pprelim2 - Pragdm.2 Praan'CQ PFOMMVIS 501 WP 0L(Ev-1PM...

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Unformatted text preview: Pragdm .2 Praan'CQ PFOMMVIS @ 501} WP 0L (Ev-1PM, ifiefiral {o cam/{ode {fig VOL/flaw of: W reqf‘wn find/V1424 aim/e 63 {he Fa-IQEOLo—Lc: 2 a=8~ x72- ?— amoi £550“; 63 Ha pajaéotowi 2—K +3 . Use "Him order (xi-:fi twang/won dtgakobn @ coma” Hm Wm m Hm gm mime MW” W pLa/Vles X+g+2=3 moi x+g+%=40:fifind £45 Vegan/v22. - .‘ J I ~ 0 flag of: "n by), W curve ® Find Hue pom/C (£0st ’E 3,1, . oi: Lrydrerseczhow. 0% Hue Mama phyla-=5 amd +142 cone 21>- qXZ +ng? MAW/e @ Ld £(X,g,2)=xy+zz rqareSMLlL fhe {’WM' «,2? each 1005sz on 79/71 sphare x2+gz+£z :29’. 2:4 ho'Ha'l we! wc’aleszi) {WI/pom. res , ’H’w $h%€(flaf £5 1910/2 iMsz/I'Seaéjon 0f 'éheFéQ/J’ZZ 0% X+€+gz0 mol W sphut, ML 50X @ Fin/ml Hm maXi’nwr/m mogul/ma #1 re Wyn/M 'Msmged i’VL 7%? Sphbfé'xz’fglqtaii wifié. (.115 559.625 {la/raw {o {42.2 warpéi'naulz [2le 0211 ’26) Find 1%: ama/riaafion L(X/g/,i)0/%e fim 02/2 £0033): Vi wsXsi'rz(y+2) ad P°{01QE¢/)*meo{cm ”2:20“ 69‘4””! fir’l’h‘? maw‘mde 05f 1%2 afar E m 7% a/pproximm‘wn «F(X,51,'Z)$L(X,g,-Z)OV€/ {he reg/£012 R: lX/éQOL, Ig/scwg /2-%/sa 0/. <7) , We wish to compute the integral " 1.01 1.01 1 ,, ., ——. 2"“? (ix dy. [.99 .99 1112 Because there is no closed form for the antiderivative of the integrand, we must estimate. (a) Find an appropriate linear approximation, L, of the integrand and then integrate L over the region. (1)) Find a bound on the error incurred by this estimation. In computing the error bounds, you may use the fact that all second partials of the integrand on the given region have absolute value less than one. -2- 8. Consider f (x, y) = k2 x — xjy sink + (6 —— 5k)x Where k is a real number. (a) Find all values of k for Which (0, O) is a critical point of the function f. (13) Can (0, 0) be a local extreme of f? 9. (a) Test f (x, y) = x3 + 3xy + y,3 for local maxima, local minima, and saddle points. 10. ll. 12. 13. 14. 15. 16. (b) Find the absolute maximum and minimum of f (x, y) = x — 430} + 2 y +1 on the closed triangular plate in the first quadrant bounded by the lines x = 0, y = 0 , and y = 1 — x. 2 Find all local maxima, local minima, and saddle points of z = x + y2 — xy — x . Locate the critical points of the function f (x, y) = x4 + xy + y4 and classify them as local maximum points, local minimum points, or saddle points. Consider the function f (s, t) = 3st +33 —t3 . (a) Find the local maxima, local minima, and saddle points for f. (b) Find the absolute maxima of the fianction f (s,t) over the region bounded by the lines s=t,s=l,andz‘=0. Find the greatest and smallest values of the fimction f (x, y) = x2); in the region x2+y2S1. The height above sea level of Gorgeous State Park is given by H(x, y) = 4xy —x4 —y2 +350 for — 4 S x S 4 and -— 4 S y S 4. Professor Pontificus goes for his daily walk in Gorgeous Park along a path given by x(t) =t y(t) = t2 Find the coordinates of all points on the path Where the Professor will be neither ascending nor descending. Consider the fimction f (x, y) = x2 + y2 — 2x — 4y on the region R bounded by the lines )2 = x, y = 3, and x = 0. Find the absolute maximum and the absolute minimum of f (x, y) on R. if ' ' ' Consider the function f (x, y) = [:0 ~ 22‘ —z‘2)dt defined on the domain at S )2 (that is, the domain is the half—plane x S y). Find the critical points of the function. What is the absolute maximum of f ? At what points does the absolute maxima occur? Explain Why the absolute maxima exist. l. -3- Double Integrals and Polar Coordinates {W l 17. Consider the integral f(x,y)=Jl_1 2 ydydx. x (a) Describe or sketch the region of integration (b) Evaluate the integral (0) Reverse the order of integration. 1 1/ 3 18. Consider the integral f (x, y) = [01:2 xy2 dxdy. (a) Evaluate the integral. (b) Sketch the region of integration. (0) Reverse the order of integration. l9. Reverse the order of integration in the following double integrals. DO NOT evaluate the integrals. ’ (a) Jg/zfnx(xsiny)dydx (b) [12 [1:(x2+ y)dydx 20. Consider the integral 2 4—x2 2y J- J‘ xe dydx 0 0 4— y (a) Sketch the region of integration. (b) Reverse the order of integration. _ 2 J4 , 1 x e"(x2+y2)dydx 0 0 (a) Change the Cartesian integral into an equivalent polar integral. l: (0) Evaluate the integral. 21. Consider the integral (b) Evaluate the integral. 22. For each of the following integrals: 1') describe (in words or with a sketch) the region of integration and ii) evaluate the integral. (a) I; [5052 + y3)dx dy (b) If Lyzyneosxdxdy (c) I: L2; ye‘xs dx dy -4, , I x _ 23.(a) Consider the integral J; frflr, y)a§1dx. Sketch the region ofintegrafion. Reverse the order of integration, Evaiuare the integral for f (x, y) = J37 + y2 . 1 x—l—p (b) Consider the integral In I: f (x, y)dydx Where p is a positive number. Sketch the x‘ 17 region of integration. Reverse the order of integrafion. Use your results in? (a) to evaluate the integral when f(x,y) zer/y—p +{y~p)2 . ExPlajn. 24. Find the area of one ieafofthe rose :- 2: sin(39) . 25 . Sketch the curve r = 3511029) . Find the intersection points of the curve and the circle r = cos 9 . 26. Consider the half-annulus given by y .>_ 0 and I: (12 + fl 3 4 as seen in the figure. The density is uniform. Calculate the y coordinate of the center of mass. (or 0} r a 27. Find the centroid (ie, the center of mass with uniform density) efme region bonded by y = 1——Jc2 and filer—axis. Fund 4411; mass 4 HM «Eh/L51 were covering Hm rz%«i,om gawm «the Wv'eg 32/3,}?! lye/7:137 Md 1350,1¥ W Msii’? £5 593$):exzit32: ...
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