Cal II- Final Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Epress the following logarithm as specified. ) ln 4. in terms of ln and ln ln B) ln ln - ln ln + ln ) Graph the inverse of the function plotted, on the same set of aes. Use a dashed curve for the inverse. ) ) 0-0 0-0 B) 0 0-0 0-0 0-0 -0 0 0-0 0-0 0-0 -0
Find the inverse of the function. ) f() = - ) f-() = - B) f-() = + f-() = + Not a one-to-one function Find the domain and range of the inverse of the given function. 4) f() = - 7 Domain: [0, ); range: [7, ) B) Domain and range: all real numbers Domain: [7, ); range: [7, ) Domain: [7, ); range: [0, ) 4) Epress as a single logarithm and, if possible, simplif. ) ln - + ln ) ln ( - ) B) ln - + ln ( - ) ln ( - ) Find the derivative of with respect to, t, or θ, as appropriate. 6) = ln (cos (ln θ)) tan (ln θ) B) tan (ln θ) - θ tan (ln θ) θ - tan (ln θ) 6) Evaluate the integral. cos d 7) + 8 sin 7) 8 ln + 8 sin + C B) ln + 8 sin + C 8 sin + C 8 ln + 8 sin + C 8) d 4 + 7 ln 8) 7 ln 7 + 4 ln + C B) 4 7 ln 4 + 7 ln + C 7 ln 4 + 7 ln + C ln 4 + 7 ln + C 4 Find the derivative of with respect to, t, or θ, as appropriate. 9) = ln ln B) ln ln 9)
Use logarithmic differentiation to find the derivative of. 0) = sin + sin + sin cos + + B) sin + cot + sin + lnsin + 6cot + cot ln( + ) + ( + ) 0) Find the derivative of with respect to, t, or θ, as appropriate. ) = 8eθ(sin θ - cos θ) 6eθ(sin θ - cos θ) B) 8eθ(sin θ - cos θ) + 8eθ 6eθ sin θ 0 ) Find d d. ) ln = e cos 7 ) e cos 7-7e sin 7 B) -7e sin 7 e sin 7 - e cos 7-7e sin 7 - e cos 7 Evaluate the integral. ) e-4 d -4e- + C B) - 4 e - + C - 4 e -4 + C e -4 + C ) Solve the initial value problem. 4) d dt = e t + 7 sin t, (0) = 0, (0) = 0 = et - 7 sin t + 6t - 4 B) = e t 4-7 sin t + t - 4 = e t 4-7 sin t = e t 4-7 sin t + 0t - 4 4) Find the derivative of with respect to the independent variable. ) = cos πθ πcos πθ ln B) cos πθ -cos πθ ln sin πθ -πcos πθ ln sin πθ ) Use logarithmic differentiation to find the derivative of with respect to the independent variable. 6) = ( + ) ln( + ) + B) ln( + ) + ( + ) ln( + ) + + + ()- 6)
Find the length of the curve. 7) = ln(sin ) from = π/6 to = π/4 - ln - B) ln - - ln( - ) ln( - ) 7) Use lʹho^pitalʹs rule to find the limit. sin 8 8) lim 0 8) B) 0 4 Find the derivative of with respect to. 9) = sin- + 9) + ( + ) B) - 9 - ( + ) 6 + ( + ) 9 - ( + ) Evaluate the integral. dt 0) t + 4t + 8 tan - t + tan- t + + C B) tan- (t - ) + C + C -t+ C 0) Find the derivative of. ) = ln (sinh 8) coth 8 B) sinh 8 8 coth 8 8 csch 8 ) 4
Find the area of the shaded region. ) f() = - + + 6 g() = 4 0 0 (4, 6) 0 (0, 0) - -4 - - - - 4 6-0 (-, -) - -0 - -0 ) - 4 B) 97 4 ) ) 8 6 4 = = - 4 - -4-6 -8 4 6 8 0 8 B) 64 Find the volume of the solid generated b revolving the region bounded b the given lines and curves about the -ais. 4) = +, = 0, = -, = 4) π B) π π π ) =, = 0, = 0, = 6 4π B) 7π 944π 7776 π ) 6) =, = 6, = 0 04 π B) 644 8 4096 π π π 6) Find the volume of the solid generated b revolving the region about the -ais. 7) The region enclosed b =, = 0, = - 4, = 4 4 8 π B) 048 64 π π π 7)
