History Bookmar1cs Profiles Tab Window ~ t Chrome File Edit - -- - - - - -- - - -- - - ----=---- - -- - - - - - -------'-------- - - - - - - - - -- - - - - - -- =----!: x \';\ 4.1 HW: Areas and Distance - x ( ) Resources for 4.1: Fal21 MATH x I O 4.1 HW: Areas and Distance • • • + View i Help 91% ~ • @ Sat 7 :07:13 PM o_ 0 -0 webassign.net/web/Student/Assignment-Responses/last?dep=27003985 1. [-/2.22 Points] MY NOTES SCALC9 4.1.001. DETAILS 11 ASK YOUR TEACH ER 11 PRACTI CE ANOTI-iER U n less specified , all approximating recta ngles are assumed to have the same w idth. T he graph of the function f Is g iven. y + 6 y = ftx) 4 2 +-------L-- -----'-- - - - - x 4 (a) 8 Ci) By reading values from the g iven graph off, use five rectangles to find a lower estimate for the area under the given graph off, the x-axis, and the lines x to o ne deci mal place.) Sketch the rectangles to Illustrate the estimate. =0 and x = 10. ( Round you r answer ~ I • File Chrome • ➔ Edit --- View History () Resources lor 4.1: Fal21 MATH x C Bookmarks O Profiles Tab 4.1 HW: Areas and Distance s/last? webassign.ne t/web/ Stude nt/Ass ignme nt-Res ponse i (a) • Windo w 14;. x Sat 7:0 8 :01 PM 4 .1 HW: Areas and Distanc e - r x 0 * dep=2 70039 85 =0 and x = 10 . ( Ro u nd your answe r te. Sketch the rectan gles to Illust rate the estima - L---i----;---- I ~ .L -r 4 y y ,- r- i 1 I y = fix) i 2 y 6 y = ftx) 8 X X X 4 0 I _ _J - - 2 (j) O 0 ·- + x -axis, and the lines x for the area und er the g iven graph of f, the use flve rectan gles to flnd a lower estima te By readin g values from t he given graph off, to one decim al p lace.) y Q. Help 4 8 (j) O 4 8 (j) _ __ C_h_ro_m _e __F_il_e__ Ed_i_t _ Vi _1e_w__H _i_ s t_ory ....:....__B__ ookm _ arks _ _P _r_of_il..:. e:.. s _ l:..:a..:. b_ W .:.:..::in.:.:d:..:o.:.: w_.:...: H=el:::P_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ r i • • • ( ) Resources for 4.1: Fal21 MATH X i O 4 .1 HW: Areas and Distance x ~ 4.1 HW: Areas and Distance - r x ,e 91% M • @ Sat 7:08:10 PM Q. O __ + C webassign.netjweb/Student/Assignment-Responses/1ast?dep=2700398 5 '-!.I y .. -+ +- 6 y = fi x ) - l + 4 4 0 8 (j) By read ing v alues f rom the given graph of f , use five rectangles to find an upper estim ate for the area under t he g iven graph of f, the x-axis, and the ll nes x to one decimal p lace. ) Sketch the rectangles to Illustrate the estimat e. y y y 6 6 y = fix) 4 y = fi x ) y 4 = ft x) =0 and x = 10. ( Rou nd your answer • • ➔ () Resources for 4.1: Fal21 MATH x / ( : 4.1 HW: Areas and Distance C i X II-Ji. 4.1 HW: Areas and Distance - I X + * webassign.net/w eb/Student/Assignment-Res p onses/ las t?dep= 27 003985 By read ing valu es f rom the given gra p h off, use five rectang les t o fi nd an u pper e st imate for the area under the given graph of f, t he x - axis, and the lin es x • 0 and x • 10. ( Round your an sw er t o one d ecim al p