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Homework 9

# Homework 9 - Vincent(3mv777 w— HW 09 — Gilbert...

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Unformatted text preview: Vincent (3mv777) w— HW 09 — Gilbert — (57495) ' 1 This printmout should have 18 questions. Multiplechoice questions may continue on the next column or page - find ail choices before answering. 001 10.0 points Determine fee: when ﬁx, 9) = ﬁle/2005271;- 1. fym m —y2(3 cos my + my sin my) 2. fw = 21.12 (3 cos my m 323; sin my) 3. fym = 2m2(3 cos my —— my sin my) 4. fm m ~\$2(3 cos my ~i~ my sin my) 5. fym = _2:22(3 sin mg + my cos my) 6. fym = 322(3 sin my —— my cos my) '7. fym m 312(33‘1I133ymmycoswy) 8. fyx = ~2y2 (3 sin my + my cos my) cor— rect Explanation: By the Product Rule, fy = 43; cos my — 233342 sin my. But then fym m ~4y2 sin 33y m 2y2 sin my — 2.93113 cos my . Consequently, fxy = —2y2(3 sin my + 331,: cos my) . 002 10.0 points Determine 82/0y when z m 2(x, y) is de— ﬁned by 3m2+2xy+5yz+z2 = 7. 1 azm 2y~55c '3ym 5\$+2y '62 ‘ mes: 2. m 2 (33; 59: + 29 3 9E — W233+5z correct ' 8y — 5y+22 4 32:“ ___ M2\$~l~52 ' 5y ” 22: 5 if _ BmMSE I 8y - 2z Explanation: Differentiating z implicitiy with respect to y, we see that 2x+52+5y§~z~+2z§£ = 0. 8y 8y Consequently, :93 W _ 2:1: + 52 6:9 m 5y + 22 ' 003 10.0 points Determine fxy when f(m, y) “:2 l:cis'ciiii"1(§m>. 2 :1: 1- fey : 3253112 2- few 7 m 3' f“? 7' 7%? ‘ 4. fm 3 % correct 5' fe c” e533??? 6. fey : 332\$:ny Explanation: Vincent Univ???) ~ KW 09 .... Gilbert _._ (57495) 3 After diﬁerentiating a second time therefore, we see that firm = 4-75: Consequently, fmfyy — (fay? = 16:13 .. 35. 006 (part 1 of 4) 10.0 points The productivity of an automobile manu- facturing company is given by the function ﬂay) m 80M . with the utilization of cc units of labor and y units of capital. (1) Find fm and ﬁg. fry : 6a fyy m 4- ll 1. f3; aim/g, fy m 40m .’ 1 I 1 2. m -——- r; 40 m...— fa: 40-1133}? fy my 33 3. f\$ = KKK/g, fy = 40\/~: correct x y i r m a 4. fx _ 80¢, fy eoﬂ 5.1;. = 401?, fy : 4055 Explanation: After differentiation ﬁrst with respect a: and then with respect to y we obtain W_ r ﬂ_ a am‘.4O\/;’. 8y—4‘Ol/r' Thus _ 3% m... 5’3 'fx—40\/;,_ fym40\/y' . 007 (part 2 of 4) 10.0 points (ii) if the company is now using 40 units of labor and 360 units of capital, ﬁnd the marginal productivity of labor. 1. marg. prod. labor m ~— 120 correct 2. marg. prod. labor 3. marg. prod. labor m 80 1 4. marg. prod. iabor m ~39 5. marg. prod. labor 2 480 Explanation: The marginal productivity of labor is fx. So when a: = 40 and y = 360, the marginal productivity of labor is given by marg. prod. labor 2 120 . ‘ 008 (part 3 of 4) 10.0 points (iii) If the company is now using 40 units of labor and 360 units of capital, find the marginal productivity of capital. 80 1. marg. prod. capital m ~7- . 40 2. marg. prod. capital : ~3— correct 3. marg. prod. capital = 120 4. marg. prod. capital m 480 5. marg. prod. capital Explanation: The marginal productivity of capital is fy. So when m = 40 and y = 360, the marginal productivity of capital is given by marg. prod. capital = "4:9?" 