Find the volume of the solid generated b revolving the region about the given line. 8) The region bounded above b the line = 6, below b the curve = 6 -, and on the right b the line = 4, about the line = 6 768 04 64 89 π B) π π π 8) Use the shell method to find the volume of the solid generated b revolving the shaded region about the indicated ais. 9) About the -ais 9) 4 = 4 = 4 = 6-4 64π B) 8 64 π π π Use the shell method to find the volume of the solid generated b revolving the region bounded b the given curves and lines about the -ais. 0) = - 4, =, = 0, for 0 0) π B) 8π 60 π 64π Use the shell method to find the volume of the solid generated b revolving the region bounded b the given curves about the given lines. ) = 4 -, = 4, = ; revolve about the line = 4 ) 4 π B) π 8 6 π π Find the length of the curve. ) = / from = 0 to = 4 ) 08 B) 9 4 7 Solve the problem. ) A vertical right circular clindrical tank measures 4 ft high and 8 ft in diameter. It is full of oil weighing 60 lb/ft. How much work does it take to pump the oil to a level ft above the top of the tank? Give our answer to the nearest ft lb.,0,. ft lb B),06,704. ft lb 868,87.7 ft lb 844,460.0 ft lb ) 6
Find the fluid force eerted against the verticall submerged flat surface depicted in the diagram. Assume arbitrar units, and call the weight-densit of the fluid w. 4) 4) 87 9 9 9 79 w B) 474w w 87w Solve the problem. ) The spring of a spring balance is 7.0 in. long when there is no weight on the balance, and it is 9. in. long with 9.0 lb hung from the balance. How much work is done in stretching it from 7.0 in. to a length of. in.? 4. lb in. B) 4 lb in. 90 lb in. 9.9 lb in. ) Find the center of mass of a thin plate of constant densit covering the given region. 6) The region bounded b = and = = 0, = 8 B) = 0, = 7 = 0, = 9 = 0, = 6) Find the centroid of the thin plate bounded b the graphs of the given functions. Use δ = and M = area of the region covered b the plate. 7) g() = and f() = + 7) =, = 8 B) = 4, = 8 =, = 8 =, = Evaluate the integral. 8) 8 sin d 8) 8 sin - cos + C B) 8 sin + 8 cos + C 8 sin - 8 cos + C 8 sin - 8 cos + C 9) e d 9) e - 4 e + 4 e + C B) e - e + 4 e + C e - e + 4 e + C e - e + C 7
Evaluate the integral b using a substitution prior to integration b parts. 40) + d 0 -.7 B) -0.94 0.9 -. 40) Evaluate the integral. π/4 4) sin 4 d 0 4) 0 B) 4 6 4) 7 sec4 d 4) 7 tan + C B) 7(sec + tan )+ C 7 tan + 7 tan + C - 7 tan + C Use an method to evaluate the integral. 4) 8 csc tan d 4) - 8 csc 4 + C B) csc4 cot + C - 8 cot + C - 8 csc + C Integrate the function. 44) dt t 8 - t - - 8 - t + C B) - 8t 8 - t + C 8t 8 - t + sin- 8t + C 8t 8 - t + C t 44) Use a trigonometric substitution to evaluate the integral. d 4) 9 - (ln ) /e -0.69 B) 0.07 0. 0.6 4) 8
Epress the integrand as a sum of partial fractions and evaluate the integral. 4 + 4 46) d + 9 + 4 ln ln ( + )7 ( + ) 6 + C B) ln ( + 7) ( + 7)7 ( + )4 ( + ) + C ln ( + 7)7 ( + 7)7 + C + C 46) Evaluate the improper integral or state that it is divergent. 47) e- d 0 0 B) - Divergent 47) Determine whether the improper integral converges or diverges. 48) e- d - Diverges B) Converges 48) Find the interval of convergence of the series. 49) ( - 4)n < B) < < < - < < 49) Determine convergence or divergence of the series. 6 0) n/ n= Converges B) Diverges 0) Find the interval of convergence of the series. ( - 4)n ) n44n -8 < < 8 B) 0 8 < 8 ) ) ( - 7)n (n)! n= B) - < < 8 6 8 ) Use the ratio test to determine if the series converges or diverges. (n)! ) n n! n= Converges B) Diverges ) 9
