In given problem, sketch the level curve z=k for the indicated values of k.

$$z=x^2+y, k=-4,-1,0,1,4$$

Let $f(x, y)=x^2 y+\sqrt{y}$. Find each value.

(a) f(2,1)

(b) f(3,0)

(c) f(1,4)

(d) $f\left(a, a^4\right)$

(e) $f\left(1 / x, x^4\right)$

(f) f(2,-4)

What is the natural domain for this function?

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $y=\sin ^{-1}\left(e^{x}\right)$

Sand is pouring onto a conical pile in such a way that at a certain instant the height is 100 inches and increasing at 3 inches per minute and the base radius is 40 inches and increasing at 2 inches per minute. How fast is the volume increasing at that instant?

Find the geometric and algebraic multiplicity of each eigenvalue of the matrix A, and determine whether A is diagonalizable. If A is diagonalizable, then find a matrix P that diagonalizes A, and find P-1AP. A = [-1 4 -2, -3 4 0, -3 1 3]

Compute the matrix A10. A = [1 0, -1 2]

Explaining How is the Department of Defense vital to national security?

The Department of Defense is considering three sites in the National Wildlife Preserve for extraction of rare metals. The cash flows associated with each site are summarized. The extraction period is limited to 5 years and the interest rate is 10% per year. Use the B/C method to determine which site, if any, is acceptable. The monetary unit is $million.$

$$\begin{array}{lccc} \text { Site } & \mathrm{A} & \mathrm{B} & \mathrm{C} \\ \hline \text { Initial cost, } \$ & 50 & 90 & 200 \\ \text { Annual cost, \$/year } & 3 & 4 & 6 \\ \text { Annual benefits, \$/year } & 20 & 29 & 61 \\ \text { Annual disbenefits, \$/year } & 0.5 & 2 & 2.1 \\ \hline \end{array}$$

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The city engineer and economic development directors are evaluating two sites for construction of a multipurpose sports arena. The sites are downtown (DT) and southwest (SW) of the metropolitan area. The city already owns enough land at the DT site; however, the land for a parking garage will cost $1 million, and construction costs will be$10 million for the parking garage, infrastructure relocation, and drainage. The SW site is 30 km from downtown, but the land will be donated by a developer who knows that an arena at this site will dramatically increase the value of the remainder of his land holdings. Because of its centralized location, there will be greater attendance at most of the events held at the DT site. This will result in more revenue to vendors and local merchants in the amount of $700,000 per year. Additionally, the average attendee will not have to travel as far, resulting in annual benefits of$400,000 per year. All other costs and revenues are expected to be the same at either site. If the city uses a discount rate of 6% per year, and will construct at one site or the other, which site should be selected? Use a 30-year study period.

How to calculate the PPP loan amount for sole proprietorship?

From the following data for a PPP project, calculate the (a) conventional, and (b) modified benefit/cost ratios using an interest rate of 6% per year and an infinite project period.

$$\begin{array}{cccc} & \text { To the People } & \text { To the Government } \\ \hline \text { Benefits: } & \$ 100,000 \text { per year } & \text { Costs: } \$ 1.8 \text { million now } \\ & \text { beginning now } & \text { and } \$ 200,000 \\ & & \text { every } 3 \text { years } \\ \text { Disbenefits: } & \$ 60,000 \text { per year } & \text { Savings: } \$ 90,000 \text { per year } \\ \hline \end{array}$$

A rural, agriculture-based city that has 17,000 households is required to install treatment systems for the removal of arsenic and other harmful chemicals from its drinking water. The annual cost is projected to be $150 per household per year. Assume that one life will be saved every 3 years as a result of the removal systems. (a) What is the B/C ratio, if a human life is valued at$4.8 million? Use an interest rate of 8% per year and assume the life is saved at the end of each 3-year period. (b) What justifies the project?

Identify the viewpoint (e.g., budget, government unit, citizen, business owner), and categorize the following cash flows as a benefit, disbenefit, or cost: (a) $600,000 annual income to area businesses from tourism created by new freshwater reservoir/recreation area (b)$450,000 per year for repainting of bridge across the Mississippi River (c) $800,000 per year maintenance by containership port authority (d) Loss of$1.6 million in salaries for border residents because of strict enforcement of immigration laws (e) Reduction of $600,000 per year in car repairs because of improved roadways (f) Expenditure of$350,000 for guardrail replacement on freeway (g) $1.8 million loss of revenue by farmers because of highway right-of-way purchases

In given problem, verify that

$$\frac{\partial^2 f}{\partial y \partial x}=\frac{\partial^2 f}{\partial x \partial y}$$

$$f(x, y)=\tan ^{-1} x y$$