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# DIFFEREN CALCULUS FOR SCIENCE M 408N

UT

GPA 3.67

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This 3 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 408N at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/181506/m-408n-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15

uppal mu2465 7 HW17 7 meth 7 54865 1 This print out should have 7 questions Multiple choice questions may continue on the next column or page 7 nd all choices before answering 001 100 points Sarah is driving her car very smoothly from Austin to Dallas one day At 900 pm she is travelling at 30 mph while 10 minutes later she is travelling at 50 mph Calculus tells us that at some time between 900 pm and and 910 pm her acceleration is exactly at mphh Determine x 1 x 120 mphh correct 2 x 116 mphh 3 x 112 mphh 4 x 118 mphh 5 x 114 mphh Explanation 1f Sarah7s speed at time t is given in mph by v vt then her acceleration vt mphh But if she is driving very smoothly her veloc ity will be continuous and differentiable as a function of 75 between 900 pm and 910 pm Thus by the MVT there will be some time c strictly between 900 pm and 910 pm at which her acceleration satis es the equation v910 7 v900 v C 1060 Consequently an 120 mphh 002 100 points The function f is de ned on 1 3 by m 335 ac 7 03 7 a At which point c in 1 3 if any does 1 3 f1 lt gt2 lt gt m hold 1 i 13 c 7 5 2 i 7 c 7 3 3 c 3 5 4 c 7 correct 2 5 no suchcexists Explanation The function f is continuous on 1 3 and differentiable on 1 3 although it is not dif ferentiable at the end points of the interval 1 3 The MVT thus applies and so f37f1 f 7 m for at least one c in 1 3 Now by the Product and Chain Rules m 3 gtfiiltgtf con rming that fl and f3 do not exist But 3 37951 1 71 Zlt71gt 713715 3377ac1 43E7137x 9174 410 7 137 x Consequently we 7 uppal mu2465 7 HW17 7 meth 7 54865 2 On the other hand f3 7 rm 7 6 Hence if m f3 N is to hold the point 0 must satisfy fc 3 26 7 c 5 3 07113 7 c1 7 and so 003 100 points As a graduation present Natalie received a sports car which she drives very fast but very smoothly She always covers the 25 miles from her home to the nearest Wal Mart in less than 24 minutes To slow her down the local DPS o icer Fred Meaney decides to change the posted speed limit Which one of the speed limits below is the highest he can post but still catch her speed ing at some point on her trip Hint Apply the MVT 1 Speed Limit 40 mph 2 Speed Limit 50 mph 3 Speed Limit 55 mph 4 Speed Limit 60 mph correct 5 Speed Limit 45 mph Explanation Let s st0 g t g 24 be the distance Natalie covers after driving 15 minutes from her home to the nearest Wal Mart The fact that she drives very smoothly means that s is a function continuous on 0 24 and differen tiable on 0 24 Since she starts at home and nishes at Wal Mart 25 miles away 30 0 324 25 By the MVT therefore there exists some 0 in 0 24 such that i 8247s0 7 s c i 24 i 24 milesmin So Natalie must be travelling at a 125 speed of 7 mph 625 mph at some point on her trip Consequently the highest speed limit the DPS o icer can post but still catch her speed ing is Speed Limit 60 mph 004 100 points Determine if the function at 12 7 x satis es the hypotheses of Rolle7s Theorem on the interval 0 l2 and if it does nd all numbers 0 satisfying the conclusion of that theorem 1 c 8 9 2 c 9 3 hypotheses not satis ed 4 c 5 5 c 8c0rrect 604 5 Explanation Rolle7s Theorem says that if f is 1 continuous on 1 b 2 differentiable on a b and 3 u fa 07 then there exists at least one a a lt c lt b such that fc 0 uppal mu2465 7 HW17 7 meth 7 54865 3 NOW the given function at acx 12 7 at is de ned for all x g 12 continuous on 700 12 and differentiable on 700 12 In addition 712 ND 7 0 In particular therefore Rolle7s theorem ap plies to f on 0 12 On the other hand by the Product and Chain Rules 2473 ac V1277L fltgt 2x127ac 2x127ac Thus there exists a 0 lt c lt 12 such that 2473c fltcgt2 m0 005 100 points in Which case Determine if the function at 7 2 7 x 7 1 satis es the hypotheses of the Mean Value Theorem MVT on the interval 73 3 If it does nd all possible values of c satis fying the conclusion of the MVT 3 1 c 7i 0 2 hypotheses not satis ed 3 3 c i 0 7 2 3 4 c i 77 2 5 c 0c0rrect 3 6 c 7 2 Explanation The MVT says that if f is 1 continuous on 1 b and 2 differentiable on a b then there exists at least one a a lt c lt b such that fa u f a i b 7 a Now at 2 7 x 7 1 is a polynomial function so continuous and differentiable everywhere in particular the MVT applies on 73 3 But flt73117 While fQE 27 Thus fc 2C7 lt5gtTlt11 71 6 Hence the only value of c satisfying the con clusion of the MVT is keywords MVT cubic Mean Value Theorem 006 100 points Determine if the function at 310 132 7 3023 satis es the hypotheses of the MVT on the interval 0 2 If it does nd all possible values of c that satisfy the conclusion of the MVT 1 i1 072 2 2 c 7c0rrect 3 301 4 hypotheses not satis ed

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