Date Period For each problem, find all points of absolute minima and maxima on the given interval.

Transcription

1 Calculus C o_0b1k5k gkbult_ai nsoo\fwtvwhairkew ULNLuCC._ ` naylflu [rhisg^h^tlsi traesgevrpvfe_dl. Final Eam Review Day 1 Name ID: 1 Date Period For each problem, find all points of absolute minima and maima on the given interval. 1) y = ; [-3, -1] ) y = ; [1, ] For each problem, find the average rate of change of the function over the given interval. 3) y = - - 1; [1, ] ) y = - - 1; [0, ] For each problem, find the: and y intercepts, -coordinates of the critical points, open intervals where the function is increasing and decreasing, -coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maima. Using this information, sketch the graph of the function. 5) y = y Given the graph of f (), sketch an approimate graph of f '(). ) f() f '() S fm0z1_5z PKruPtFaA ESVoRfUtawWarPem slxl^cw.g N eaalplv frniugbhstks_ jrie`szerlvqe_dw.s T ]Mea[dueF SwOiitahy VIknHfEi_nBiOtFep \CFaklzcZuel`uvsw. -1-

2 For each problem, find the open intervals where the function is concave up and concave down. 7) y = ) y = + + For each problem, find the open intervals where the function is increasing and decreasing. 9) y = ) y = For each problem, find the values of c that satisfy the Mean Value Theorem. 11) y = - + ; [3, 7] 1) y = ; [-5, -3] A particle moves along a horizontal line. Its position function is s(t) for t ³ 0. For each problem, find the velocity function v(t) and the acceleration function a(t). 13) s(t) = t 3 - t + 1t 1) s(t) = t 3 - t - 5t Solve each optimization problem. 15) A supermarket employee wants to construct an open-top bo from a 10 by 1 in piece of cardboard. To do this, the employee plans to cut out squares of equal size from the four corners so the four sides can be bent upwards. What size should the squares be in order to create a bo with the largest possible volume? Solve each related rate problem. 1) A crowd gathers around a movie star, forming a circle. The radius of the crowd increases at a rate of 7 ft/sec. How fast is the area taken up by the crowd increasing when the radius is 1 ft? D EN0J1^5v jksuotwac XSioofotMwBaarne^ LLHLfCO.k r SlDlA r]irglhptbs[ brmeksheirdvgeqdc.q [ AMMatdceO ewyittshl \IXnDfhirnaiBtNe\ _CUafl^czutlfuvsZ. --

3 17) Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius of the spill increases at a rate of m/min. How fast is the area of the spill increasing when the radius is 10 m? For each problem, find all points of relative minima and maima. 1) y = ) y = For each problem, find the values of c that satisfy Rolle's Theorem. 0) y = ; [-3, -1] 1) y = - - 5; [0, ] For each problem, find the derivative of the function at the given value. ) y = at = 3 3) y = at = -3 For each problem, use implicit differentiation to find dy in terms of and y. d ) 1 = 3-5y 5) - y = 3 Differentiate each function with respect to. ) y = sin ) y = sin -1 - ) y = ln 3 9) y = ln 3 30) y = log 5 31) y = 35 3) y = - 33) y = 5 3 3) y = ( ) ) y = - (- 5 + ) O Rg0i1h5j HKcuitwal _SXomfGtowNaer`eX TLZLhCV.T A _AXlPlo zrmihgzhstcsb LrEeHsnecrXvbepde.Y B MMCaodpeF uwijtpho siwnffpiknfitrez ycqaflocludllunsm. -3-

4 3) y = - 37) y = ) y = cos 39) y = cos 3 For each problem, find the indicated derivative with respect to. 0) y = 5 Find d y d 1) y = - 5 Find d y d Evaluate each limit. ) lim ) lim ) lim - ( ) 5) lim - ( ) ) lim ) lim ) lim 3 (- + 5) 9) lim 3 K tw0[1g5i AKwuBtvat [S\oIfKtWwPaNrdew glrlgcw.f d QAzlQlb lrjisg_ht\sf \raews^errcvfepd\.b L gmgaddzel qwbiqho iilncffiinyictuei QCkaelOcLurlduJsG. --

5 1) Absolute minima: ( -3, - 1 ), ( -1, - 1 ) Absolute maimum: (-, 0) 3) -3 ) - 5) ) Answers to Final Eam Review Day 1 (ID: 1) y -intercepts at = 3-3, y-intercept at y = 3 Critical point at: = 3 Increasing: (-, 3) Decreasing: (3, ) No inflection points eist. Concave up: No intervals eist. Concave down: (-, ) No relative minima. Relative maimum: (3, ) f '() ) Absolute minimum: (1, -5) Absolute maimum: (, ) - - 7) Concave up: No intervals eist. Concave down: (-, ) ) Concave up: (-, ) Concave down: No intervals eist. 9) Increasing: (, ) Decreasing: (-, ) 10) Increasing: (-, ) Decreasing: (, ) 11) {5} 1) {-} 13) v(t) = 3t - t + 1, a(t) = t - 1) v(t) = 3t - t - 5, a(t) = t - 15) in 1) A = area of circle r = radius t = time Equation: A = pr r = 1 = pr dr Given rate: dr = 19p ft²/sec = 7 Find: 17) A = area of circle r = radius t = time Equation: A = pr Given rate: dr = Find: r = 10 = pr dr = 0p m²/min 1) No relative minima. Relative maimum: (-, 1) 0) {-} 1) {1} ) dy d = 5y 5) dy d = y Z Ol0F1n5H gkouutmar QS`oZfRthwXa^rdeS flllycd.r S halldlb prdi]gmhuthsy urwegszekrhvbefdl.e b UM[asdPeX lw`iotbh` oitnwfyikngiotiem zcvaol[c^uulpuzsb. -5- r = 1 r = 10 19) No relative minima. Relative maimum: (-, 1) ) dy = - 3) dy d d = 3 ) dy d = (- 5 ) -5 = = -3 =

6 7) dy d = (- ) -3 = - 30) dy d = 1 = 33) dy d = ln 5 ln 5 ) dy d = 1 3 = 3 31) dy d = 35 ln 15 3) dy d = (55 + 1) = ) dy d = (- 5 + ) - 3 = 3-3 3) 37) dy d = (3-5) ( 3-5) = ) dy d = = 3 3) dy d = - dy d = ( - ) - 3 ( - ) = ) dy d = -sin 1 3 = -1 3 sin 39) dy d = -sin 3 1 d 0) y d = 0 1) d y d = -03 = -1 sin 3 ) - 3) ) 5) ) - 1 7) 7 ) -1 9) 3 q R^0D1W5s `K]uqtHat _SQoSfatnwBaRrgez llflcc].s Y takl]lo Rr]iegzhHsi SrJehsOeNrLvBeg.` H jmpaudwe[ KwKiBtJhy pignmfmicnuimtoem fcqabllcruileucsh. -