009 (part 4 of 4) 10.0 points Vincent (jmv777) m HW 09 W Giibert w (57495) 5 Where A2: = 0.04 and Ag 2 0.07. But 8,3 8,? —— m _____ = 2 . 6m 1032, 33.! y Consequently, A2 m 40xAm+6xAy m 2.02 .' keywords: differentials, multi—variable estiw mate, polynomial function 012 10.0 points Find the linearization, L(r1:, y), of ﬂat, y) 2: V9 + 3:132 “— fly? at the pOint 19(1, —2). 1 3 1. L(;t,y) = +-~i~-~m~§~2y 2 2 9 3 . L = —- — m2 ' 1 3 . m — — —-2 3 L(\$,y) +2+2x y 1 4. L(:L',y) a: +§+3mm2y 9 3 5. L(;c,y) = §+§m+2y correct 6. L(3:,y) = 2-3—3234-23; Explanation: The Iinearization of z = ﬂm, y) at the point P(a, b) is given by L(x,y) 2 f(a’>b) + ”2“":ch“ (was w a) + gin» minty“ b) Now when m y) = 9+ 32:2 - 2142, we see that 8f __ 3m while if M -Jm 3y 9 + 3x2 — 2y2 ' Thus at P, jaw W2) : 21 While iii 1' 5’: if. _ 2 63: (1,m2) ” 2’ 8y (1,—2) _ ' So at P the linearization of f is 3 WW) = 2+§(\$-1)+2(y+2), which after rearrangement becomes ,9 3 L(;‘t,y) = §+§m+2y . keywords: iinearization, partial derivative, radical function, square root function, 013 10.0 points Find an equation for the tangent plane to the graph of Hm) = m at the point P(i, 2,f(i, 2)). 1. 3m+2y+22~11= 0 2. 3m—2y-22+5 = 0 3. 2m+3y—2z~3 m 0 4. 2\$+3y—22——ii == 0 5-.m3mw2yi—22—3 m Ocorrect 6.2m—3y+22+5 0 Explanation: Vincent (jmv'???) — KW 09 — Gilbert m (57495) 7 By the Chain Rule for Partial Differentia— tion, ' ﬁe __ ﬁeﬁe gee 332113 dt " dealt aydt azdt' _ When 11) — may/z when a: = 252, y = 24, z a: 2+7t, therefore, cite a: 73:1; __ 2 2t y/z__ y/z___._. y/z. dt 3 28 9:2 e Consequentiy, keywords: partial differentiation, Chain Rule keywords: partial differentiation, Chain Rule keywords: partial differentiation, Chain Rule keywords: keywords: 016 100 points Use partial differentiation and the Chain Rule applied to F(a:, y) m 0 to determine 'dy/da: when F(:r:, y) w {308(3: ~ 5y) — :3er = 0. dy _ sin(a: — 5y) — 2\$€2y dm 2 8111(3: m 53;) m 5er dy sin(a: — 5y) — 2623’ 2. m = W dm 2 ein(m —— 5y) —- 5x62?! 3 fig _ sin(.r m 53;) + 62y '- dzz: — 556829 m 23in(a: m 5y) . _ - 2y 4. 013} — 3111(5: 5y) 4 8 correct dm .— 5 sin(\$ w 53;) w 22:63? dy sin(a3 — 5y) «I— 829 ' 5. W .—- M aim 2338231 m 5 sin(ac — 511) gig sin(x —— 5y) ~i— 2.73629 6. d2: 5 sin(m ~— 5y) — 62?! H Explanation: Applying the Chain Rule to both sides of the equation F(m, y) m D, we see that gee gee at d3: (93; dcc 3F 69de w 5:64.333:ng —- 0. Thus ' 8F it 2 “Es; 32 01:6 _3__F_ Pg 39 When F(:r, y) = cos(;t w 53;) ~— xe2y m 0, therefore, gig Z: _ —sin(;z: w 53;) w e29 05\$ 5 sin(m — 5y) — 232629 ' Consequentiy, 35E _ sin(a: —- 5y) + 82y 03:13 5811101: —— 5y) — 23:62?! ' 017 10.0 points The temperature at a point (:23, y) in the plane is T(:c, 3;) measured in degrees Celsius. if a bug crawls so that its position in the plane after t minutes is given by 2 m3 —t y +3} determine how fast is the temperature rising on the bug’s path after 3 minutes when :1: =1 6+t, mes) = is, 1;,(3, 5) = 6. 1. rate = 10° C/rnin ...
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Homework 9 - Vincent(3mv777 w— HW 09 — Gilbert...

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