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Chapter 12: Problem 16
Evaluate the following limits. $$\lim _{(x, y) \rightarrow\left(e^{2}, 4\right)} \ln \sqrt{x y}$$
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Economists model the output of manufacturing systems using production
functions that have many of the same properties as utility functions. The
family of Cobb-Douglas production functions has the form \(P=f(K, L)=C K^{a}
L^{1-a},\) where \(K\) represents capital, \(L\) represents labor, and C and a are
positive real numbers with \(0
A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.$$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=x\left(x^{2}-3 y^{2}\right)$$
Use the definition of the gradient (in two or three dimensions), assume that \(f\) and \(g\) are differentiable functions on \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\), and let \(c\) be a constant. Prove the following gradient rules. a. Constants Rule: \(\nabla(c f)=c \nabla f\) b. Sum Rule: \(\nabla(f+g)=\nabla f+\nabla g\) c. Product Rule: \(\nabla(f g)=(\nabla f) g+f \nabla g\) d. Quotient Rule: \(\nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}}\) e. Chain Rule: \(\nabla(f \circ g)=f^{\prime}(g) \nabla g,\) where \(f\) is a function of one variable
Economists model the output of manufacturing systems using production
functions that have many of the same properties as utility functions. The
family of Cobb-Douglas production functions has the form \(P=f(K, L)=C K^{a}
L^{1-a},\) where \(K\) represents capital, \(L\) represents labor, and C and a are
positive real numbers with \(0
Batting averages in baseball are defined by \(A=x / y,\) where \(x \geq 0\) is the total number of hits and \(y>0\) is the total number of at bats. Treat \(x\) and \(y\) as positive real numbers and note that \(0 \leq A \leq 1.\) a. Use differentials to estimate the change in the batting average if the number of hits increases from 60 to 62 and the number of at bats increases from 175 to 180 . b. If a batter currently has a batting average of \(A=0.350,\) does the average decrease if the batter fails to get a hit more than it increases if the batter gets a hit? c. Does the answer to part (b) depend on the current batting average? Explain.
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