Math 4A03: Practice problems on Multivariable Calculus

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1 Mat 4A0: Practice problems on Mltiariable Calcls Problem Consider te mapping f, ) : R R defined by fx, y) e y + x, e x y) x, y) R a) Is it possible to express x, y) as a differentiable fnction of, ) near te origin x 0, y 0 ) 0, 0)? b) Compte,,, at x, y) 0, 0), ie wen, ), ) Problem Consider te mapping f, ) : R R defined by fx, y) x y, x y), x, y) R a) Sow tat te range of f is R tat if 0, 0 ) 0, 0), tere are exactly two points in R tat are mapped to 0, 0 ) by f b) Sow tat te mapping f is locally inertible at te point x 0, y 0 ), ) Find an explicit formla for its local inerse g, ) defined in a neigborood of f, ) 0, ) Problem Consider te system of eqations { w x y z 0 w 4 + x 4 + y 4 + z 4 8 a) Is it possible to express x, y) as a differentiable fnction of w, z) near te soltion w, x, y, z), 0,, )? Use te implicit fnction teorem to answer tis qestion) b) If so, wat are w, ) z, )? Use te implicit fnction teorem to answer tis qestion) c) Compte explicitely te differentiable fnction of w, z) in part a) erify yor answers in part b) by a direct comptation ie witot sing te implicit fnction teorem) Problem 4 Determine if te fnction f : R R defined by { x + y, x y, fx, y) x + x, x y is differentiable at 0, 0) sing te definition of differentiablity

2 Soltion We first compte te first-order partial deriaties at 0, 0) f 0, 0) lim 0 f 0, 0) lim 0 f, 0) f0, 0) f0, ) f0, 0) We know tat if f 0, 0) exists ten [ f f f 0, 0) 0, 0) lim 0 lim 0 0, 0) [ To see if f is differentiable at 0, 0), we need to ceck if f, ) f0, 0) f 0, 0) f 0, 0) + f, ) f0, 0) 0, + as, ) 0, 0) If, we ae f, ) + since f0, 0) 0, f, ) f0, 0) 0 so, clearly, te preios qotient goes to 0 if, ) 0, 0) wit On te oter if, we ae f, ) f0, 0) + + tis large expression does not conerge to 0 as 0 Hence, f is not differentiable at 0, 0) Problem 5 Let E R n be open let f : E R be a fnction aing partial deriaties f j, j,, n, bonded on E Proe tat f is continos on E Hint: Proceed as in te proof done in class tat, if tese partial deriaties are continos on E, ten f C E) Soltion Let x E coose r > 0 small enog so tat {y R n, x y < r} E were denotes te sal eclidean norm on R n let n j j e j wit < r were e,, e n is te stard ortonormal basis in R n ) Define 0 0 k k j j e j for k, n We can ts write Define fx + ) fx) n [fx + j ) fx + j ) j g j t) fx + j + t j j )) fx + j + t j e j ), 0 t,

3 for j,, n Ten, g jt) f j x + j + t j e j ) j By te mean ale teorem, tere exists θ j wit 0 < θ j < sc tat or, eqialently, g j ) g j 0) g jθ j ), fx + j ) fx + j ) f j x + j + θ j j e j ) j Since all first-order partial deriaties are bonded by M on E, it follows tat fx + j ) fx + j ) f j x + j + θ j j e j ) j M j Tis leads to n fx + ) fx) fx + j ) fx + j ) M j n j 0, j as 0 Tis sows tat wic means tat f is continos at x lim fx + ) fx), 0 Problem 6 Let f : R R be te mapping defined by a) Wat is te range of f fx, y) e x cos y, e x sin y), x, y) R b) Sow tat f is locally one-to-one on R ie one-to-one on a neigborood of eery point x 0, y 0 ) R ), bt not globally one-to-one ie not one-toone on R ) c) Let g be te continos inerse of f defined in a neigborood of, ) f0, π ) Find an explicit formla for g Compte f ) ) 0, π g erify tat g, ) f 0, π )) d) Wat are te images nder f of lines parallel to te coodinate axes?,

