Learning Materials
Features
Discover
Chapter 12: Problem 57
Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{x^{2}+x y-x z-y z}{x-z}$$
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
Identify and briefly describe the surfaces defined by the following equations. $$9 x^{2}+y^{2}-4 z^{2}+2 y=0$$
The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K,\) is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b} .\) Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=1\). a. Evaluate the partial derivatives \(Q_{L}\) and \(Q_{K}\). b. Suppose \(L=10\) is fixed and \(K\) increases from \(K=20\) to \(K=20.5 .\) Use linear approximation to estimate the change in \(Q\). c. Suppose \(K=20\) is fixed and \(L\) decreases from \(L=10\) to \(L=9.5 .\) Use linear approximation to estimate the change in \(\bar{Q}\). d. Graph the level curves of the production function in the first quadrant of the \(L K\) -plane for \(Q=1,2,\) and 3. e. Use the graph of part (d). If you move along the vertical line \(L=2\) in the positive \(K\) -direction, how does \(Q\) change? Is this consistent with \(Q_{K}\) computed in part (a)? f. Use the graph of part (d). If you move along the horizontal line \(K=2\) in the positive \(L\) -direction, how does \(Q\) change? Is this consistent with \(Q_{L}\) computed in part (a)?
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
Let \(f\) be a differentiable function of one or more variables that is positive on its domain. a. Show that \(d(\ln f)=\frac{d f}{f}.\) b. Use part (a) to explain the statement that the absolute change in \(\ln f\) is approximately equal to the relative change in \(f.\) c. Let \(f(x, y)=x y,\) note that \(\ln f=\ln x+\ln y,\) and show that relative changes add; that is, \(d f / f=d x / x+d y / y.\) d. Let \(f(x, y)=x / y,\) note that \(\ln f=\ln x-\ln y,\) and show that relative changes subtract; that is \(d f / f=d x / x-d y / y.\) e. Show that in a product of \(n\) numbers, \(f=x_{1} x_{2} \cdots x_{n},\) the relative change in \(f\) is approximately equal to the sum of the relative changes in the variables.
Let \(x, y,\) and \(z\) be non-negative numbers with \(x+y+z=200\) a. Find the values of \(x, y,\) and \(z\) that minimize \(x^{2}+y^{2}+z^{2}\) b. Find the values of \(x, y,\) and \(z\) that minimize \(\sqrt{x^{2}+y^{2}+z^{2}}\). c. Find the values of \(x, y,\) and \(z\) that maximize \(x y z\) d. Find the values of \(x, y,\) and \(z\) that maximize \(x^{2} y^{2} z^{2}\).
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine