\begin{tabular}{l} 4.3 - Riemann Sums \& The Definite Integral as Area \\ For each problem, approximate the area under the given function and above the \( x \)-axis on the given \\ interval using the specified number of rectangles/trapezoids. You may use your calculator. \\ 1.) \( f(x)=x^{2}-3 x+4,[1,4], n=6 \) \\ \begin{tabular}{|l|}\hline a) Left Rectangles \\ b) Right Rectangles \\ \hline c) Midpoint Rectangles \\ \hline d) Trapezoids \\ \hline\end{tabular} \\ \hline\end{tabular}

Find the following limit or state that it does not exist \[ \lim _{x \rightarrow-1} \frac{4(2 x-2)^{2}-64}{x+1} \]

\( \int _ { - 1 } ^ { 1 } \frac { x \cos ^ { 2 } x } { 1 + x ^ { 2 } } d x \)

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

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.

\( \int _ { - 1 } ^ { 1 } \frac { x \cos ^ { 2 } x } { 1 + x ^ { 2 } } d x \)