Calculus 4 Final Exam Review / Winter 2009

Transcription

1 Calculus 4 Final Eam Review / Winter 9 (.) Set-up an iterate triple integral for the volume of the soli enclose between the surfaces: 4 an 4. DO NOT EVALUATE THE INTEGRAL! [Hint: The graphs of both surfaces an the projection of their curve of intersection are shown in the graph below.] Surfaces & Intersection The volume is boune above b the surface: 4 an below b the surface: 4. These two surfaces intersect along a curve line whose projection in the -plane is foun as follows. Page of 7

2 4 4 V 4 4 (.) Epress the given integral as an equivalent integral in which the -integration is performe first, the -integration secon, an the -integration last. DO NOT EVALUATE THE INTEGRAL! [Hint: The graphs of all si surfaces an the projection of the curve of intersection of the re surface: an the blue surface: 4 are shown in the graph below.] 4 4 Intersecting Surfaces Page of 7

3 Here are the equations of the si surfaces, one each for each of the si limits of integration. 4 4 Since the -integration is to be performe first, we must integrate from surface to surface with repect to "". Therefore, we nee the equations of the two surfaces that inclue "", namel the surfaces: an 4. We solve for "" an thus obtain the two epressions we nee for the surfaces that boun "" from above: 4 an below:. These become our limits of integration for "". upper 4 lower Since our first integration is from surface-to-surface parallel to the -ais, our secon integration must be from curve-to-curve in the coorinate plane that is perpenicular to the -ais, namel the -plane, whose equation is. Here is a graph of the showing the trace of the projection of the curve of intersection of the two surfaces that boun "", the curve of interesection of the two surfaces that boun "" an the curve of intersection of the two surfaces that boun "". Taken together, these three curves constitute the bounar of the region in the -plane that the values of "" an "" are constraine to remain within or on. Bounar Region Page of 7

4 Fin the projection of the curve of intersection of the surfaces that boun "" Fin the projection onto the -plane curve of intersection of the surfaces that boun "". Fin the projection onto the -plane of the curve of intersection of the surfaces that boun "". Fin the points of intersection of the three bounar curves. 4 (, ) (, 4) The secon integration is to be performe with respect to "". We thus integrate from curve-to-curve in the "" irection. upper 4 lower Page 4 of 7

5 The thir an final integration is to be performe with respect to "". We thus integrate from number-to-number in the "" irection. upper lower Now that we know all si limits of integration, we can write the triple integral featuring the orer of integration that was specifie. 4 4 (.) Set-up a ouble integral in Polar Coorinates that equals the volume above the cone: an below the circular parabaloi : (, ) 4. Circular Parabaloi & Cone Page 5 of 7

6 Fin the intersection of the surfaces. The projection of that circle onto the -plane is the polar region of integration. 4 Convert to polar coorinates. r 4 r Solve for "r". r r 4 r This is a circle of raius r. V ( 4 r ) ( r ) r r θ (4.) Set-up a ouble integral or integrals in Polar Coorinates that equals the AREA of the escibe region. (a.) The AREA of the region common to all three circles: r( θ), r( θ) sin( θ) an r( θ) cos( θ). 6 A c sin θ r r θ 6 r r θ cos θ r r θ Page 6 of 7

7 Polar Areas 9 r( θ) 5 45 r( θ) r( θ) θ (b.) The AREA of the region insie the circles: r( θ), r( θ) sin( θ) an outsie the circle: r( θ) cos( θ). A 6 6 r r θ sin θ r r θ A c Here " A c " is the area common to all three circles, B computer, A c.44. A This, of course, is equivalent to Shengi (Ma) Yuan's quarter-circle brilliant insight. But thanks to Ben Rockstroh's ver clever observation that I coul simpl subtract the common area, I was able to get it right the har wa. Thanks to ou both. (5.) You are given this function:f(, ) region 9 below.] 4 an the elliptical 6. [ The graphs of f(, ) an the region: "R " are shown 4 Page 7 of 7