lace.) Sketch t h e rectangl es t o Illustrate the est imate. y ,- y y 6 y = fix) 4 2 0 y 8 X X X 4 4 8 (i) O 4 8 • • • ~ ➔ 0 Resources for 4.1: Fal21 MATH C i x j () 4.1 HW: Areas and Distance x "'A 4.1 HW: Areas and Distance - x webassign.net/web/Student/Assignment-Responses/last?dep=27003985 X 4 0 y + 8 X (j)O 4 ~-t----+----t-- 1 0 ( b) + 4 8 Ci) Find new estimates using ten rectangles in each case. (Round your answers to one decimal place. ) lower estimate upper estimate 8 • f- • • ➔ () Resources for 4.1: Fal21 MAT!-< C i x I C) x 4.1 HW. Areas and Distance V-,.. d .1 HW: Areas and Distance - x + webassign.net/web/Student/Assignment-Responses/last?dep=2700 398 5 2. [-/2.22 Points] DETAILS SCALC9 4.1 .002. M Y N O TE S Th e graph of the fu nction f Is given. y + i 32 24 y = ft. x ) 16 8 4-0- 0- -+---8- - -l-6- -2-4- -3-2- - - 4-8- X Ci) ( a) Use six rectangles to fi nd estimates of each type for the a rea under t he g iv e n graph o f f, t h e x -ax is, a nd the lines x (I) L 6 ( sample points are l eft e ndpoints) L6 =1~ - - - ~ (II) R 6 (sample points are right e ndpoints) R6 = ~i_ _ __ (111) M 6 (sample points are midpoints) M6 = =o and x ~ 1 1. 11 ASK YOUR TE ACHER J [ ! "RAC rl C[ J\l'l • • • 0 Resources for 4.1: Fal21 MATH x i I (j 4.1 HW: Areas and Distance x '1\ 4.1 HW: Areas and Distance - F x + webassign.net/web/Student/Assignment-Responses/last?dep=27003985 (a) Use six rectangles to find estimates of each type for the area under the given graph off, the x-axis, and the lines x (1) L 6 O O and x "' 48. (sample points are leh endpoints) L6 ""'~'- - ~ (Ii) R 6 (sample points are right endpoints) R6= ~'- - ~ (ill) M 6 (sample points are midpoints) M6"' ~'- - ~ (b) Is L 6 an underestimate or overestimate of the true area? 0 0 (c) (d) Is R overestimate underestimate 6 an underest,l mate or overest_lmate of the true area? 0 overestimate 0 underesti mate Which of the numbers L , R 6 , or M 6 gives the best estimate of the true area ? 6 Q L6 Q M6 Q R6 3. [-/2.22 Po ints] DETAILS ;:;iiiiiii.~~ ~ I SCALC9 4.1.504.XP .MI. a~ rel~ si 0 [ MY NOTES I G • • • Q Resources for 4.1: Fal21 MATH X i IQ ...,._ x 4.1 HW: Areas and Distance - + web assign.net/web/Student /Assignment-Responses /1ast?dep=2700398 5 3. [- /2.22 Points] DETAILS =I * SCALC9 4.1 .504.XP.MI. (a) Estimate t he area und e r the graph of f(x) R4 x 4.1 HW: Areas and Distance I = 10 MY NOTE S cos(x) from x = Oto x = ,r/2 using four approximating y ftx) = 10 C0 S(X) ftX) 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 7f 11 3 7f 7f 8 4 8 2 y ; -- X 0 - - ~ ~11 8 ---11~- - = 10 C0S( X) - 4 -::3 ~,r- - - -,r "- X 8 2 y /(X) = 10 CO S(X) 10 9 8 ASK YOUR TEACHER I\ PRACTICE AN OTHER rectangles and right e ndpoints. (Round your answ ers to four decimal p lace s .) Sketch t he graph and t h e r ectangles. 10 I\ /( X) = 10 C0 S( X) I z@. 