Practice Prelim - Math 321 practice problems for first...

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Unformatted text preview: Math 321 practice 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 s h 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 a Q HF RP IGE S D 3. Let R R and R R be smooth functions. Show that " $ 10) ' $" (&%#! C 4 " 7 [email protected] 32 $ ' 7" 4 2 (#9865 32 2 2. Let R R be a smooth function and assume 1. Let R -form . R be a smooth function and define x sin x x . Find the . Show that be ...
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