Review Test 2. c ) is a local maximum. ) < 0, then the graph of f has a saddle point at ( c,, (, c ) = 0, no conclusion can be reached by this test.

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1 eview Test I. Finding local maima and minima for a function = f, : a) Find the critical points of f b solving simultaneousl the equations f, = and f, =. b) Use the Second Derivative Test for determining local etrema and saddle points. Let D, = f, f, [ f, ] ) If D c, c ) > and f c, c ) >, then f c, c ) is a local minimum. ) If D c, c ) > and f c, c ) <, then f c, c ) is a local maimum. 3) If D c, c ) <, then the graph of f has a saddle point at c,,, c f c c )). 4) If D c, c ) =, no conclusion can be reached b this test. II. Finding absolute etrema for the continuous function = f, on the closed and bounded region : The points whose functional values need to be checked as possible points where the absolute ma or absolute min ma occur are: ) the interior critical points ) the points where the "boundar curves" intersect 3) interior critical points along the boundar curves Check the functional values of each of these points. The largest of those values will be the absolute maimum of f on and the smallest of those values will be the absolute minimum of f on. Note: You do not need the function D, at all in these problems if ou are just looking for absolute etrema. III. Double Integrals ) Evaluation of double integrals ) Setting up double integrals for tpe I and tpe II regions 3) eversing the order of integration 4) Applications - a) volume under = f, and above the region in the -plane, V = f, da b) area of the region A = da c) mass and center of mass and moments of a lamina in the shape of the region, δ, da, = δ, da, = = δ, da

2 d) average value of the function = f, defined on, av f ) = f, da area of V. Triple Integrals ) Setting up of triple integrals - deciding on the order and finding the limits ) Volume of the three-dimensional region, V = dv 3) ass and center of mass = ass δ,, ) dv = oment with respect to the -plane δ,, ) dv = oment with respect to the -plane δ,, ) dv = oment with respect to the -plane δ,, ) dv Center of mass is,, ) where: =, =, = Practice Problems. If f, = , find the local etrema and saddle points of f.. Find the coordinates of all points whose functional values must be checked to be assured of finding the maimum and minimum values of f, = on the region triangular region with vertices,),,), and,- 4). How would ou then find the maimum and minimum values of f on? 3. A thin metal plate is in the shape of the region bounded b the graphs of = and = and has densit functionδ, = + ) + ). Which of the following represents the mass of the plate.

3 a) + ) + ) d d b) + ) + ) d d c) + ) + ) d d 4 4. Evaluate: a) + dd d) + ) + ) d d π sin b) e cos d d 5. Write the given integral or sum of integrals as an iterated integral with the order of integration reversed: a) f, d d 4 3 b) f, d d + f, d d 4 6. Find the volume of the solid in the first octant bounded b the graphs of + = 9, =, =, =. 7. Set up a double integral to find the volume of the solid bounded b the graphs of = + 4, = 4 -, + =, =. 8. A thin metal plate has the shape of the region bounded b the graphs =, 3 = 4, and the -ais. The densit function is δ, =. Set up the integrals needed to find the center of mass Use polar coordinates to evaluate the integral cos + ) d d. Set up as a double integral in polar coordinates: + d d. a) Find using polar coordinates the mass of a lamina in the shape of the upper quarter of the circle of radius two centered at the origin if the densit at an point is equal to three times the distance from the origin to the point. b) Find using polar coordinates the mass of a lamina in the shape of the region bounded b the lines =, = 3, =, = if the densit at an point is δ, =. Hint: sec ln sec tan + t dt = t + t + C 3. Evaluate + + 4) d d d 3. Set up the triple integral needed to find the volume of the region bounded b the graphs of =, =, + = 4, and =.

4 4. Set up a triple integral to find the volume of the region bounded b the graphs of = 4, + + =, =, and =. 5. Set up the triple integral needed to find the volume of a solid in the shape of the region bounded b the graphs of = and = 5. Answers to eview Problems. Local min at 3,), local ma at -,-), saddle points at 3,-) and -,)..,),,),,-4), 4/9,/9),/7,/7),,-),4/9,-6,9) 3) B 4. a) 6/3 b) e- 5. a) f, d d 4 3 b) f, d d 6. 8 cu. units / ) dd 9. πsin π/ cosθ π / 4. r drdθ = d d, = d d, d d = 4. a) 4π b) ln + 3). 39/ 3. d d d d d d d d d and d d d 4 5. V = d d d

