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Problem 1

Calculate $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\left.\frac{\partial f}{\partial x}\right|_{(1,-1)}$, and \(\left.\frac{\partial f}{\partial y}\right|_{(1,-1)}\) when defined. HINT [See Quick Examples page 1098.] $$ f(x, y)=10,000-40 x+20 y $$

Expert verified

The partial derivatives of the function \(f(x, y) = 10000 - 40x + 20y\) are \(\frac{\partial f}{\partial x} = -40\) and \(\frac{\partial f}{\partial y} = 20\). Since they are constants and do not depend on the point, their values at \((1, -1)\) are \(\left.\frac{\partial f}{\partial x}\right|_{(1,-1)} = -40\) and \(\left.\frac{\partial f}{\partial y}\right|_{(1,-1)} = 20\).

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Chapter 15

Compute the integrals. HINT [See Example 1.] $$ \int_{0}^{1} \int_{0}^{2-y} y d x d y $$

Chapter 15

Locate and classify all the critical points of the functions. HINT [See Example 2.] $$ f(x, y)=x y+\frac{2}{x}+\frac{2}{y} $$

Chapter 15

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

Chapter 15

Sketch the region over which you are integrating, and then write down the integral with the order of integration reversed (changing the limits of integration as necessary). $$ \int_{-1}^{1} \int_{0}^{\sqrt{1-y}} f(x, y) d x d y $$

Chapter 15

Sketch the region over which you are integrating, and then write down the integral with the order of integration reversed (changing the limits of integration as necessary). $$ \int_{1}^{e^{2}} \int_{0}^{\ln x} f(x, y) d y d x $$

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