Find the Talor polnomial of order generated b f at a. 4) f() = + 4, a = - - 9 + ( - ) 7 - ( - ) 8 - + + ( + ) - ( + ) 6 B) - + 9 + ( + ) 7 - ( + ) 8 - - + ( - ) - ( - ) 6 4) Find the Maclaurin series for the given function. ) + ) (-)n n+ n B) (-)n n n+ n= (-)n n n (-)n n n+ 6) - 6 6) n B) 6n n n - - 6n+ 6n+ n 6n 7) e- (-)n n n n! B) n n n! n= n= (-)n n n n! n n n! 7) Find the Talor series generated b f at = a. 8) f() =, a = 7 8) (-)n ( - 7)n 7n+ B) ( - 7)n 7n ( - 7)n 7n+ (-)n ( - 7)n 7n Find the quadratic approimation of f at = 0. 9) f() = esin Q() = - B) Q() = - + Q() = + Q() = + + 9) 0
Find the first four terms of the binomial series for the given function. 60) ( - 6)/ - - 9-7 B) - + 9-7 - - 9-7 4 + + 9-7 4 60) Solve the problem. 6) Obtain the Talor series for ( - ) from the binomial series for -. + + + 4 B) - + - 4 + + 4 + 4 + + 4 + 76 6) Find a Cartesian equation for the parametric equations. 6) = t -, = t + = + 6 + 4 B) = + 4 = - 4 = - 6-4 6) = 6 cos t, = 8 sin t 6 + 8 = B) 6-64 = 6 + 64 = 64 + 6 = 6) 6) Graph the pair of parametric equations in the rectangular coordinate sstem. 64) = t -, = t + 7; -4 t 4 0 64) -0 0-0 B) 0 0-0 0-0 0-0 -0
0 0-0 0-0 0-0 -0 Solve the problem. 6) Use the Alternating Series Estimation Theorem to estimate the error that results from replacing e- b - + when 0 < < 0.. 0.00 B) 0.0007-0.00 0.0067 6) Graph the pair of parametric equations in the rectangular coordinate sstem. 66) = + 6 sin t, = + 6 cos t; 0 t π 66) 0-0 - 0 - -0 B) 0 0-0 - 0-0 - 0 - - -0-0
0 0-0 - 0-0 - 0 - - -0-0 Find the Talor polnomial of lowest degree that will approimate F() throughout the given interval with an error of magnitude less than than 0-. 67) F() = sin t dt, [0, ] 67) 0 4 4-0 B) - 9 4 60 6 4-8 - 7 48 6 68) F() = e-t dt, [0, ] 0 - + 0-7 4 + 9 6 - + - 7 7 B) - + 0-7 4 - + - 7 7 + 9 9 68) Find the first four terms of the binomial series for the given function. 69) + - 69) - + 4 9-9 B) - + - 4 9 - + - 4 7 - + 4 9-8 7 For the given polar equation, write an equivalent rectangular equation. 70) r = 0 rcos θ - 6r sin θ - 9 ( - 0) + ( + ) = 9 B) 0-6 = 9 ( + 0) + ( - 6) = 9 ( - 0) + ( + ) = 00 7) r = 6 cos θ - 8 sin θ 6-8 = B) 6-8 = 6 + 8 = 6-8 = 70) 7)
For the given rectangular equation, write an equivalent polar equation. 7) + - 4 = 0 r = 4 cos θ B) r = 4 sin θ r cosθ = 4 sin θ r sinθ = 4 cos θ 7) Use the root test to determine if the series converges or diverges. ln n n 7) n + 7 n= Converges B) Diverges 7) Use substitution to find the Talor series at = 0 of the given function. 74) e-8 (-)n(8)8n8n (8)nn B) n! n! (-)n(8)nn (-)nn n! (8)nn! 74) Find the Talor series generated b f at = a. 7) f() =, a = 9 7) ( - 9)n 9n+ B) ( - 9)n 9n (-)n ( - 9)n 9n+ (-)n ( - 9)n 9n Find the seriesʹ radius of convergence. 76) ( - )n n = 0 B) 0, for all 76) 77) ( - 8)n nn n = 0 4 B) 0, for all 77) Find the vertices and foci of the ellipse. 78) + 6 = Vertices: (±, 0); Foci: (±, 0) B) Vertices: (0, ±); Foci: (0, ±) Vertices: (0, ±); Foci: (0, ±4) Vertices: (±, 0); Foci: (±4, 0) 78) 4
Find the focus and directri of the parabola. 79) - 8 = 0 (7, 0); = -7 B) (7, 0); = 7 (7, 0); = -7 (0, 7); = -7 79)
Answer Ke Testname: CAL II FINAL REVIEW ) C ) B ) C 4) A ) A 6) C 7) D 8) C 9) D 0) B ) C ) D ) C 4) B ) D 6) C 7) B 8) D 9) D 0) A ) C ) B ) B 4) C ) D 6) D 7) A 8) B 9) C 0) D ) A ) A ) A 4) A ) B 6) C 7) C 8) C 9) C 40) C 4) B 4) C 4) D 44) C 4) D 46) A 47) C 48) B 49) B 0) B 6
Answer Ke Testname: CAL II FINAL REVIEW ) B ) B ) B 4) D ) C 6) C 7) A 8) A 9) D 60) A 6) A 6) A 6) C 64) C 6) A 66) B 67) A 68) A 69) C 70) D 7) B 7) A 7) A 74) C 7) B 76) A 77) C 78) A 79) A 7