2) y = x2 + 2x. 4) f (x) = 1 2 x 4. 6) y = (-4-4x -5 )(-x 2-3) -1-

Transcription

1 Calculus e Et0WV6z lkvuhtlap HSQopfUtdwhaYrve^ vlplcco.o d naflml[ wrnibgohatzsz irpevssexr]viefd\. Derivative Review Differentiate each function with respect to x. ) f (x) 5 x5 + x x ) y 4 5 x + x + 5 x4 ) f (x) 5 x 4 x 5x 4) f (x) x x 5) y ( 5x 5) (x + ) 6) y (4 4x 5 )(x ) 7) f (x) ( x )(5x ) 8) f (x) ( 4 x 4) (4x4 ) Z C]0PE6^ RKGuNtdaO usuomfetowvaermek ylblkcg.` V haxlmlo orsikgph`tjss zrfejs^ewrxvjerds.x M tmuagdxeb nwhiutjhy si[ncfhisn`irtvem KCXaplBcwuUlcu\sO.

2 9) f (x) x + 5 x 4 0) f (x) x x + ) y x4 + 4x + 5 x ) y x 5 4 x + ) y (x 5 + ) 4) y (4x 5 5) 5) f (x) (5x 5 ) 5 6) f (x) (x 4 5) ` xw0ph6` WKRuPtray vs]omfetswfabrte] aluldcx.` g fazlcly SrAihgIhwtyss argeospeqrkvqecdq.[ b GMcaFdUeJ AwQiotQht PIInFfeiPnqigt]ez TCxafltcOuAlVuYsw.

3 7) f (x) (5x + ) (x 4 5) 8) f (x) (5x4 ) (4x 5 ) 9) y (4x + 5) 5 (x ) 0) f (x) (5x 5 4) (4x ) 5 f bv0gj6g JKQuft`aB `SOohfStowoadrlel PLALTCV.S B aarlxlh ]rniigwhythsh ]rqeiskeorevwexdz.z L rmqaddbee SwRiWtRhF mibnxfviynriztweh actadlvcmukljuosh.

4 For each problem, find the indicated derivative with respect to x. ) y x 4 Find d y dx ) f (x) 4x x Find f (4) ) f (x) 5x 5 + 5x 4 x Find f ''' 4) f (x) x 4 Find f '' Differentiate each function with respect to x. 5) f (x) sin x 6) y sin x 5 7) f (x) x + sin x 5 \ NZ0va6q GKSuVtnaM bszoufjtgwgaqrgeo ]LoLpCi.v D MAplTl\ Grmixgehwt_sf `rregsneqr^vfecd`.u x [MxaOdCe_ rwxiotzhi iinnjfqiwnmiotyen ZCzaml_c\u^l]u]sA. 4

5 8) f (x) sin x 5 x + 4 For each problem, find the derivative of the function at the given value. 9) y x + x + at x 0) y csc (x) at x p 4 ) y x x + at x 0 ) y 6x x + 6 at x ) y tan (x) at x p 4) y x 9 at x 0 u ^g0ih6\ ckbuhteap ASgoxfPtmw_acrfea VLTLNCx.q M RAilxlg hrlimgqhhtusc jryegsneqrfvxegd[.u e TMzaRdueL OweiYtgh_ `I`nxfji]nbittxei ecjaflgcxuzlnuksb. 5

6 Solve each optimization problem. 5) Engineers are designing a boxshaped aquarium with a square bottom and an open top. The aquarium must hold 08 ft³ of water. What dimensions should they use to create an acceptable aquarium with the least amount of glass? 6) Two vertical poles, one ft high and the other 4 ft high, stand 5 feet apart on a flat field. A worker wants to support both poles by running rope from the ground to the top of each post. If the worker wants to stake both ropes in the ground at the same point, where should the stake be placed to use the least amount of rope? 7) A farmer wants to construct a rectangular pigpen using 00 ft of fencing. The pen will be built next to an existing stone wall, so only three sides of fencing need to be constructed to enclose the pen. What dimensions should the farmer use to construct the pen with the largest possible area? 8) A geometry student wants to draw a rectangle inscribed in a semicircle of radius 8. If one side must be on the semicircle's diameter, what is the area of the largest rectangle that the student can draw? H Y[0Jx6O ZKBuZtEaV vsqojfrtkwkaorzet PLvLzCM.q m JAGljls hroirgmhdtesa Nr_e^s[ekrpvaejdm.N P QM^aCdoeG fwvi_tyh` sifnefsibnmihtdek AC]aAlkcWu\lruWsP. 6

