The productivity (measured in the dollar value of goods produced) of a certain country is given by the function $\displaystyle f(x,y)=160x^{3/4}y^{1/4},$ where $x$ denotes the amount (in dollars) invested in labor, and $y$ is the amount invested in capital. Recall that the marginal productivity of labor (respectively capital) is the rate of change of $f$ with respect to $x$ (respectively $y$ ), holding $y$ (respectively $x$ ) fixed. Find all vectors $\|{\textbf {v}}\|=\|\nabla f(81,16)\|$ that point in the direction of no change in the productivity from its current value.

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[show] Hint
The gradient vector $\displaystyle \nabla f(81,16)$ points to the direction of maximal increase of productivity. Any vector perpendicular to that one will have no change of productivity (it's the direction of a level curve, that is a curve where f is constant).

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[show] Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. In part (a) we computed the partial derivatives, allowing us to write the gradient vector $\displaystyle \nabla f(x,y)=(120x^{-1/4}y^{1/4},40x^{3/4}y^{-3/4})$ hence (as calculated previously) $\displaystyle \nabla f(81,16)=(80,135)$ Recall that a perpendicular vector to (a,b) with the same length is is (b,-a) and (-b,a). Therefore, the perpendicular vectors (of the same length as the gradient itself) are the vectors $\displaystyle (-135,80)\quad {\text{and}}\quad (135,-80)$