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< 03 Test 4 910 November 2005
Math 119, Section 1, Fall 2005 Jason Grout
No calculators, notes, or books. Instructions: Read the questions carefully. Put your answers in the provided boxes. In order
to receive full credit, you will need to neatly show your work on these pages and simplify your
answers appropriately. Do not attach extra pages. If the provided space is insufficient, use the
blank sides of adjacent pages. 1. (5 pts) Which of the given graphs corresponds to the function 2 = $4 — 312? Circle the
correct graph. (Warning: do not pay attention to the scale of the axes in the graphs
below). 2. (5 pts each) Let f (it), y) = —m2 — 41:31; + 63””? Find the following. (a) f(—1,2) (b) Mar, .71) (e) fszv, y) (f) Z—yﬁ"<x,y> ~2 ’L/x 1'2 6 I:
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®
(B 4 3
(3a ?x+25 3x +23 X ,\ , .
H + 4
“V I: “r
gs _ . W 15 3. (5 pts each) Let f($,y) = (3y — :5)? (a) Find fm(2, 1). Write a sentence or two interpreting how this value affects the graph
of f (9:, y) .gttzaejx) 14H (2,0, as M MMM W )éidt‘rM/Iém [13,} [mpg/Jab ﬂea X'ﬂwl‘s Wd‘ﬁfugml'
\Y\ 'WFOEJ'VW X' W‘J’M (b) Find fm(2,1). Write a sentence or two interpreting how this value affects the
graph of f(a:,y). w
XX Cgllxﬂé MW lh4l¢£ xiél/I‘I‘ﬁh‘l Jrcrl’t‘g WWVW' L :
;.'£Mc§/.w ﬂu laxMM;pr j>// #wﬂ «malice Mp2. n; L. 4. (4 pts) Suppose fz(a:,y) = 2x — 6y and fy(x,y) = 12y2 — 6513. Find the critical points
of f. Do not test to see if those critical points are relative extrema or saddle points. ’; : X33¢ (0)0)
Mm 0 > I 5. (8 pts) Suppose we have the following table of values for the partial derivatives of
f(:r, y). For each of these points (a, b), fz(a, b) = 0 and fy(a, b) = 0. Find and
classify the relative extrema and saddle points. If there is not enough information to determine whether a point is a relative extrema or saddle point, then put “Not enough
information” (61,19) fxz<aab) fyy(avb) fmy(aab)
1 (—1, ) 0 1 —3
(2,1) —1 —4 (3,2) 3 —2 (1,4) 2 2 l wa ./ ,‘,_ :— ' (—1,1)2
D{_"\) ,0 ‘1 740 5AM? T. D c2,» = Lnwu'fw <2a1>= NatW74 a; 7 Mal) ': 36$) ’06) <0 (3 2
' V6. 91 L0 21*! 9370 (1,4); Kala/>3 \, 7
70 g‘
6. (8 pts) Let {2” W n
3
f($’y)=$2—2y$+y§+22. Find and classify the locations of relative extrema and saddlepoints of f(a:, (DID) 5,4DDL6 1 2y '2? :0 '5) X'Tj £1,13ﬂﬁbkﬂrlg MIN.
x 7. (8 pts) Using Lagrange multipliers, ﬁnd the location of possible extrema of
f(:v,y,2) = 332 +312 + 22
subject to the constraint 32: + 2y + z = 14. A "L 'L’ l got/mt) ’ 3x myrz ""l _ , . ,,x
lull/m MD a) V2" W2 5
: 21 :
«if*5?
11*0 ‘19 3% ’r 1C?) ‘LCEXYH
Uﬁ‘ﬂl /
{j :7 l"\ :[L( 90 XZB'
/ 2
3 {hr/l
fly/“VA 8. (3 pts) The rate of change of velocity v of a boat in water is proportional to 100 minus
the current velocity. Set up a differential equation modeling the speed of the boat. dy _ 3$2+1
d$_ y 9. (8 pts) Use Euler’s method to approximate the value of y(6) if and y(0) = 1. Use a stepsize of h = 3. 10. (8 pts) Find the general solution to the differential equation giftq = yﬁ.
d2:
Please show every step of your work.
if;
1.. : It 5 ’ m6
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\ Ax ‘ w / 5 a M ' 3 r
*L/
, if; . I;
l“ L U 2 m zﬁ &
l \‘ 5 6°
‘5 M wk N‘ I
0‘ V” M” fl 037?; w le’k gl “8 ’ 11. (8 pts) Find the particular solution of the initial value problem dy a: _ _ _
EE~E, y—2whenx—1. You do not have to solve for y in your solution. ‘5
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J‘—
24W M '1
x C/
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 Calculus

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