7 9) Which point on the graph of y x is closest to the point (4, 0)? 40) A rancher wants to construct two identical rectangular corrals using 00 ft of fencing. The rancher decides to build them adjacent to each other, so they share fencing on one side. What dimensions should the rancher use to construct each corral so that together, they will enclose the largest possible area? A particle moves along a horizontal line. Its position function is s(t) for t ³ 0. For each problem, find the position, velocity, and acceleration at the given value for t. 4) s(t) t 4 + 9t ; at t 5 4) s(t) t + t 40t; at t 5 4) s(t) t 4 t ; at t 44) s(t) t 0t 9; at t 8 T ys0kx6v skquatkan msbolfgtwwoa_rber glaldc^.[ n cacl`lh crwimgfhltxsf Ur_e_slezrVv_egdd.` V ZMRaHd\eu GwfiwtYhx QITnUfIisnVivtUe] QCXalldcOuYlVuOsq. 7

8 Solve each related rate problem. 45) A hypothetical square grows so that the length of its sides are increasing at a rate of 4 m/min. How fast is the area of the square increasing when the sides are m each? 46) A hypothetical square grows so that the length of its diagonals are increasing at a rate of 6 m/min. How fast is the area of the square increasing when the diagonals are m each? 47) A crowd gathers around a movie star, forming a circle. The radius of the crowd increases at a rate of 4 ft/sec. How fast is the area taken up by the crowd increasing when the radius is 4 ft? 48) 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 6 m/min. How fast is the area of the spill increasing when the radius is 0 m? p m`0ie6m JKQuFtOaD hscoaf^t]waaqrlem ylml[cw.l W naqlyly kriicghhftzsb mreejspekrfvye`dg.y H ZMwapdFeV kwjixtdhp iionzffiyntictxey MCAaKlzcsuLlnuSsx. 8

9 ) f '(x) 5 x4 + 8x + 5 x 5x4 + 8x + 4) f '(x) 5 x 5 4 x4 5x 5 4x 4 5x Answers to Derivative Review ) dy dx 8 5 x + x 8 5 x5 8x 5 + 5) ( dy dx 5x 6) dy dx ( 4 4x 5 ) x + (x ) 0x 6 8x x 60 4 x 6 D lm0tg6x dk`uutrac oshogfjtwwvafrleb RLfLcCa.o F OAjl[l] brgikgohpt^s] irmessaearovbezdo.p i ^M`ahdaeI pweiltehj wimnzf_ianhitttex ^CNaGlxc[uilsuhsD. 80x 8) f '(x) ( 4 x 4 ) 6x + (4x 4 ) 4x 5 0) f '(x) 64x x 5 ( 5x 5 + ) x (x 5) x 4 5 ( 5x 5 + ) 9x + 4x x 5 + 0x + 4x 9 ) dy dx ( + 5x )(4x + 8x) (x 4 + 4x ) 0x ( + 5x ) x7 + 54x x ) dy dx ( x 9x 4 + 0x ) x (x 5) x 4 7x + 8x ( x 4 + ) 8x x + 8x 4) dy dx (4x 5 5) 0x 4 60x 4 (4x 5 5) 7 x 9 8 5x 5 5) 4x + (x + ) 0 5 0x 0 x ) f '(x) 9 5 x 4 5 x x 9x x 7) f '(x) ( x ) 5x + (5x ) 4x 5x x 9) f '(x) ( 5 x 4 ) 9x (x + ) 8x 5 ( 5 x 4 ) 45x0 4x 6 8x 5x 8 0x ) dy dx (x 5 + ) 4 5x 4 5) f '(x) 5(5x 5 ) 6 5x 4 5x4 (5x 5 ) 6 5x4 (x 5 + ) 4

