Semester Exam Review Answer Key PDF

S e m e s t e r | Exam R e v i e w 1. The table gives selected values for a continuous function. Based on the data, w h a t is the best approximation for lim_ f(x)? j i m , £@) D i n £ix) = 4...

S e m e s t e r | Exam R e v i e w 1. The table gives selected values for a continuous function. Based on the data, w h a t is the best approximation for lim_ f(x)? j i m , £@) D i n £ix) = 4 KX to ? _ je) | 3018[2007 | 9000 [ase| 7902_| 7907 | 2. T h e t a b l e gives s e l e c t e d v a l u e s f o r a c o n t i n u o u s f u n c t i o n. Based on t h e data, w h a t is t h e b e s t a p p r o x i m a t i o n f o r BmT@? Lim B Y DNE XA Lo se [ 9 8 [958 [9899 [ao000[soa | 100 [0a 3. T h e t a b l e g i v e s selected v a l u e s f o r a c o n t i n u o u s f u n c t i o n. Based o n t h e d a t a , w h a t is t h e b e s t a p p r o x i m a t i o n f o r HMO? Liem tex) = 0 X10 [ 3 389 5 [9999 ?_ ] 4. Find the derivative: y = 2xV3x + 5 5. Find the derivative: 3x + 3 = Scoty 1 Y= a x ( 3 x r s ) 3 3 a S e e c y aylay ' -7/3 Y = RCax+s)"F4 a x ( S l s x + 5 ) 3 ) 3 ?Sescy 7 = 2° x e 4 | OX ( 3 x + s y 7% J dx ~_. 2 a = qe pBE AIO Extn work = siey = ae Vexes_ * 1 Keely o. y vu 6. Given the grapho f f?, sketch a possible f? and f. Based ev araph of ( M 5 0 @-a, a tom - 5 4 @-2> - a + £ min O £' has m i n @2a7x4°- t' Was max  a x ny 22 3 x 4 5 * = , 7 Bxts * 2 (3ure N e x t _2% y 3x4 * (3x +s 2 (3x45) + 2x Bx +10 aG s x e s = ) a l z = 2.328 x = THe + 47 8. The graph of f?, the first derivative of the function f is shown. Evaluate the following: a) ( f a c e Je t2w+ Se = /is+2n) b) Given f ( 0 ) = 10, Find f ( 5 ) tee (d+> tle = [Fe+m) O u a r d e v civelse 32am e m 3 9. A s p h e r i c a l b a l l o o n is d e f l a t i n g s u c h t h a t i t s v o l u m e is d e c r e a s i n g a t a r a t e o f J sec? A t w h a t r a t e i s t h e r a d i u s o f the balloon changing when the radius is 9cm? ( v = 0. Find the intervals where the particle is speeding up and slowing down. ° Velocity = vie) 2 - 3 t * 4 2t + 5 6 Accel =a (+) = ~ G t + n © = -3t240t + S BD = - & t +2 3t7-ak - S ? 2 0 wtea t= is (Ot-14)(4e4)=0 vay Speeding vp? tM/3, -4 act) Co, Ya) C i «s0 ) Slow: dewn' ( l z , M/s)  11. Evaluate: i 2x? li Inx _ 100 r e r ?1?x fi 200 n F. ¥.. 200 ? ?oe Lien 2x? =O L a m 4 - \ - ¥ =O h i m LnX = eo Z i m , x = eo J i m X s oo S o E=00 X o x20 X e y A 4, 100: a CHSpital fom te ot tin ; X00 a00,"7 HLM Doge I Sim BL 2 4 x e Svertvally f o n t ant = aim OU < Rme e - | X90 2? = o ) =[0) 12. Differentiate the following, then find the x-coordinate(s) where the curve has a horizontal tangent. x3 - 6 x 4 + 3 y 3 = 3 2 2 J a toria t o 2a 2 '* 2 Toy =0 ~ 3x74 =O x a 13. Given f ° f(x)dx =5, f ° g(x)dx = ?8 Find f ° 3f(x) + 2g(x)dx. 3ls)+ 2(-8) = IS?Ie =[-1) 14. Given f ° f ( x ) d x = 2, f2° f ( x ) d x = 7 Find 7 ° Sf(x)dx and J f ( x ) d x 19 a) S s f e d d x = s ( 2 m ) = 4 5 f t u d d e = O ' 4 i, Pa [oofos Poo as P20) Pedro 07hos oa for Let y = f ( x ) be the solution to the differential equation d y = f ' ( x ) with initial condition f (1) = 5. Selected dx values of f?