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Problem 10

Let \(g(x, y, z)=0.01 x+0.02 y-0.03 z-0.05 .\) Complete the following sentences. a. g ___ by ___ units for every 1 unit of increase in z. b. g ___ by ___ units for every 1 unit of increase in x. c. ______ by 0.02 units for every _______.

Expert verified

a. g decreases by 0.03 units for every 1 unit of increase in z.
b. g increases by 0.01 units for every 1 unit of increase in x.
c. g increases by 0.02 units for every 1 unit of increase in y.

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Chapter 15

The Gym Shirt Company manufactures cotton socks. Production is partially automated through the use of robots. Daily operating costs amount to $$\$ 150$$ per laborer and $$\$ 60$$ per robot. The number of pairs of socks the company can manufacture in a day is given by a Cobb-Douglas production formula $$ q=50 n^{0.6} r^{0.4} $$ where \(q\) is the number of pairs of socks that can be manufactured by \(n\) laborers and \(r\) robots. Assuming that the company has a daily operating budget of $$\$ 1,500$$ and wishes to maximize productivity, how many laborers and how many robots should it use? What is the productivity at these levels? HINT [See Example 5.]

Chapter 15

Compute the integrals. HINT [See Example 1.] $$ \begin{aligned} &\int_{0}^{1} \int_{y}^{y+2} \frac{1}{\sqrt{x+y}} d x d y\\\ &\text { HINT [See Example 2.] } \end{aligned} $$

Chapter 15

Sketch the graph of a function that has one saddle point and one extremum.

Chapter 15

Find the volume of the tetrahedron with corners at \((0,0,0)\), $(a, 0,0),(0, b, 0)\(, and \)(0,0, c)$

Chapter 15

If the partial derivatives of a function of several variables are never 0, is it possible for the function to have relative extrema on some domain? Explain your answer.

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