MATH 122B AND 125 FINAL EXAM REVIEW PACKET ANSWERS (Fall 2016) t f () t 1/2 3/4 5/4 7/4 2
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1 MATH B AND FINAL EXAM REVIEW PACKET ANSWERS (Fall 6) f () / / / 7/ f( + h) f(). lim h h The slope of f a = f (6) The average rae of change of f from = o = dy = 8. a) f ( a) b) f ( a) + f( a). a) g( ) =, g ( ) = b) g( ) =, g ( ) =. a) For f () = : If a person weighs pounds, he dose should be milligrams. For f () = : The dose for a pound person would need o be approimaely milligrams higher han a dose for a pound person. b) milligrams. 6. Graph of f( ) Graph of f f() f ' () 7. a) False b) True c) False d) False e) False f) True 8. y a) b) + e c) d) e) Does no eis.
2 . ( + h) () e e lim = f () where h h f = e. 6 =. f () e. V ( ) = (.8), V () = ln(.8)(.8) 9.7 dollars per year.. a) =, =, = b) =, = c) =. a) (, ), (, ) b) (, ) dy dy a. a) = b) = + ( a + ) a a d) ( ) ( + ) dy dy = ln a a + a e) = cos a a c) f) dy = a sec ( a) an( a) dy a = e a ln( a). a) y = 7 h = f 6 f f b) Decreasing g () = g = 9 ( ) ( ) ( f ) c) k () = k = f ( ) d) m () = e f m = e ( ) f e) Concave down j = f f + f f 6. =, = π π π π 7. θ =, π,,,,, π y ( θ ) = 6cos( θ )sin( θ ) a) dm mv o dv = c v c ( ( )) z z < z 9 = z < z < dz z z > The derivaive is no defined a z = and z =. d b) ( ) 9. A =, B = < f + = < <
3 . f( h) = h+. k =. a) dy dy y sin ( π) y + + π = b) 7 y =. a) = d + 9 b) = ± c) 9 = ± (, ln() 8). a) ( e, ) c) = ± are inflecion poins + b) = is a local maimum y = + y = + ( ). a) Yes. E( R) = lim E( r) = kr b) No. The slope is k for r < R, bu he slope E r + r R c) d) R r approaches k as r approaches R from he righ. k r < R de = kr dr r > R r 6. a) = is a local minimum, (, 7) b) = is he inflecion poin, (, ) c) = is he global maimum, = is he global minimum 7. a) ( a,a + a ) is he local maimum, ( a,a a ) b) (, a ) is he inflecion poin. π 8. a) lim = π Use L Hopial s rule. π sin sin( θ ) b) lim = Use L Hopial s rule. θ sin(7 θ ) 7 c) lim arcan = π is he local minimum.
4 d) lim+ y = y e y Ge common denominaor hen use L Hopial s rule wice. 9. A = 6, B = Use f () = and f () = o find A and B.. a) ( a,) e b) a = e is a local maimum.. The maimum volume is L 8 V =. The minimum cos is L L L L cubic inches. The dimensions are 78 (9.6)V dollars. The dimensions are. V V V. (.8) by (.8) by.6.v C ( ) = + 6. The wavelengh is w= c k w c c V ( w) = + c w c w π 7. The maimum value of I is, occurs a =. ω π The minimum of value of I is -, occurs a = ω 8. The coordinaes ha maimize he area of he riangle are (, ) A ( ) = ( ) 9. = The oal cos will be maimized when Cr ( ) r 6r π r (approimaely 7.87 fee) and approimaely $7.7 r = 6 + π π h = (approimaely.78 fee). Maimum oal cos is 6 + π. a) b =, b = b) b = b Ab ( ) = ( b + ). a) f is he graph ha looks linear. b) = is a local minimum. = is a local maimum. g = f( e ) + e f, g = when f = f.
