ࡱ> %'$ 3bjbj 46||4 22$L.mmmQSSSSSS$<wQmmmmmw???mQ?mQ???|v?=0???nmm?mmmmmwwpmmmmmmmmmmmmmmmm2 R: SOLUTIONS Relationship between the Derivative and the Integral (Essay #4 AB Students) 1. In your paper, you wrote about both forms of the Fundamental Theorem of Calculus. Please solve the following problems and discuss how they relate to the Fundamental Theorem. a)  EMBED Equation.3  You must find the anti-derivative, then evaluate it at b and a as follows: -1/3cos(3b) - -1/3cos(3a) = -1/3cos(3b)+1/3cos(3a) The fundamental theorem would say F(b) F(a) where F is the antiderivative of sin(3x) b) If g(x) = EMBED Equation.3 , find g(x) This is the second form of the FTC. (Students might mention that the derivative of an anti-derivative would undo one another.) (They might also integrate and then take the derivative, so be ready for either.) Answer is sin(5x2)* 10x You might ask why the 10x is there. (Because of the chain rule) 2. In your paper, you discussed various ways that the derivative and integral could be used in a variety of contexts. Please discuss the following problem: Water is draining out of a tank at a variable rate as given by the graph and chart below. tR(t) gal/min0055102020303015350  INCLUDEPICTURE "C:\\Program Files\\TI Education\\TI InterActive!\\TIIimagefile3458.gif" \* MERGEFORMATINET  a) Approximate the volume of water that has leaked from the tank for 0 d" t d" 35 using the Trapezoid Rule with 5 intervals as indicated by the data in the chart. (0 + 5)*5 + (5+20)*5 + (20+30)*10 + (30+15)*10 +1/2(15+0)*5 because the intervals are not the same, this should be the setup. After the show the set up, you can just tell them that the numerical answer is 650 gallons. Ask them why the units are gallons (because you are finding the product of gal/min * min) b) The rate of the leak can be modeled by Q(t) = 16.78 sin (0.15x 1.25) + 14.6 How can you find the actual volume of water that has leaked from the tank for 0 d" t d" 35? Compare your answer with your answer to part a.  INCLUDEPICTURE "C:\\Program Files\\TI Education\\TI InterActive!\\TIIimagefile5316.gif" \* MERGEFORMATINET  Answer:  EMBED Equation.3  You can just tell them that the numerical answer is 619.39 gallons. Here students should discuss the fact that the actual answer is slightly smaller than the trapezoid rule, because trapezoid rule made straight segments when the graph is actually curved. Students should talk about the fact that the trapezoid rule is just an approximation but the definite integral gives the exact area. c) How can you find the average rate of the leak over the time interval from  GTUV  # $ % & * + V u w     # $ % & 3  ! ) * L M j k Ⱦ}sih=5B*phh"5B*phh]h]5B*H*phjXh"h-EHUjOQ h-CJUVaJh]h]5B*phhY)g5B*phh]5B*phh]jh"h"EHUjWOQ h"CJUVaJjh"Uh|;hv h"hG:) VW + w  3 * j k d e g t $P6&#$/IfgdiRl $a$gdQw  M R W \ e      $&(RT2u4>|~ҾҶҩྀ!jh]h]5B*Uphjhp&UjhTQUhYHuh^\h]h]5B*phhY)gh]5hY)g5B*phhC5B*phh]5B*phh3LhTQjH hp&Ujhv UhB5h"hv 1t u w y oS4$$P6&#$/Ifa$gdiRl $P6&#$/IfgdiRl kd$$Ifl0\ t 6P60644 lapytiRy z | ~ oS4$$P6&#$/Ifa$gdiRl $P6&#$/IfgdiRl kdj$$Ifl0\ t 6P60644 lapytiR~  oS4$$P6&#$/Ifa$gdiRl $P6&#$/IfgdiRl kd$$Ifl0\ t 6P60644 lapytiR oS4$$P6&#$/Ifa$gdiRl $P6&#$/IfgdiRl kd$$Ifl0\ t 6P60644 lapytiR oS4$$P6&#$/Ifa$gdiRl $P6&#$/IfgdiRl kdY$$Ifl0\ t 6P60644 lapytiR oS4$$P6&#$/Ifa$gdiRl $P6&#$/IfgdiRl kd$$Ifl0\ t 6P60644 lapytiR   &(tuojhhhhhhhhhhgdv kd$$Ifl0\ t 6P60644 lapytiR '()jk..b/d/0Z1B2D2T3V33gdDgd gdTQ'()UX_ry Mhijk...$.&.L.N.¸¸n_j*4PQ h CJUVaJ!jh]h 5B*UphUhhTQh]h]h 5B*phh=5B*phh*35B*phhY)g5B*phh 5B*phh]5B*ph!jh]h]5B*Uph%j:%h]hY)g5B*EHUphjģQQ hY)gCJUVaJ0 d" t d" 35? Find  EMBED Equation.3  and divide by 35. (Students might mention that this is the Mean Value Theorem for Integrals, A.K.A. the average value for integrals.) d) The average rate of the leak from 0 d" t d" 35 is 17.7 gal/min. Illustrate your understanding of the Mean Value Theorem for Integrals by showing this on the graph. INCLUDEPICTURE "../2011%20Materials/TIIimagefile29925.gif" \* MERGEFORMATINET  The area under the curve from 0 to 35 is the same as the area of the rectangle formed by drawing a line at y = 17.7 That is, is you averaged the rate of leakage from the time interval of 0 to 35 minutes; the average rate of leak would be 17.7 gal/min. 5. 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