Show your work on all problems using methods from the course. Test is open notes / open book. A scientific (nongraphing) calculator is allowed. You may NOT use the help of other students, people, other electronic devices, or the internet (exam is scored with these restrictions in mind.) Print and complete by hand (or complete on paper/tablet) and scan your work – save as a single pdf file and submit in Canvas within the time limit.
There are 50 possible points.
1. (6 points) Evaluate the indefinite integral. (Use the integration by parts method, show all your steps.)
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2. (8 points) Suppose a particular company has expected income c(t) over the following 8 years to be
?=3,000+5,000? , 0≤?≤8
If we assume the inflation rate is 5%, what is the present value of this income? (Show all steps, round to 2 digits past the decimal.)
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3. (8 points) Evaluate the following improper integrals. Be sure to specify if the integral is convergent or divergent. Show all your steps to justify your solutions.
?) ∫ ?2√2+?3 ??∞2
?) ∫ 6?4?∞3 ??
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4. (6 points) Suppose we have the following Cobb-Douglas model for production, given by the function ?(?,?)=500?0.65?0.35. When x = 800 and y = 1600, find the marginal productivity of labor (??/??) and marginal productivity of capital (??/??). (Round each to 2 digits past the decimal.)
5. (8 points) Find all the first partial derivatives of the given functions.
?) ?(?,?,?)=2??2+??3−??? (be sure to provide partials for x, y, and z )
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6. (8 points) For the given function, locate the relative extrema and the saddle points. (Show all steps of the process and use the second partial derivatives test to classify extrema as relative maximum, minimum, or saddle points. Be sure to also give the function value at any relative extrema.)
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7. (6 points) For the given function, find the extrema indicated using the method of Lagrange multipliers. Show all steps.