5 . a) miles per gallon during he firs 7 miles. / 6.67 miles per gallon in he ne miles. b) The oal gallons used hours ino he rip. k (.) =. gallons. c) k (.) =.8, k (.) =.8 gallons per hour. Gas consumpion is beer in he firs 7 miles, bu gas is being consumed more quickly.. The camera is roaing a radians per minue. = anθ, dθ =. d. The heigh is growing a π fee per minue. ( ) V = π h h.. The resisance is decreasing a ohms per minue. dv dr di = I + R d d d 6. a) Populaion is increasing when < P< L and decreasing when P> L. When P= L he, populaion remains consan. d P L b) = ( )( ) kpl P L P P=, P= L, P= d 7. dm dr K = d + r d 8. The volume is decreasing a a rae of 8π cm per hour. The surface area is decreasing a a rae of π cm per hour. Noe: he formula for he surface area of a sphere would be given in he problem. S = π r. 9. a) The lower esimae is.8. The upper esimae is.8..6 b) Using he Fundamenal Theorem of Calculus, f ( ) d = f (.6) f (.) =.... g ( k ) k = 9 g ( k ) k = n lim ( ) gk n k =. a) Objec B b) Objecs A and D c) Objecs A and D d) Objec C. a) lengh b) slope of he line
6 f( b) f( a) f( b) f( a) c) area d) lengh average. a) b c + + b) bln + + c c) ln b c arcan b + c b + + d) ( ). a) ( ) + + = ( ) = b) ( ) ( ). people will be added o he ciy. ( ) + d = 7 < 6. r () = ( ) d + ( ( ) + 7) d = gallons 7. g() (, 8) (8, 8) (, ) (6, -) (, -7) (, -) 8. a) V ( ) = 6+ 8 b) fee c) s ( ) = v() 8 v() s()
7 d 9. a) F() = e d = b) F = e d e = c) Increasing. d) Concave down. F = e 6. π π = cos π π 6. a) ( ) ln + cos = π ln f( ) is an even funcion. π π b) ( ) ln + cos( ) =π ln Le u = and change he endpoins. 6. a) True b) True c) False d) True e) False 6. a) 7 6 b) c) 7 d) M 6. a) gc ( ) > gd ( ) b) g ( B) > g ( C) c) g ( A) > g ( B) 6. a) F = e + ( ) ( ) b) Using he Fundamenal Theorem ln() e ( + ) d = e ( + ) = 7e 9 ln() ln() 66. a) Posiive look a signed area b) Negaive look a y value c) Posiive look a slope r() d 67. a) Approimaely 77 algae per hour. b) The populaion decreased by approimaely 8 algae.. r() d K = 8 9 K = a) sin( θ) cos ( θ) dθ = w dw = cos ( θ) + C cos( v) b) dv = dv dw ln sin( v ) C an( v) = = + sin( v) w
8 w w w c) e d = e dw = e dw = e = ( e ) 7. a) g d = g( w) dw = g = g( w) dw = 6 b) ( ) 7. a) s () = + b) π θ = arcan + 7. The hickness of he ice is decreasing by cubic inches per minue. π ( ) dv dt V = π + T 6 π() 6 = π ( + T ) d d 7. a) people per dollars, negaive, his is he rae a which he number of people purchasing he skaeboard changes as he price of he skaeboard increases (he number of people who will no buy he skaeboard if is price was increased from $ o $). b) people, posiive, his is he number of people willing o have he skaeboard if he skaeboard was free. c) people, negaive, his is he change in he number of people purchasing he skaeboard if he price changed from $ o $. 7. fee per second D = + (6 ) D dd = (6 ) d d d y 7. a) b) y
9 76. a) () + () = 8 new subscribers b),, c) negaive d) r(8) r() = π + g 77. ( ) π g 78. =, = = + 6 = ( + ) 8. (8 f ) + (8 g) 8. A. lim sin( ) d = and lim an d = B. d sin( ) sin( ) d d sin( ) cos( ) lim = lim = lim = lim = an( ) d d an( ) d an sec Yes.
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