10 6) f '(x) (x4 5) 8x 7) f '(x) 8) f '(x) 9) dy 8x (x 4 5) (x 5) 4 (5x + ) 4 5 (5x + ) 4 (x 5) 4 6x (5x x) 5 (x 5) 4 (5x + ) 4 ( (x 5) 4) (4x 5 ) (5x4 ) 0x (5x 4 ) 0x (6x 5 + 8x + ) (5x 4 ) (4x 5 ) ((4x 5 ) ) dx (4x + 5) 5 4 (x5 + ) 4 5x 4 + (x 5 + ) 5x (44x 5 + 5x 44) 4(x 5 + ) 4 (4x + 5) 6 (4x 5 ) 0x 4 4 5(4x + 5) 6 x 0) f '(x) (5x 5 4) 5(4x ) 4 x + (4x ) 5 (5x5 4) 5x 4 5x (4x ) 4 (00x x ) (5x 5 4) ) d y dx 6x ) f (4) (x) 0 ) f '''(x) 00x + 0x 4) f ''(x) x 5) f '(x) cos x 9x 9x cos x 6) dy dx cos x5 5x 4 7) f '(x) sin x5 x (x + ) cos x 5 5x 4 sin x 5 x (sin x 5 + 5x 5 cos x 5 0x cos x 5 ) sin x 5 x 8) f '(x) 5 cos x + 4 (x + 4) 5x 4 x 5 6x 9) dy (x + 4) dx x 4 x 5 cos x + 4 (9x + 0) (x + 4) x 5x 4 cos x 5 7 0) dy dx x p 4 G d`0wr6m tkuuhtsar zswo^fit^wpairfeg ILPLoCd.t \ oa^lslg ErwilgWhEtas` zrwexsgebrjv_eudw.] w tm\aad]e_ awui^tehp BIJnyfdiUnXiLtWeO RCSatlUcbuolLuJs[. 0

11 ) dy dx x 0 ) dy dx x E lf0\a6a IKtujtnaF NS\ohfYtYwIa\rVew ALBLxCh.y F [AKl^ln qrniqgrhctpsd BrXessLelrRvQeQd`.h Q rmladdcec ^wiiutjhy pirnzfyienpiftjen zcvablhcaullxuysa. ) dy dx x p 4) dy dx 5) A the area of the glass x the length of the sides of the square bottom Function to minimize: A x + 4x 08 x where 0 < x < Dimensions of the aquarium: 6 ft by 6 ft by ft tall 6) L the total length of rope x the horizontal distance from the short pole to the stake Function to minimize: L x + + (5 x) + 4 where 0 x 5 Stake should be placed: 5 ft from the short pole (or 0 ft from the long pole) 7) A the area of the pigpen x the length of the sides perpendicular to the stone wall Function to maximize: A x(00 x) where 0 < x < 50 Dimensions of the pigpen: 75 ft (perpendicular to wall) by 50 ft (parallel to wall) 8) A the area of the rectangle x half the base of the rectangle Function to maximize: A x 8 x where 0 < x < 8 Area of largest rectangle: 64 9) d the distance from point (4, 0) to a point on the curve x the xcoordinate of a point on the curve Function to minimize: d (x 4) + ( x) where < x < Point on the curve that is closest to the point (4, 0): ( 7, 4 ) 40) A the total area of the two corrals x the length of the nonadjacent sides of each corral 00 4x Function to maximize: A x where 0 < x < 5 Dimensions of each corall: 5 50 ft (nonadjecent sides) by ft (adjacent sides) 4) s(5) 500, v(5) 75, a(5) 0 4) s(5) 0, v(5) 5, a(5) 4 4) s() 4, v() 6, a() 08 44) s(8) 55, v(8) 6, a(8) 45) A area of square s length of sides t time Equation: A s s s ds Given rate: ds 4 96 m²/min Find: s 46) A area of square x length of diagonals t time Equation: A x x x dx Given rate: dx 6 78 m²/min 47) A area of circle r radius t time Equation: A pr Given rate: dr 4 r 4 pr dr p ft²/sec Find: Find: x r 4 x 0 0

12 48) A area of circle r radius t time Equation: A pr Given rate: dr 6 r 0 pr dr 0p m²/min Find: r 0 b Sg0_m6s AKfuEt\ai ismoefltkwjafrmer qlwljcn.y E vadlxlo ErVingLhKttsv lrdews[e]rdvhey.t Q smea`djej ]wpictihz IIDn^fJiKnSiBtveG FCcaPlmcduWlauqsp.