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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
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- Fall '06
- PANTANO
- Multivariable Calculus