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Chapter 13: Problem 53
Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1, \ln 2,3)} z e^{x y}$$
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Find an equation of the plane that passes through the point \(P_{0}\) and contains the line \(\ell\) a. \(P_{0}(1,-2,3) ; \ell: \mathbf{r}=\langle t,-t, 2 t\rangle\) b. \(P_{0}(-4,1,2) ; \ell: \mathbf{r}=\langle 2 t,-2 t,-4 t\rangle\)
Absolute maximum and minimum values Find the absolute maximum and minimum values of the following functions over the given regions \(R\). Use Lagrange multipliers to check for extreme points on the boundary. $$f(x, y)=(x-1)^{2}+(y+1)^{2} ; R=\left\\{(x, y): x^{2}+y^{2} \leq 4\right\\}$$
A clothing company makes a profit of \(\$ 10\) on its long-sleeved T-shirts and \(\$ 5\) on its short-sleeved T-shirts. Assuming there is a \(\$ 200\) setup cost, the profit on \(\mathrm{T}\) -shirt sales is \(z=10 x+5 y-200,\) where \(x\) is the number of long-sleeved T-shirts sold and \(y\) is the number of short-sleeved T-shirts sold. Assume \(x\) and \(y\) are nonnegative. a. Graph the plane that gives the profit using the window $$ [0,40] \times[0,40] \times[-400,400] $$ b. If \(x=20\) and \(y=10,\) is the profit positive or negative? c. Describe the values of \(x\) and \(y\) for which the company breaks even (for which the profit is zero). Mark this set on your graph.
Production functions 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
Use the formal definition of a limit to prove that $$\lim _{(x, y) \rightarrow(a, b)}(f(x, y)+g(x, y))=\lim _{(x, y) \rightarrow(a, b)} f(x, y)+$$ $$\lim _{(x, y) \rightarrow(a, b)} g(x, y)$$
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