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Math 254 - Written Homework 7
发布时间：2024-05-16
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Math 254 - Written Homework 7
Grading
Some written homework problems are graded for correctness. To earn full credit for each problem you need to clearly and neatly show your work or provide a written explanation when appropriate. Partial credit will be awarded for logical, organized progress toward a solution when work is required and shown. Correct answers with no supporting work will not receive full credit.
Work Instructions
You may submit your work using one of the following methods:
(1) Print this document and do all of you work on this paper, or
(2) Complete your homework on a tablet by digitally writing or typing on this PDF, or
(3) Do the work on your own paper with the problems laid out the same way as on this template sheet.
Merge and Upload Instructions
Follow these instructions carefully.
If your PDF is not in the correct format, it will not be graded and you will receive a 0. (1) If you used method (1):
. Scan the printed pages and create a single PDF ile in the same format (and with the same number of pages) as the template.
(2) If you used method (2):
. The resulted PDF is in the correct format.
(3) If you used method (3):
. Use CamScanner (or app/website of your choosing) to
– take pictures of your work (including the cover page and blank problems), and
– merge your pictures into a single PDF ile.
. Your PDF should be in the same format as the template, i.e., a cover page followed by one page per problem:
– If the solution to a problem is short (or blank), you may get lots of white space in its page.
– If a problem is long, you may need multiple pages to write your answer. How- ever, you need to merge all those pages into a single page (like a collage) and
use the resulted page in your inal submission.
Upload into Gradescope under Written Homework 7.
(1) Find an equation of the tangent plane to each surface at the indicated point. (a) z = f (x, y) = x2 - y3 at (2, 1).
(b) z = f (x, y) deined implicitly by xy + xz + yz - ln (z2 + 1) = 4 at (2, 2, 0).
(c) The surface x2 + y2 - z2 = 0 at P (3, 4, 5).
(2) (a) Find an equation of the tangent planeto 
at (1; 2).
(b) Use this tangent plane equation, which is a linear approximation of 
at (1; 2), to estimate the value of z at (:95; 2:05).
(3) For each given surface, ind and classify all critical points.
(a) g(x, y) = xye-x2 -y2
(b) f (x, y) with partial derivatives fx = 6 (x2 - 1) (y + 1) and fy = (2x) (x2 - 3).
(4) Find the absolute maximum and minimum for f (x, y) = x2 + y2 - 2x - 2y on the closed region (= containing all the boundary points) R, where R is bounded by the triangle with vertices (0, 0), (2, 0), and (0, 2).
(5) Rectangular boxes with a volume of 10 m3 are made of two materials. The material for the top and bottom of the box costs $10 /m2 and the material for the sides of the box costs $1/ m2. What are the dimensions of the box with the minimum cost?
