Use the Comparison Test to determine whether the series converges. \( \sum_{k=1}^{\infty} \frac{1}{k^{\frac{7}{4}}+2} \) The Comparison Test with \( \sum_{k=1}^{\infty} \square \) shows that the series

\( \int \frac { 3 x ^ { 5 } } { \sqrt { 7 - 12 x ^ { 6 } } } d x = \square \)

49. Marginal Productivity A manufacturer estimates that produc- tion (in hundreds of units) is a function of the amounts \( x \) and \( y \) of labor and capital used, as follows. \[ f(x, y)=\left(\frac{1}{4} x^{-1 / 4}+\frac{3}{4} y^{-1 / 4}\right)^{-4} \] a. Find the number of units produced when 16 units of labor and 81 units of capital are utilized. b. Find and interpret \( f_{x}(16,81) \) and \( f_{y}(16,81) \). c. What would be the approximate effect on production of increasing labor by 1 unit from 16 units of labor with 81 units of capital? 50. Marginal Productivity The production function \( z \) for the United States was once estimated as where \( x \) stands for the amount of labor and \( y \) the amount of capital. Find the marginal productivity of labor and of capital.

Resuelve la ecuación diferencial \( \frac{dy}{dx} = 3y^{2} \) dado que \( y(0) = 1 \).

Evaluate the following definite integral using the Fundamental Theorem of Calculus. \( \frac{1}{7} \int_{0}^{\ln 4} e^{x} d x \)

Given the following velocity function of an object moving along a line, find the position function with the given initial position.