Chapter 3 Practice Test

Transcription

1 -- 0 4W0Cu gkujtda UScohfwtKwcaZrYe0 LBLTCT.W V CATlrlZ wrdigthhtmsg yrbeysjetrhvede.r l kmhasdfel YwEi9tqh8 vikncfminoirtkeb WCAa8lnc8uPlXuusA.4 Worksheet by Kuta Software LLC Calculus BC 0 Name Chapter Practice Test r d609y KQultaC 9SloDfntPwfaCrOeV llyltcu.a s EAmlrle kroitgehmtds7 YrbeJsvegr7voesdU.q For each problem, find the average rate of change of the function over the given interval. ) y = x + ; [, ] ) y = x ; [, ] Date Period Use the definition of the derivative to find the derivative of each function with respect to x. ) y = 4x 4) y = x For each problem, find the equation of the tangent line to the function at the given point. ) f (x) = x ; (, ) 6) f (x) = x + x ; (0, ) Differentiate each function with respect to the given variable. Problems may contain constants a, b, and c. 7) f = rb r 4 r 8) f = x a + 4a + 9) s = tb + 4ct + 4 t 0) h = 4 s bs + 4s

2 -- 9 km0ocb 0KSuut7am qsopfitkwsayrle FL7LqCT. F uavlil hrbibghhqt7se krteosnenrmvjex.i X mmsaodkea wmihtlhl 9IpnfKiOn9i7tOed CCaulzctuylku4sp.u Worksheet by Kuta Software LLC For each problem, find the indicated derivative with respect to x. ) y = x + x Find d y ) y = x + 4 x Find d dx dx y Differentiate each function with respect to x. ) y = ( x )( x ) 4) y = ( x 4x 4 + ) (x ) ) y = x4 + x x + 6) y = x 4x + 7) y = ( x 4 + ) ( x + ) 8) y = (x + )(x + )

3 -- p M0qV LKGuEtYaI NSxotfStFwsaCreq LILFC. Y maglulo Qrsi6g7hGtbsd rrbeesheurhvueyd.h j 7MraldbeE MwxiPtRhI xifnufzinnxi6tteo mcaclncputlputsl.d Worksheet by Kuta Software LLC 9) y = ( x + ) 0) y = (x + ) (4x 4 + ) ) y = x 4 + tan x ) y = sin x ) y = csc (x + ) 4) y = cos 4x (x + ) For each problem, use implicit differentiation to find in terms of x and y. dx ) x = y + 6) x = y +

4 -4- l fu0qr eknuateav 6SmoQfCtYwcaTr0eB ELxLoCJ.M d natlolb irwixgqhstosn r6evs9emruvqebdi.g d dmha0dten zwrictchy LIincfAiYnsiKtBem 9CDaQlwccujlsuSsE.e Worksheet by Kuta Software LLC For each problem, use implicit differentiation to find d y in terms of x and y. dx 7) = x y 8) x y = Solve each related rate problem. 9) An observer stands 00 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of ft/sec, where a is the altitude of the rocket. a Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 400 ft from the ground? 0) Water slowly evaporates from a circular shaped puddle. The area (A) of the puddle decreases at a rate of 6π in²/hr. Assuming the puddle retains its circular shape, at what rate is the radius A of the puddle changing when the radius is in?

5 -- y rz0iyl YKFuhtFah 0SofatBwiamrBem KLXLFCB.F q GAglblc OrBiPg4hptso Srqe4sqekr KvYeidq.y v KMcaFdAeL wbiktphs bixn8f7icneistfex UCXaolcSuplupst.6 Worksheet by Kuta Software LLC ) Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius (r) of the spill increases at a rate of r m/min. How fast is the area of the spill increasing when the radius is m? ) A conical paper cup is 0 cm tall with a radius of 0 cm. The bottom of the cup is punctured so that the water level goes down at a rate of cm/sec. At what rate is the volume of water in the cup changing when the water level is 8 cm? For each problem, find a linear approximation of the given quantity. ) ) sin 44