(x) are given in the table above. What is the approximation for f ( 2 ) if Euler's method is used with } a step size of 0.5, starting at x = 1? ? t f u s ) = S+-Slo-s) = 525 os o-4 £ 2 ) = 5. 2 6 4. s(o-+4) 2 / 6 4 5 )  o h Tannents om yoa x i s Ax-axis ~y 40 x / y C a n t 16. be on bette \. \ ay ? a 8 So e r t € oD : dy oy @( 3 , 3) - \ (B) w i s @ (-3,3)+ ~ dy ¥ co a. de = XY 17. W h a t is t h e f o r m u l a f o r a l o g i s t i c differential equation? (6.4) H o w can y o u tell t h e c a r r y i n g c a p a c i t y ? ay ? Get The d i f f e r e n t i a l i n Hee v i g h t f o r m at = 4 (L-y) and i d e n h t y Le t h e t fs H t ma, 18. A t t h e s t a r t o f an e x p e r i m e n t , there are 100 bacteria. On day 3, t h e r e a r e 250. I f t h e n u m b e r o f b a c t e r i a is i n c r e a s i n g a t a p r o p o r t i o n a l rate a n d f o l l o w i n g an exponential g r o w t h pattern, h o w m a n y b a c t e r i a w i l l t h e r e b e on day 8 s o n e | «sos €o,100) €3,asv) y tes OR ys love 3k y = l o o * * 2 9 1 = loose y= woo S o t y = Yup. 4 3 4 2 s : 2 wactcria > 4 4 0 bacterin ANAT = BK Rowmd : 4 e o ?Ke In2. 19. A p o n d contains 30 gallons o f w a t e r at time t = 3. The w a t e r level changes at a rate o f P(t) gallons p e r h o u r. Use the table b e l o w and a r i g h t Riemann sum w i t h 3 subintervals to a p p r o x i m a t e the n u m b e ro f g a l l o n s a t t = 8. W h a t w o u l d the a p p r o x i m a t i o n be w i t h a l e f t Riemann sum? 2 1 2 |c g a t |o 88 n |s 82 m | m49| o u43n| Rpt? ZO+ 2(6-2) +1l4.a)*Al42) =/53.4 gallons Left: BO+ 2(5-8) + 1(e-2) + 24.9) =(?Seie aetleas S  20. f 2 e=r + 7 dx UE Q E ? du = R e dx a k ( 2R e3 +2> ? dx * a s a n = P F a l a e ' s | + c } a 21. Evaluate: lim h>0 sin(Z+h)-sin? h jim 1G)? l i m in(oen)-tn2 Elx)= sinx £0x) = t a n x t 4 x ) = 2nx T's ensx l = s e c t x© G a the. t -° ( 4 ) - & t C%/3) = T i o =[4\ £Us) » Ve 22. Use the table of values to evaluate the following: 1 3 41 1 a) Given h (x) = f ( x ) g(x), find h?(3) = b) Given h (x) = o find h'(2) = c) Given h (x) = f ( g ( x ) ) , find h?(1) = f-£.° a. 4) Wed'g+ git bv) he v o Ps) h ~ # % ). 5 wiade 9 2 ) #U2)- £l2)e (e) Wa) = £'(2) 92) * 4 (2)402). ha) = Plgen) 911) = - 1 ( 2 ) + (-1)1 (ge = ¢' z= -a-| = 3¢1)-acie) ( ) - a a2 = l e a = ] - 2 @ ) =[E3) s e  23. Given 5x2 + 2y2 = 4, find 43y - stay 2y (-5) - (-54) (2 d2) 10x +4y dif =0 i t : (Zy)' t y : - 5X 3-104 -2934 - = - 10y 2-25x z ay ty' dy's 24. Given f(x) = 5x3 + 5x - 3, evaluate (F-1)' (7) f= (1, 2) 5x3+5-3=7 (49% 1) = FrienD = F5 : I 5x3+5x-10=0 f'+ (2,1) (Calculator) {'(x) =1547+5 × = f'61) =20 25. Find the total distance a particle travels from t = O to t = 4 given the velocity of the particle is 0(1) = -3 Distances SI vet) st D: 51 5354 at = 12.192 (calculator) 26. At time t=0, there are 100 reindeer. If the rate of growth of a population of reindeer is modeled by the equation P(t) = 30000.19t reindeer per year, approximate the number of reindeer after 5 years. 5 Population @5yrs= 100+S PLA)dt=|25137 reindeer 27. Evaluate: 9x2 4=x3 45x2 ? =3x3-1 157+128+40dx du = 3231x - dx 125-502x 1 V3x3-1 du=9x"ax 1=4 a = 5 5=36 * +12X4 36+4 d x :5 S dx (*+6)2+4 ?