Math 3210practice problems for first prelimFall 2009There are more problems here than many of you can do in fifty minutes, but they arerepresentative of the type of problems you might expect on the exam. On the exam youmay 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 homeworkassignment.)1.Letf:Rn→Rbe a smooth function and defineg(x) =sin(f(x)2). Find the1-formd f.2.Letf:R2→Rbe a smooth function and assumef(x,y) =-f(y,x). Showthat∂f∂x(a,b) =-∂f∂y(b,a).3.Letf:R2→Randg:R→Rbe smooth functions. Show thatddxZg(x)0f(x,y)dy=f(x,g(x))g(x) +Zg(x)0∂f(x,y)∂xdy.(Use the fundamental theorem of calculus, the chain rule and differentiate underthe integral sign.)4.Thermodynamicists like to use rules such as∂y∂x∂z∂y∂x∂z=-1.Explain the rule and show that it is correct.(Assume that the variables are
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