6 -6- H Go0E0Q wkbuvtmao GSXojfhtuwPaSrmev PLnLZCa.7 C AZljl4 ernimguhit0st IrwefsseErvvjeudw.R T um0akdoel GwWittihL HI 9nffpiPnwigtKen CWaflPcbuAlkuasV.M Worksheet by Kuta Software LLC Use differentials to solve each problem. ) The hypotenuse of a right triangle is known to be exactly 9 ft. One of the acute angles is measured to be 4, with a possible error of ±. Estimate the possible propagated error in the side adjacent to the measured angle. 6) The hypotenuse of a right triangle is known to be exactly 7 in. One of the acute angles is measured to be 60, with a possible error of ±. Estimate the possible propagated error in the side adjacent to the measured angle.

7 -- e FZ0bqL KuLtzaR SCoRfatcwaErYeK OLNLMC.N X oalli VrLibgvhwtZs4 krzejsse8rrvxeydg.y K PMracdAeV lwzit9hh oiqnef9iyn7i4tzed pcyaul4cquwlduss7.j Worksheet by Kuta Software LLC Calculus BC 0 Name Chapter Practice Test I AC0J I E 0KXuotIaV TSoIfetcwaZrIeT LL7LRC9.o s 0APlXlJ xrdi glh htysn NreevsoeMrqv6ebdU.Y For each problem, find the average rate of change of the function over the given interval. Date Period ) y = x + ; [, ] ) y = x ; [, ] 4 Use the definition of the derivative to find the derivative of each function with respect to x. ) y = 4x dx = 4x 4) y = x dx = x 4x + For each problem, find the equation of the tangent line to the function at the given point. ) f (x) = y = x + 8 x ; (, ) 6) f (x) = x + x ; (0, ) y = x Differentiate each function with respect to the given variable. Problems may contain constants a, b, and c. 7) f = rb r 4 r df dr = r4 brb 4r 4 8) f = x a + 4a + df dx = 6axa 9) s = tb + 4ct + 4 t 0) h = 4 s bs + 4s ds = 0btb c t dh ds = s + 0b s s 4

8 -- k u90b9r DKPubtJaq hszolfetgwoaerceu alal7cn.m P faolhlc xrimgghqts9 br0edsie0rrvxedb.u X dm7adlen zwait4ho si ZnsfiGntiJt9e tc7aslscbuo lufsz.0 Worksheet by Kuta Software LLC For each problem, find the indicated derivative with respect to x. ) y = x d y dx = 0 + x Find d y dx 7x 8 0 x 8 ) y = x + 4 x Find d y dx d y dx = 64x 4 Differentiate each function with respect to x. ) y = ( x )( x ) dx = ( x ) x + ( x ) 4x = x + 4 x 4) y = ( x 4x ( dx = x 4x 4 + ) (x ) 4 + ) x + (x )( 6x x 4 = 60x 8x 4 + 7x + x 4 ) ) y = x4 + x x + dx = (x + )(4x + x ) (x 4 + x ) x 4 (x + ) = x8 0x 7 + 4x + x x 0 + 6x + 6) y = x 4x + dx = (4x + ) x 4 (x ) 8x (4x + ) = 60x6 + x 4 + 4x 6x x + 7) y = ( x 4 + ) ( x + ) 8) y = (x + )(x + ) dx = ( x4 + ) 0x + ( x + ) ( x 4 + ) x dx = (x + ) (x + ) 4 + (x + ) 6x = x( x 4 + ) (0x 4 8x ) = (x + ) 4 (x + + 9x)

9 -- j Be0I0o GKtuutrae VSofZtDwVaUrseZ xlulacf.x h Alslh yrbiygchrtxss RraeOs7evr vvecdo.o h CMmaRdTem TwMiOthM 7InDfjisnTiQtneA tchailic4uxlruism.a Worksheet by Kuta Software LLC 9) y = ( x + ) 0) y = (x + ) (4x 4 + ) dx = ( x + ) x = x ( x + ) dx = (x + ) (4x4 + ) 6x + (4x 4 + ) = x (x + )(0x x) (4x 4 + ) (x + ) y = x 4 + tan x ) y = sin x dx = (x4 + ) sec x x + tan x (x4 + ) 8x dx = cos x x 4 = x = x (6x 4 sec x + 9 sec x + 4xtan x ) 4 cos x (x 4 + ) ) y = csc (x + ) dx = csc (x + ) cot (x + ) (x + ) x = 6xcsc (x + ) cot (x + ) (x + ) 4) y = cos 4x (x + ) dx = cos 4x 6x + (x + ) sin 4x 0x 4 = x (cos 4x 0x sin 4x 0x sin 4x ) For each problem, use implicit differentiation to find in terms of x and y. dx ) x = y + dx = 6y 6) x = y + dx = x y

