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Course: MATH 321, Fall 2008
School: Cornell
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321 practice Math problems for first prelim Fall 2006 There are more problems here than many of you can do in fifty minutes, but they are representative of the type of problems you might expect on the exam. On the exam you may use a one-sided letter-size crib sheet, but no books, notes, or calculators. From notes: 2.7, 2.8, 2.13(i). (These problems will also be on the next homework assignment.) (Use the...

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321 practice Math problems for first prelim Fall 2006 There are more problems here than many of you can do in fifty minutes, but they are representative of the type of problems you might expect on the exam. On the exam you may use a one-sided letter-size crib sheet, but no books, notes, or calculators. From notes: 2.7, 2.8, 2.13(i). (These problems will also be on the next homework assignment.) (Use the fundamental theorem of calculus, the chain rule and differentiate under the integral sign.) 4. Thermodynamicists like to use rules such as Explain the rule and show that it is correct. (Assume that the variables are subject to a relation defining functions , , , and differentiate implicitly.) Naively cancelling numerators against denominators gives the wrong answer! stants. Deduce from the chain rule that j for all x. Here is a positive constant and s jr 2 4 ! kl m l 32 4 RRC 4 l l 4 C C h s gdtRRB" 8dIG%B i f "C C C e "s 6. Suppose that q y q q y (a) (b) (c) , , , ; ; . R R is a smooth function satisfying x , , , are arbitrary real conx. x x v s 3DwDDD r x DIDw3DDwD 5ry 3DwDuY DtGrVq s x v Y x v s x v s 5. Let , forms on R . Calculate and \$" i &%#!p a `hg\$ 0) f " " \$ a a CT\$U 2 \$" &%` Q HF 32 RP IGE S Y W " #!X88#!I#! C ' b 32 a 2 32 a \$ 2 \$ 2 2 \$ \$" VUT%! d " \$" eA 0%`Gc ...

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