52. Marginal Productivity A manufacturer of automobile batter- ies estimates that his total production (in thousands of units) is given by \[ f(x, y)=3 x^{1 / 3} y^{2 / 3} \text {, } \] where \( x \) is the number of units of labor and \( y \) is the number of units of capital utilized. a. Find and interpret \( f_{x}(64,125) \) and \( f_{y}(64,125) \) if the current level of production uses 64 units of labor and 125 units of capital. b. Use your answer from part a to calculate the approximate effect on production of increasing labor to 65 units while holding capital at the current level. c. Suppose that sales have been good and management wants to increase either capital or labor by 1 unit. Which option would result in a larger increase in production?

Hallar el volumen del sólido limitado arriba por \( x=1+x^{2}+y^{2} \) y abajo \( z=2 \sqrt{x^{2}+y^{2}} \)

Describe cómo se utiliza un campo de pendientes para analizar la ecuación diferencial \( \frac{dy}{dx} = x + y \).

Estimate the area under the graph of \( f(x)=\frac{1}{x+4} \) over the interval \( [1,3] \) using five approximating rectangles and right endpoints. \( R_{n}=\square \) Repeat the approximation using left endpoints. \( L_{n}=\square \)

A coal-burning electrical generating plant emits sulfur dioxide into the surrounding air. The concentration \( C(x) \), in parts per million, is approximately given by the function \[ C(x)=\frac{0.4}{x^{2}} \] where \( x \) is the distance away from the plant in miles. The instantaneous rate of change of the sulfur dioxide concentration: (A) 2 miles from the plant = (B) 11 miles from the plant =

Between 0 degrees Celsius and 30 degrees Celsius, the volume V (in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula \( V=999.87-0.06426 T+0.0085043 T^{2}-0.0000679 T^{3} \) Find the temperature at which water has its maximum density. Temperature \( = \)