5 ath 4 June 3, Eam Wells Answer the ultiple Choice section on this test paper. Write everthing else that ou wish to be graded on separate paper. You must show all steps to our solution for the Written esponse section in order to receive full credit; however, ultiple Choice questions do not require eplanation. I ma give partial credit on ultiple Choice responses provided our work is shown and ou had the correct concepts in mind when attempting to answer the problem. Don't be afraid to use complete English sentences. If I were ou, I would de nitel take a moment to look at all 4 pages of this test. Although, I ma seem harsh, I am not without compassion of some form.) Alwas use correct mathematical notation and terminolog. You ma use calculators on an portion of this test; however, ou ma not use an smbolic or graphing abilities that our particular calculator ma have, i.e. di erentiation is a smbolic function. Furthermore, decimal approimations do not qualif as justi cation. When ou are nished, please sign the Honor Pledge. ultiple Choice 5 points each). Which of the following equations is not equivalent to the others. Wasn't that a game on Sesame Street?) a. cot Á = 3 b. = r p 3 c. = p d. Á = ¼ 3 ; ¼ 3 e. =jrj p 3. Convert p 3; ¼ 3 ; ¼ ) from the Spherical Coordinate Sstem to the Clindrical Coordinate Sstem assuming r > and µ ¼. a. 3 ;p 3 ; 3¼ ) b. p 3; ¼ 3 ; ¼ ) c. 3 ; 3¼ ; p 3 ) d. 3 ; 3¼ ; ¼ ) e. ; 3 ; p 3 ) 3. The equation of the tangent plane to the surface = e at the point ; ; e) is: a. + + e = b. t) = + t t) = t t) = e t e c. + = 4e d. e = e. t) = t t) = t t) = e + t

6 ath 4, Eam Wells Page 4. The rate of change of f;;) = ln at the point P; 4; ) in the direction of ~v = 8~i 4~j + ~ k is: a. 9 b. 6 p 5 c. d. 7 e. 5. Which of the following is moderatel eas for us to integrate b hand without an change of coordinate sstems or change in order of integration? You need circle onl one. Ultimatel, the're all integrable, but some are de nitel more di±cult than others.) Z ¼ Z ¼ sin a. d d Z p 4 Z p + b. c. d. e. Z Written esponse points each) p 4 Z 6 Z 4 Z Z Z Z p d d d p + cos 6¼ 5 d d p ) d d e d d 6. Set up the integration one would need to nd the mass of a thin plate bounded b = and = p 4 if the densit at an point in the plate is equal to that point's distance from the origin. You ma use either applicable coordinate sstem, but remember, the Cartesian Coordinate Sstem onl uses,, and as its variables, while the Polar Coordinate Sstem onl usesrand µ as its variables. You can't mi and match variables between coordinate sstems. 7. Assuming the same region as the previous problem but constant densit, set up and evaluate the integration one would need to evaluate the moment about the -ais. Hint: one particular coordinate sstem has de nite advantages over the other.) 8. Evaluate Z Z Ã sin +! 3 +) d d 9. Using the Cartesian Coordinate Sstem, set up the integrations) necessar to calculate the average densit of the solid formed b the region cut from the clinder + = 4 b the plane = and the plane + = 3 if the densit at point ;;) is given b ±;;) = 4. Hint: Think about it... we've onl done one thing in recent histor involving averages, but ou could approach this problem di erentl if ou'd like.)

7 ath 4 Second Summer Test Name Directions: Show ALL work on separate paper. Number each problem as ou work it and print our name on each sheet. No credit will be given for unsupported answers.. Convert, / 4, π / ) q p). π from spherical to clindrical coordinates. r > and. Find an equation of the plane tangent to the surface = ln ) at the point where = and = Given f, = , make a list of all candidates for absolute ma and min on the closed triangle bounded b the lines, =, =, and = in the first quadrant. 4. Set up the integration in rectangular coordinates necessar to find the mass of a solid bounded b = 6 and = + if the desit at an point is equal to the distance from the origin. Do not evaluate. 5. epeat problem 4 using clindrical coordinates. Do not evaluate. 6. Evaluate carefull. Show all steps, leave an eact answer. 3 e d d 7. Convert to an equivalent integral in polar coordinates. Do not evaluate everse the order of integration. Do not evaluate. 4 e d d 9. Find the formula for the volume of a sphere of radius a b setting up and then evaluating a triple integral in spherical coordinates.