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Chapter 13: Problem 14
Evaluate the following limits. $$\lim _{(x, y) \rightarrow(2,-1)}\left(x y^{8}-3 x^{2} y^{3}\right)$$
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Use the formal definition of a limit to prove that $$\lim _{(x, y) \rightarrow(a, b)}(f(x, y)+g(x, y))=\lim _{(x, y) \rightarrow(a, b)} f(x, y)+$$ $$\lim _{(x, y) \rightarrow(a, b)} g(x, y)$$
Rectangular boxes with a volume of \(10 \mathrm{m}^{3}\) are made of two materials. The material for the top and bottom of the box costs \(\$ 10 / \mathrm{m}^{2}\) and the material for the sides of the box costs \(\$ 1 / \mathrm{m}^{2}\). What are the dimensions of the box that minimize the cost of the box?
Evaluate the following limits. $$\lim _{(x, y) \rightarrow(0, \pi / 2)} \frac{1-\cos x y}{4 x^{2} y^{3}}$$
Use the definition of differentiability to prove that the following functions are differentiable at \((0,0) .\) You must produce functions \(\varepsilon_{1}\) and \(\varepsilon_{2}\) with the required properties. $$f(x, y)=x y$$
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.
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