CALCULUS 4 QUIZ #2 REVIEW / SPRING 09
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1 CALCULUS QUIZ # REVIEW / SPRING 09 (.) Determine the following about the given quaric surfaces. (a.) Ientif & Sketch the quaric surface: +. 9 In planes parallel to the -plane an planes parallel to the -plane, the traces are parabolic. Thus, this a parabaloi. In planes parallel to the -plane, the traces are ellipses. Therefore, this an elliptic paraboloi This "sketch" is computer-generate. Elliptic Parabaloi Page of 0
2 This "sketch" is han-generate. (b.) Ientif & Sketch the quaric surface the quaric surface:. In planes parallel to the -plane an planes parallel to the -plane, the traces are parabolic. Thus, this a parabaloi. In planes parallel to the -plane, the traces are hperbolas. Therefore, this a hperbolic paraboloi This "sketch" is computer-generate. Page of 0
3 Hperbolic Parabaloi This "sketch" is han-generate. Page of 0
4 (c.) Ientif & Sketch the quaric surface the quaric surface: + 6. In planes parallel to the -plane an planes parallel to the -plane, the traces are hperbolic. Thus, this a hperboloi. In planes parallel to the -plane, the traces are circles. This, nonetheless, is calle a hperboloi of sheet. This "sketch" is computer-generate. Hperboloi of One Sheet This "sketch" is han-generate. Page of 0
5 (.) Ientif & Sketch the quaric surface: + 0. In planes parallel to the -plane an planes parallel to the -plane, the traces are straight lines. Thus, this a cone. In planes parallel to the -plane, the traces are circles. Therefore, this is a circular cone, which is a special case of the elliptic cone. This "sketch" is computer-generate. Page 5 of 0
6 Elliptic(Circular) Cone This "sketch" is han-generate. Page 6 of 0
7 (e.) Ientif & Sketch the quaric surface:. In planes parallel to the -plane an planes parallel to the -plane, the traces are hperbolic. Thus, this a hperboloi. In planes parallel to the -plane, the traces are circles. Since > 0, this a hperboloi of sheets. This "sketch" is computer-generate. Hperboloi of Two Sheets Page 7 of 0
8 This "sketch" is han-generate. (f.) Ientif & Sketch the quaric surface: + +. In planes parallel to the -plane, planes parallel to the -plane, an planes parallel to the -plane the traces are all ellipses (circular in planes parallel to the -plane which is a specical case of ellitic). Thus, this a specicialie ellipsoi ( prolate sphereoi.). This "sketch" is computer-generate. Page 8 of 0
9 Ellipsoi(Prolate Sphereoi) This "sketch" is han-generate. Page 9 of 0
10 (a..) Fin traces in the coorinate planes of the quaric surface: + + an sketch trace in the -coorinate 9 5 sstem. Page 0 of 0
11 plane plane plane Page of 0
12 This is an ellipsoi. Here is a sketch. Page of 0
13 (b..) Fin traces in the coorinate planes of the quaric surface: + an sketch trace in the -coorinate sstem. plane plane The trace is the origin. plane Page of 0
14 This is an elliptic paraboloi. Here is a sketch. Page of 0
15 (c..) Fin traces in the coorinate planes of the quaric surface: + an sketch trace in the -coorinate 9 6 sstem. plane plane plane Page 5 of 0
16 This is a hperboloi of one sheet. Here is a sketch. (.) Determine the following about the given partial erivatives. (a.) Fin f (, ) an f (, ), where f(, ) e. f e + e f e + e f (, ) e + e f (, ) e + e f (, ) e f (, ) e Page 6 of 0
17 (b.) A point moves along the intersection of the given elliptic paraboloi (, ) + an plane. At what rate is " " changing with " " when the point is at (,, )? Also, fin the equation of the tangent line at that point. Paraboloi & Plane (, ) + (, ) The tangent line lies in the plane. Therefore, "" is fie. Accoringl, these are the equations of the tangent line to (, ) at the point (,, ) in the "" irection. t () + t t () t () + t Page 7 of 0
18 (c.) Calculate ln + ( ) (, ). using implicit ifferentiation where ( ( )) ln + ( ) ( ( ) + ) + ( ) (.) Let f (, ) cos( ). Fin f, f, f, an f. f cos( ) f sin( ) Page 8 of 0
