Final Exam Key

# Final Exam Key - M408D-AP Final Exam[I ﬂy Name Printed...

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Unformatted text preview: M408D-AP Final Exam [I ﬂy Name - Printed Show your work on problems 3—13. Put all answers on this test paper and circle your answers. No calculators, etc.. Maximum score = 100 1. Find the limits if they exist , nglnn a) hm 2n _—__: O (4) b) lim nZ—ntan_ln :0 n—voo 2 (4) 2. Classify as absolutely convergent, conditiona . . . r divergent _1 n3” (7) lax/42:3 "A, W2. ' maﬁa {71w perv-VB V\ 3” A VA 3 D 3 V\ C99 #3 n “‘ ' — (“U DU X: “S 5U!“ EU) 7 \ HAS m _L Ch“) ' 4. Let (3 = <2,—1,—1>, 5: (3,2,1) and let 0 2 angle between (1' and Find tanQ expressed as a fraction (e.g., 2/3). 6 ~ - ()Zv‘ TOLLHQH HQHCosé‘ 3 3:44—. : wmaﬂ ‘0 ““9‘W ﬁ‘ﬁs ‘ 5. Find the area of the triangle with vertices (1, 2, 3), (—2, 1, 2), (5, O, 2) (6) ,7 7 (/1351) ) l)—\7 a 5:4”75‘ ' A7 (30):.) C5 : <4) «1)‘\> 6. Find 0086 where 9 is the angle of intersection between 230 — y + z = 4 and F(t)=(t+1,2t—1,2t+1) #7 \> 7. Let m) :<1nt,\/E>, 2 g t g 4 a) Find a Cartesian equation for the curve (3) x 7f : 6 AA} /2_ 9:472“: b) Sketch the curve (label the end points) 2, / (£722) 42 Ami) c) Express as a deﬁnite integral (do not evaluate) the length of the curve. Simplify the integrand. 8. Find the equation of the tangent plane to so — z = 4aretan(yz) at (1 + 7r, 1, 1). j(xjg)%) :_ IM‘FQCC— Is 3093)?) F0’ .4) :O has —— (WWI—[>3 + 1mm +3(i 9— . 1 1 . , 1 1 9. Find the max and min of f(a:,y) : — + ~ subject to the constraint — + —2 = 1. CE 2:2 (8) 3;} L - “L‘l’iz y y 3 X133“ x1 7 . ' 1 _1A 2A> Vftxjv): /\ Ujjtx 57) U’C’MJ X“) 37") 4X3) ‘9” .1. ... 2A 10 r 3:? a PM Jo x:7 of: >934“) .L -22. ' 5:2) 31. k33 I? A] :1) 10. Express as an iterated triple integral in order “dz da: dy” the volume of bounded above by z : (1132—1—342)3/2 and below by the region D = {(m, y) : 0 _<_ y S V323 and 11:2 + 1/2 S 9}. Do not evaluate. (7) (3/1)3?)7(Z l”; 3 cl 11. Express as an iterated double integral in order “do: dy” the surface area of — 3/2 that lies above the my plane (do not evaluate). z=4~—31:2 12. Let //f(:z,y)dA= f f(m,y)dy>dw D 7, a) SketChQ 1‘ (2) E ) (5) e , e /1) (030.: (2)8 :‘e’)< c "J ...
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Final Exam Key - M408D-AP Final Exam[I ﬂy Name Printed...

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