8. The sum of the height of a cylindrical can and its circumference is 120 cm. Use your knowledge of two-variable calculus to prove that the biggest volume enclosed by the can like this is 64,000/m cubic centimeters. 9. The linear approximation of a function of two variables at the origin involves the value of the function and two first partial derivatives at the origin. A quadratic approximation of a function of two variables will involve the values of the function and six of its derivatives at that point. What are those derivatives? Clarify that before attempting your best to give some idea of how you could find a quadratic approximation to sinxsiny at the origin from the perspective of two-variable calculus. If you recall your calculus I knowledge, where you know that when x is very close to 0, that sin x ~ x, and so it makes sense to assume that sinxsiny ~ xy. Does the two-variable calculus agree?For problems 1 to 3, let f(x, y) = excos y. Or feel free to use 2 = excos y. 1. Show that a normal vector of the plane tangent to the surface at the point (0, 0, l) is the vector i + k. 2. Use linear approximation to approximate the value of e'0'026cos(0.003). Take the number 0974330705104 to be the exact value of e'0'026cos(0.003). Explain Why someone might casually say \"just add the exponent to one and that's the value\" but explain precisely the mathematics happening, in your own words. (If English is not your rst language, feel free to answer that last part in the language you speak. I will be able to read it.) 3. Using the information in 2, write out in words how you would explain to someone how you would estimate the change in excos y (by that, you mean \"the change in z\") by differentials. The actual change in z, from (0, 0) to (0.026, 0.003) is 0.025669294896, so be sure to include that in your discussion. (The best written answer to this contains a clear understanding of notation by including it in both symbols and words.) 5. This problem got me both in trouble with bosses but closer than ever to a group of students one year, and it became a problem that the students appreciated and remembered, and also brought laughter to our course. (The students created a competition to write what would be the best, which to them meant \"the funniest.\" I have to keep the one that won a secret between my students and I.) Do not trouble yourself if you cannot figure it out. Skip it if you do not find it worth thinking about. This will never appear on an exam again, but the inner city school students I had, who had no access to computer technology, learned to love them as they were learning to speak and write both English and mathematics, and returned these kinds of problems to me as they learned calculus, when I asked them what kind of problems did they want to see on their exams, perhaps the most exciting time I have been more educated by students than students educated by me. It is a humbling experience to face where you know you must improve in your work. A problem, as written by my students, was: Tea is a function of sea and why, sea is a function of pee and jay, and why is a function of pee and jay. What is the rate of change of tea to pee? Students did their final exams in groups that year, and had to help each other explain their answers either out loud, supporting each other, or written, or both. One of the answers a student gave me, a shy young lady with a talent for Opera and humor, sang \"del tea del sea del sea del pee will be added to del tea del why del why del pee.\" Was she correct? ... . . . ..-. H .. 1 . nn/ \\ 1'}, 7' 71*