4 4 Soltion a) Since any point, ) R can be written in polar coordinate as, ) r cosy), r siny)) were r 0 r 0 if only, ) 0, 0) wile r e x, for some real x if only if r > 0 or, ) 0, 0), it follows tat te range of f is te set R \ {0, 0)} b) If x 0, y 0 ) R, we ae f x 0, y 0 ) [ f x 0, y 0 ) f x 0, y 0 ) f x 0, y 0 ) f x 0, y 0 ) [ e x 0 cosy 0 ) e x0 siny 0 ) e x0 siny 0 ) e x0 cosy 0 ) Since det f x 0, y 0 )) e x 0 cos x 0 ) + sin x 0 )) e x0 0, it follows tat f x 0, y 0 ) is inertible te inerse fnction teorem sows tat f is oneto-one on a neigborood of x 0, y 0 ) Neerteless, f is not one-to-one on R, since fx, y + π) e x cosy + π), e x siny + π)) e x cosy), e x siny)) fx, y) c) Since f0, π ), ), we need to sole te system { e x cosy) e x siny) for x, y) in term of, ), for, ) close to, ) Since + e x, it follows tat x ln + ) Since tany), we ae y tan ) Note tat tan ) tan ) π ) Te inerse mapping g is ts gien by g, ) Letting g g, g ), we ae We ae g 0, 0 ) [ g [ f 0, π ) f 0, π ) wic sows tat g, 0, 0 ) g 0, 0 ) ln + ), tan ) ) ) [ g, g 0, 0 ) g 0, 0 ), g, ) [ + + ) [ [ f 0, π )) + + [ 0 0

5 5 d) Te line x x 0 as for image te circle of radis e x 0 centered at 0, 0) wile te line y y 0 as for image a ray starting at te origin bt not inclding it) tat makes an angle y 0 wit te positie x-axis Problem 7 Soltion Problem 8 Consider te mapping f, ) : R R defined by a) Sow tat te range of f is te set fx, y) x + y, x y), x, y) R L {, ) R, 0} Hint: Sow first tat f maps R into L ten tat eery point in L is te image nder f of some point in R b) 9 pts) Sow tat te mapping f is locally inertible at te point x 0, y 0 ), ) Find an explicit formla for its local inerse g, ) defined in a neigborood of f, ), 4) c) 5 pts) Compte g, 4) Soltion a) Let x + y x y, ten x+y) 4 x y x + x y+y 4 x y x x y+y x y) 0 Tis sows tat te range of f is contained in L Now let, ) belong to L, so 0 If 0, we ae f0, ), 0) or f, 0), 0)) sowing tat, 0) belongs to te range of f for any ale of If 0 we ae y x x+ x or x x+ 0, so x ± Tere are ts two points mapped to, ) by f, namely x, y) x, y) +,, + Ts, ) also belongs to te range of f wen, ) L 0 b) We ae [ f x, y) y x [ ) )

6 6 Since te first-order partial deriaties of are continos, f belongs to C R ) Since [ f, ) , te inerse mapping teorem sows tat f is locally inertible at te point, ) Using part a), te local inerse is gien by + ) g, ), + +, ) c) [ g, 4) f, )) 6 Eqialently, we can se te explicit expression of g to compte g We ae [ [ g, ) + 6 [ g, 4) 6 6 Problem 9 a) Sow tat te system { x + y 0 x + y 0 can be soled for x, y in terms of, near te point x, y,, ), 0,, ) b) Compte te partial deriatie, ) Soltion Let fx, y,, ) f, f ) x + y, x + y ) Note tat f belongs to C R ) [ f x y x, y,, ) x y [ f f f f f f f f

7 7 For te system to be solable for x, y in terms of,, we need to erify, according to te implicit fnction teorem, tat te matrix [ f f f f ealated at te point x, y,, ), 0,, ) is inertible Tis matrix is [ [ x y 0,0,, ) wic as determinant 0 is ts inertible b) Since f x, ), y, ),, ) 0, we ae Similarly, f f + f + f Tis can be written is matrix form as [ f f f f + f 0 + f 0 [ [ f f from wic we dedce tat [ [ f f f f [ f f Ealating te preios expression at te point x, y,, ), 0,, ), we obtain [ [ [ [ 0 [ [ 0 0, ) In particlar,, )