10 -4- t qp0jer OKcuVt7aV TShofAtJwoajrie LL9LWCf.m f marlelw CrRiSgxhTtYsT SrIeDsPeRrkvne.P e MMTa9dIev KwyihtXhn EInWf0iLnfietheu qclanlhcmu9l0ugs4.z Worksheet by Kuta Software LLC For each problem, use implicit differentiation to find d y in terms of x and y. dx 7) = x y d y dx = 0y 4x y 8) x y = d y dx = 4y Solve each related rate problem. 9) An observer stands 00 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of ft/sec, where a is the altitude of the rocket. a Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 400 ft from the ground? a = altitute of rocket z = distance from observer to rocket t = time Equation: a = z Given rate: da = Find: dz a dz a = 400 = a z da = 40 ft/sec a = 400 0) Water slowly evaporates from a circular shaped puddle. The area (A) of the puddle decreases at a rate of 6π in²/hr. Assuming the puddle retains its circular shape, at what rate is the radius A of the puddle changing when the radius is in? A = area of circle r = radius t = time Equation: A = πr Given rate: da = 6π A dr r = = πr da = π in/hr Find: dr r =

11 -- s 7C0C8 rkhuvttaq MSMoNfrtNw6aYrner 0LSLlCW.v L 8AFlJl UrAizgwh0ttsd Sr0eKs0eBrwvgeMdA.s o xmzavdeel zwgitmh6 liunvfti6nvihtfe acxamlyc6unlpuxs6.n Worksheet by Kuta Software LLC ) Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The radius (r) of the spill increases at a rate of r m/min. How fast is the area of the spill increasing when the radius is m? A = area of circle r = radius t = time Equation: A = πr Given rate: dr = r da r = = πr dr = 6π m²/min Find: da r = ) A conical paper cup is 0 cm tall with a radius of 0 cm. The bottom of the cup is punctured so that the water level goes down at a rate of cm/sec. At what rate is the volume of water in the cup changing when the water level is 8 cm? V = volume of material in cone h = height t = time Equation: V = πh Given rate: dh = dv Find: dv h = 8 = πh dh = 8π cm³/sec h = 8 For each problem, find a linear approximation of the given quantity. ) f (x) = x 4, f '(x) = 4x x 0 = 8, x = 0.0 f (x 0 + x) f (x 0 ) + f '(x 0 ) x = 096 = ) sin 44 f (x) = sin x, f '(x) = cos x x 0 = π 4 radians, x = π 80 radians f (x 0 + x) f (x 0 ) + f '(x 0 ) x = (80 π)

12 -6-9 ep0a4 KLuxtgar XSnoTfYtHw4air eew kl7luc9.z 4Aolw l jri4glhhtmst ErCeqs6eerGvfeLdh.G P imkaadcer Uwimt0hi mi dnkfui6nigtret CCavlrcpuYluysG.E Worksheet by Kuta Software LLC Use differentials to solve each problem. ) The hypotenuse of a right triangle is known to be exactly 9 ft. One of the acute angles is measured to be 4, with a possible error of ±. Estimate the possible propagated error in the side adjacent to the measured angle. x = 9cos θ, dx = 9sin θ dθ θ = π 4 radians, dθ = ± π 60 radians x dx = ± π 40 ±0. ft 6) The hypotenuse of a right triangle is known to be exactly 7 in. One of the acute angles is measured to be 60, with a possible error of ±. Estimate the possible propagated error in the side adjacent to the measured angle. x = 7cos θ, dx = 7sin θ dθ θ = π radians, dθ = ± π 80 radians x dx = ± 7π 60 ±0.08 in