20Emidterm2 - 1 Consider t he f unction f(x y = sin(xy cos...

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1. Consider the function f(x, y) = sin(xy) + cos( x y) . a) (7pts) Find the second-order Taylor approximation to f at (0,0). b) (3pts) Use the formula from a) to approximate f(O.l, -0.1) ' -1' (0) 0') = \ lA\ Jx (>' :, ~ \ ~ ~ to s (j.~\ - ~JI~ (x~\ ) :L. (0) 0) = 0 J 1 ()(I~) -== j. to.5 ( Xj ') - X 6 \h()<~) ') J:; (u, 0) = 0 . j.;l..j... (~ ~ ) =:-y7.. jl ' '()[X:J' - yLWS(X~) ') Jxx (v/o):= 0 J~j (X ) j) c= - x"\\v\'(}{J\ - x7 to~ l~~ )) j~ j C O; 0) = 0 J ,, ~ (X, j) co J~" (X, ~ ') ~ to s )(~ '\ - )(~ S; Yl \)( Ji) - 5. \0 (;( ~\ - Xj t.o s ( X~ '\ ) ..f ><~(o) 0) ::= { ~ x ( oJ 0) ;;:::. \ Jet..or0 o\~~ L (j t o( ttf'~('ox,~b ~ ~ (D-1b)\S T( )( lj ) ~ jlQ(8') + jx(O,b)()(;-CL) -l- jj( ~ t1)(~-b, . , ~ ~x(a. I ~\ (x -o..~ + ±f~ a l t;) (j- ~) ">- + J'<j (0. 1 6 ,\(>( - «..)( x - I, 1 SO ~L ~I CX/y\ .= \ ~ x ~ b ') j ( 0 • \ ) _ o. ~\ ~ T ( o. \ ) - 0 I \\ - I + ( 0 . \\ (.- 0, \ =- 0 ~ 9 q 2
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2. a) (5pts) Sh ow th at the pat h is c(t) = (sin t, cos t, e t ) a fl ow line of the vector fie ld F(x, y, z) = (y , -x , z ). b) (5pts) Show that F = (: r2 + y2) i - 2xyj is not a gradient fi el d. ~ \ r f C(+) n ~ -}J v w I i ~ fu V\.. F (c f-\- ')\ -::: C. i ~ ') F ( C (-\- ') ') = < ~ st ) '- 5 ~ fl-t ) ~ '> c/ <t ') =- < Cos t ) - 5~ Y\.t ) e..-t > t-(c f+)') = C /
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