19 f sin( ) f sin( ) f cos( ) f cos( ) (.) Determine the following about ifferentials. (a.) Fin the local linear approimation " L (, )" to f (, ) + at the point P(,, 5) an compare the error in approimating "f" b "L" at Q(,, 5) with the istance between "P" & "Q". L (, ) f 0, 0 L (, ) 0 L (, ) 5 ( ) f ( 0, 0 ) ( 0 ) ( ) + ( 0 ) ( ) ( ) + + f 0, ( ) + 5 ( ) 0 + ( 0 ) ( ) ( 0 ) 0 ( 0 ) + 0 ( ) ( 0 ) L (, ) ( ) + 5 ( ) f (, ) 5 5 Page 9 of 0
20 PQ L (, ) f(, ) PQ 5 (b.) A function f(, ) an its Linear Approimation at some point P 0, 0 (, ) + 8. Determine the point "P". L (, ) 0 ( ) 0 ( ) is L ( ) ( ) ( 0 ) ( 0 ) 0 ( ) ( ) ( ) ( 0 ) 0 0, 0 0 P (, ) (c.) Accoring to the ieal gas law, the pressure, temperature, kt an volume of a confine gas are relate b P where "k" V is a constant. Use ifferentials to approimate the percentage change in pressure if the temperature of the gas is increase b % an the volume is increase 5%. Page 0 of 0
21 P T P T + V P V T P k V V P kt V P V k PV T T P PV T V P T V P P V P ΔP P T T + P T ΔT + P V P V V ΔV ΔP P ΔT T ΔV V 0.0 T 0.05 V 0.0 T V ΔP P 0.0 ΔP P 00% % Page of 0
22 (.) Determine the following about the Chain Rule. (a.) Fin t an then fin the value of t at t 0where t + an u + v, u v, u t +, v +. ( ( u() t, vt ()), ut ( (), vt ())) t u t u + v t v + u t u + v t v u v u v t u t v t u t u v t v t u 6 t u v t v t t 0 t t 0 t ( ) 0 Page of 0
23 (b.) Fin u an v where e sin( ), u v, v. ( ( u, v), v ( )) u u + u u 0 u u v v + v e sin( ) cos( ) ( ) e sin( ) cos( ) ( ) ( ) ( ) e sin( ) cos( ) ( ) v cos u v e sin u v ( ) ( ) e sin( ) cos( ) ( ) u v cos u v e sin u v u u v v u u 0 v v v Page of 0
24 u ( ) ( ) u v cos u v e sin u v (c.) Fin u an ( ( ) v where ln cos, v tan( u). ( ( u, v) ) u u v v ( ( )) ( ) cos( sin ) ( ) tan v tan( u) tan v ( tan( u) ) u v ( sec( u) ) v tan( u) u v tan( u) ( sec( u) ) tan v ( tan( u) ) v v ( tan( u) ) tan v ( tan( u) ) Page of 0
25 (.) Determine the following about Directional Derivatives. (a.) You are given the equation of the sphere: + + an the point: (,, ). f(,) & Steepest Tangent (a..) Fin a unit vector in the irection in which " " increases most rapil at the point:,, ( ) Since the point is locate in the upper hemisphere, f (, ) (, ). Δf(, ) f(, ) i + f(, ) j Page 5 of 0
26 We can calculate the partial erivatives b ifferentiating f (, ) (, ) eplicitel or implicitel. I chose to o it implicitl. ( ) + + ( ) + + Δf(, ) i + j Δf(, ) i + Δf(, ) i + Δf(, ) j j (a..) Fin the rate of change of " f (, ) " in that irection at the point:,, ( ) s f (, ) i + j i + j s f (, ) Page 6 of 0
27 (a..) Fin parametric equations of the tangent line to " f(, ) " in that irection at the given point:,, ( ). t () t t () t t () + t t () + t t () + t t () t (a..) Fin a unit vector, u 0, in the irection in which " " has a slope of ero at the point: (,, ). A ero slope is in the irection of the level curve at the given point Since an are both positive in the neighborhoo of the given point, the trace of the level curve is given b r ( ). r ( ) i ( ) ( ) ( ) + j + k r ( ) i ( ) + j r ( ) i ( ) + j ( ) u 0 i + j or u 0 i + j Page 7 of 0
28 Δf(, ) u 0 0 Alternativel, we coul have etermine u 0 as follows. u 0 u u Δf(, ) u i + j i u + j u 0 ( ) u u u u i ( ) + j ( ) u 0 u u u i ( ) + j ( ) i u + j or u 0 i + j (a.5.) Fin parametric equations of the tangent line to " f(, ) " in that irection at the given point:,, ( ). t () + t t () t t () Page 8 of 0
29 Steepest & Shallowest Tangent Lines (a.6.) Fin the rate of change of " f (, ) " in the irection a i + j at the point: (,, ). s f(, ) Δf(, ) a a i + j ( i + j ) 0 Page 9 of 0
30 s f(, ) 5 (a.7.) Fin the parametric equations of the tangent line to " f(, ) " in the irection a i + j at the point: (,, ). t () + t t () + t t () t Steepest, Shallowest & Representative Tangent Lines Page 0 of 0
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