5.1 How do we Measure Distance Traveled given Velocity? Student Notes


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1 . How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the xxis & yxis for the grph of these points. Drw rectngles using the lefthnd velocities s the heights of the rectngles. Wht unit is ssocited with the width & height of ech rectngle? Wht does ech rectngle re represent nd wht is its unit? Since ech rectngle re represents, the sum of the rectngle res my e used to estimte the totl distnce the cr hs trveled. distnce LHS We cll this LeftHnd Riemnn Sum estimte for totl distnce trveled. B) Agin, lel the xxis & yxis for the grph. This time drw rectngles using the righthnd velocities s the heights of the rectngles. distnce RHS We cll this RightHnd Riemnn Sum estimte for totl distnce trveled. C) Another wy to estimte the totl distnce trveled is y using trpezoidl res insted of rectngles. Agin, lel the xxis & yxis for the grph. This time connecting ech height with segment to crete trpezoid. Find the re of ech trpezoid nd then the sum of these res. Recll the re formul of trpezoid: distnce TRAP D) Find the verge of the LHS nd the RHS: How does this compre to the Trpezoid Riemnn Sum? Wht is the estimte of the totl distnce trveled in the first five seconds?
2 Let s look t the nottion tht represents the exct vlue for the totl distnce trveled rther thn the Riemnn Sum estimtes (LHS, RHS nd TRAP) for the res tht we found on the previous pge. v() t dt We red this s the definite integrl of v(t) with respect to t on the intervl from t= to t=. We interpret this s the re etween the curve v(t) nd the txis from t= to t=. Ech product v() t dt represents we llow dt, the width of ech rectngle, to get infinitesimlly smll then the numer of rectngles will ecome infinitely lrge. Sum up ll of the res of these rectngles nd we will hve n exct vlue for the distnce trveled. The integrl symol mens. The integrl symol hs limits of integrtion with the lower limit t = nd the upper limit t = which represent the left nd right ounds of the re we re finding. EX ) A polr er is moving through the wter, followed y kyk of Eskimos, during 3 minute time intervl. A tle of speeds of the er, vt (), t (min) 3 is shown for minute intervls of time, t. vt () ft/min A) Plot the points from the tle on the grph elow then pproximte v ( t) dt with Riemnn sum, using the midpoints of 3 suintervls of equl lengths. Using correct units, explin the mening of this integrl. NOTE: You must use dt in the tle. You cnnot invent dt to e midpoint height vlues. Midpoint Riemnn Sum: MID3= B) Find the other 3 Riemnn Sum estimtes i) LHS6 ii) RHS6 iii) TRAP6 In this unit we will lern how these estimtes compre to ech other nd to the ctul integrl sum.
3 . & 7. Comprison of Riemnn Sum Approximtions Student Notes For ech Riemnn Sum: () drw rectngles or trpezoids representing ech geometric pproximtion. () Show the clcultion to estimte the definite integrl. (c) Is the Riemnn Sum pproximtion n underestimte or overestimte? Figure : f ( x) x Figure : f ( x) x Use LHS to pproximte ( x ) dx. Use RHS to pproximte ( x ) dx Under or Over estimte? Under or Over estimte? Figure 3: f ( x) x Figure : f ( x) x Use TRAP to pproximte ( x ) dx Use MID to pproximte ( x ) dx Under or Over estimte? Under or Over estimte? Find the verge of LHS & RHS. Cn you explin lgericlly nd geometriclly why the verge is equl to TRAP pproximtion. 3
4 . Definite Integrl & Riemnn Sums Find n pproximtion for the definite integrl y using Riemnn sums with suintervls using left endpoints, right endpoints, midpoints nd the trpezoidl rule.. dx x x LHS = RHS = MID = TRAP =. 6 3x dx 3 x LHS = RHS = MID = TRAP =
5 . An Introduction to the Definite Integrl Student Notes Directions: Use the grphs for f x sin x shown on the next pge for the questions which follow.. Estimte the re under the grph of the sine curve over the intervl, your work. y counting locks. Show. Now, use 3 prtitions find left hnd sum LHS3 nd right hnd sum RHS3. Shde your LHS in lue nd RHS in red. Show computtions in the spce provided elow. Averge LHS3 nd RHS3 to improve your estimte of the re under the grph of the sine curve. This verge is the TRAP3
6 3. Now, use 6 prtitions find LHS6 nd RHS6. Shde the LHS in lue nd the RHS in red. Show computtions in the spce elow.. Finlly, use prtitions find LHS nd RHS ut NO DRAWING nd show the setup nd finl nswer only. Use your clcultor!. Wht seems to e hppening to the estimte for the re s you increse the numer of prtitions used in clculting the Riemnn Sums in the previous 3 questions? 6. Write definite integrl to represent the ctul re under the sine curve over the intervl,. 7. Use the clcultor feture MATH 9: to evlute the definite integrl you wrote in the previous question. 8. Wht do you think the re under the sine curve would e over the intervl[, ]? Verify using MATH 9: 9. Wht do you think the re under the sine curve would e over the intervl[, ].. Find the re under the sine curve over the intervl [, ] using MATH 9: Wht hppens?????? Why? 6
7 . & 7. Summry of Under/Over Estimtes of Numericl Approximtions Student Notes For the LHS & RHS the pproximted integrl s reltion to the exct integrl vlue depends on whether the curve is incresing or decresing. Exmine the four functions elow which illustrte the four comintions of incresing/decresing nd concve up/down functions. Red rectngles re used for LHS. Blue rectngles re used for RHS. Complete ech sttement to show the LHS or RHS reltive to the ctul integrl vlue. f x is decresing on,, then dx x f x is decresing on,, then. x dx f x is incresing on,, then. x dx f x is incresing on,, then x dx CONCLUSIONS: f x is incresing on, f, then x dx f x is decresing on, f, then x dx 7
8 For the TRAP & MID the pproximted integrl s reltion to the exct integrl vlue depends on whether the curve is concve up or concve down. Blue trpezoids re used for TRAP. Red rectngles with tngent segments t midpoints re used for MID. f x is concve up on, f, then x dx f x is concve down on, f, then x dx TRAP & MID re etter pproximtions for the ctul integrl vlue thn LHS & RHS pproximtions. FINAL CONCLUSIONS: In ech inequlity, fill in the lnk with LHS, RHS, MID or TRAP to show their pproximtions reltive to the ctul integrl when the ehvior of the grph is known. f x is incresing & concve up on, then, f x dx f x is incresing & concve down on, then, f x dx f x is decresing & concve up on, then, f x dx f x is decresing & concve down on, then, f x dx 8
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