= x*6 du =dx =35 Jame = 5 S 4-13 du a= 2 35 473 to t a -i ( x + 6 ) + 6 2 : 15 (3x31)3/3 2 + C  28. For the function f ( x ) = ?x3 ? 2x? - x + 4, determine the following: a) Intervals of increasing/decreasing + i ) = ? Bx 4 x -| £ ° G ) =?lox - 4 b) Relative minimum/maximum points 2 c) Intervals of concavity O = ? B x " - 4 y -1 z?bx-4. ® X-coordinates of pointsofinflection B x F a x al 2 0 b x 2 - 4 ¢' 7 $ 7 0 8 f l - ) = 4 C3x41)lx+1) # 0 % 2-2/3 - £ l s ) > 4.148 x 2 - Y 3 - 1 -\ V3 | g o t o e a) Ineveasiny: - \ , - l g ) D e e : C-20,-1) ( - ' , 0 ) ~2)3 v) B y s r Deviv d e s t = m i n ( - 1 , 4 ) m e x ( Y s , 4.148) ¢ ) concave u p : ( - 6 0 , ~ 2 / 3 ) concave d o u n hs, ey) A) R o t of intlction x =-2/3 5 29, E v a l u a t e : o R In(t? + 3 ) d t ant Ere = 5x4. tn()*+3) = x t tate 3 ) 30. T h e a c c e l e r a t i o no f a p a r t i c l e f o r ¢ > 0 is g i v e n by a ( t ) = 1 8 t. A t t i m e t = 0, t h e v e l o c i t y o f t h e p a r t i c l e is 5 a n d t h e p o s i t i o no f t h e p a r t i c l e is 7. F i n d a f u n c t i o n t h a t r e p r e s e n t s t h e v e l o c i t yo f t h e p a r t i c l e , a n d a f u n c t i o n t h a t represents the position of the particle. v i t ) = velo c y pts) - Po h o vlo)=S plo) 7 S a a t= f i a t a t Svanat = Satta sat vG)= 4474 elad= aheste a c V Co) = 5S = Ao) +e plo)= 7 = B l o ) + Slo)ee ers er?  ax. + !. Lin 31. Evaluate the following limits: X u e 7% t s +¥z lim 2% = l i 5x2 i 9x+1 xo 2sinx \ a m 7x 2 O zim, 5x*+6 * Covld vse Uteyital D e a r e t oF danom e f i A x ! 3 Xa-00 4 S x 4 y + t Sut wig e s t ee vraaer t a n + e a | 1s H e B a n e * [ I B i s - 9 - r s = {|}.019 33. Find the average value of the function f ( x ) = 3x? + 4 on the interval [10, 20] Z2o 20-10 wo 3 10 atte = Te C a t e ] seh (eone -tow)] o = Fro) 34. Solve the differential equation given by 2 = 2xy w i t h initial condition y ( 0 ) = 3. 23 a n d f o a l e O83 y JAK AX y r t e ~ | 2 Anlul =x 7 +c = e " ? ly| * | 35. Determine the values of x that would make the function have a removable discontinuity x? + 8 x - 9 C 9 ) Cx-1) f@)= 2 I +v2 7 3x? + 3 0+ 2 x7 3 (ufaXxer) Removable @ X=-% N o n - vemovelbe @ X = |- 36, Find the tgtal area of the region enclosed by the curves y = 4 - 3 and y = 2x + 3 from x = -4 to x =1. ? - 3 = 2x +3 From -4-7-2 Parabola on top -241 lime on top *-2x-6=0 ×7-4x-12=0 Areas: (45-3) -(28#3) 44+5 (2943)-555-34 (X-6) (x+2) -4 X = 6 - 2 2 2. 8 3 3 37. The region R is in the first quadrant bound by the x-axis, the y-axis, y = ex, and x = 1. Find the volume given the following cross sections: lry=x a) Squares perpendicular to the x-axis A = S (e*) 'de = 3.195 * ones o b) Equilateral Triangles perpendicular to the x-axis to y-axis: from y=0-7 yal c) Squares perpendicular to the y-axis Equation/Base is I From y =1 7y =e d) Semi-Circles perpendicular to the x-axis Equation Base = 1-lny A = S= (+*) Lx = 1.254 e) Equilateral Triangles perpendicular to the y-axis 38. Find an expression for the length of the curve f(x) = 5x3-2*-1 from the point (0, -0.143) to the point (2, 5). Use vour calculator to evaluate. f'(x) = 15x%- 2 § =ST + (158355) ax 7 = 6.062