8 f(,) & Region: "R" (a) Locate ALL relative maima, relative minima, an sale points both within an outsie the Region "R". f(, ) 4 6 f 4 f,, There are horiontal tangent planes at the points: (,, ), (,, ), (,, ). Now we etermine their natures. Note that all three points are within the bounar of the ellipse. Page 8 of 7

9 f 4 f f f (, ) f (, ) f (, ) D(, ) f (, ) f (, ) f (, ) > Therefore, the point: (,, ) is a relative maimum or minimum. Since f < an/or f < the point: (,, ) is a relative maimum. f (, ) f (, ) f (, ) D (, ) f (, ) f (, ) f (, ) < Therefore, the point: (,, ) is a sale point. f (, ) f (, ) f (, ) D (, ) f (, ) f (, ) f (, ) > Therefore, the point: (,, ) is a relative maimum or minimum. Since f < an/or f < the point: (,, ) is a relative maimum. page 9 of 7

10 (b) Fin all bounar points at which the absolute etrema can occur. We break up the ellipse into an upper an lower branch an eam each branch separatel for etremals. u l 9 9 f(, ) 4 6 v f, u v f, u v 4 9, 5, 4 5 We get the same values for "" for the lower branch. Here are the points: (,, ), (,, ), 5,.76,.5, 5,.76,.5, 5,.76,.5, 5,.76,.5 Now we inclue the points: (,,.56), (,,.56) that we broke the ellipse into branches. pts. (c) Use the results ou obtaine in parts (a.) an (b.) to etermine the points at which the absolute maimum an absolute minimum occur. page of 7

11 Here are the caniates: (,, ), (,, ), 5,.76,.5, 5,.76,.5, 5,.76,.5, 5,.76,.5, (,,.56), (,,.56), (,, ), (,, ), an (,, ). Comparison of the "" values of all 4 points reveals that the absolute minimum occurs at the point: (,, ), (,, ) an the absolute maimum occurs at the points: (,, ), (,, ). (6.) You are given this function:f(, ) 4. 6 f(,) page of 7

12 (a) Determine the equation of the tangent plane to f(, ) at the point: (,, ). f, ( ) f, ( ) f(, ) 4 6 f 4 f f(, ) f(, ) ( ) (b) Determine the parametric equations of the normal line to f(, ) at the point: (,, ). h(,, ) 4 6 Δh 4 i j k Δh(,, ) i j k Here are the parametric equations of the normal line to the surface. t () t () t t () t page of 7

13 (c.) Fin a unit vector in the irection in which " f(, ) " increases most rapil at the point: (,, ) Δf(, ) f(, ) i f(, ) j Δf(, ) 4 i j Δf(, ) Δf(, ) Δf(, ) i j i j j (.) Fin the rate of change of " f(, ) " in that irection at the point: (,, ). s f(, ) ( j) j s f(, ) (e.) Fin parametric equations of the tangent line to " f(, ) " in that irection at the given point: (,, ). t () t () t t () t Here are all the relevant graphs. page of 7

14 f(,) an Particulars at (,,-) (7.) Fin the equation of the plane that contains the line L: t t tan is parallel to the intersection of the planes: an. If we take the cross prouct of the normals of both planes, we obtain a vector: n that is parallel to both planes. n ( i ) n n j k j k i j k page 4 of 7

15 The vector: v i j k is parallel to the line containe b the unknown plane. The vector: v n is perpenicular to the unknown plane. ( i ) v n i j k j k i 5j k Choose a point: (,, ) on the line that is containe b the unknown plane. Here is the equation of the esire plane. 5( ) 5 5 Parallel Plane to Line of Intersection page 5 of 7

16 (8.) Evaluate the iterate integrals. (a.) u u u upper u lower u u u u page 6 of 7

17 (b.) 6 cos cos sin sin sin cos 6 cos v u u v u v u u v sin v cos cos sin cos 6 cos 5 page 7 of 7