0 ➔ C 1au Edit File View HistorY Resources for 4.1: Fal21 MATH X x I () 4.1 HW: Areas and Distance ~ 4.1 HW: Areas and Distance - f >< i webassign.netJweb/StudentJAssignment-Responses/last?dep=2700 3985 y y 10 ftx) 10 ftx) + = 10 cos(x ) = 10 cos(x) 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 ,r ,r 3 ,r ,r 8 4 8 2 X ,r 0 ~8 ,r 3 ,r 4 8 Is your estimate an underestim ate or an overestim a te? 0 underestim ate 0 overestima te (b) Repeat part (a) u sing left endpoints. L4 = I I Sketch the graph and the rectangles. y y 10 9 8 ftx) = 10 COS ( X ) ft X) = 10 COS( X) ;r X 2 • • • Q Resources for 4.1: Fal21 MATH x i lI I () 4.1 HW: Area s and Distance webassign.net/web/Student/Assign Vji. 4.1 HW: Areas and Distance - x ment-Respo nses/1ast?dep=270 + f X 03985 (b) Rep eat part (a) usin g left end poin ts. L4 = I I Ske tch the grap h and the rect ang les. y y 10 f{X) 10 = 10 COS ( X) 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 9 8 7 6 = 10 COS( X) ~,r ,r 3,r Tr 8 4 8 2 X 0 y 10 f{X ) ,r ,r 3 ,r Fr 8 4 8 2 y JlX) = 10 COS( X) 10 9 8 7 6 /(X) = 10 COS( X) X •• • ➔ 0 C Resou rces for 4.1: Fal21 MATH x i J () 4.1 HW: Areas and Distance X ~ 4.1 HW: Areas and Distance - f x + webassign.netjweb/Student/Assignment-Responses/1ast?dep=270 03985 ] 0 1l 1l 37f 8 4 8 \ \I X 1l 2 0 y 3 ,r ,r 8 4 8 2 10 ft..x) = 10 COS( X) 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 1l 1l 3 ,r ,r 8 4 8 2 Is your estimate an underesti mate or an overestimate? 0 underestimate 0 overestimate ,r X y 10 0 r. * X 0 ft..X) = 10 COS( X) ~- ,r ,r B 4 3 71" 8 ,r 2 X • f- • • ➔ () Resources for 4.1: Fal21 MATI-' C I (j 4.1 HW: Areas and Distance x l'J\ 4.1 HW: Areas and Distance - f x 0 + webassign.net/web/Stud ent/Assignment- Responses/last?dep= 27003985 i 4. [-/2.22 Points] Let f (x) (a) x =6 DETAILS MY NOTES SCALC9 4.1 .005. + 4x 2 . Estimate the area under the graph off, the x - axis, and the lines x R3 = 11 ASK YOUR TEACHER 11 PRACTI CE ANOTH ER = -1 and x = 2 using three rectangles and a right endpoints. IL___ _, Improve your est imat e by using six rectangles. R5 = L I_ _ ___, Sketch the cu rve and t he approximating rectangles for R 3 . y y y 20 20 20 15 15 15 10 1-- - - --f 10 ·- - - -- - 5 5 5 )I 2 X X X 1 (j) 0 -1 1 2 2 (j) File Chrome Edit History View ( ) Resources for 4 .1: Fal 21 MATH ' • C ➔ i Bookmarks Profiles Tab X / ( ) 4.1 HW: Areas and Distance Window X Q Help --- l'i,. 4.1 HW: Areas and Distance - x + webassign.net/ web/Student/Assignment- Responses/last?dep=27003985 I I I I I o-1 1 2 (j) I Sketch the curve and the approximating rectangles for R 6 . y y 90% [ii]• ~ Sat 7 : 1 Q:O1 PM Q. 0 0 y I * y l. 1 chrome l l~■lt■iFaIlllrlliPlila-.--- ---~--•----.:i-.....;.;.~.......------ ------ ------ File Edit History View Bookmarlcs Profiles Tab Window Help - -- -- -------- ---------------- -------- -------- ----0 • C ➔ Resources for 4.1: Fal21 MATH X / ( ) 4.1 HW: Areas and Distance x '1\ 4.1 HW: Areas and Distance - r x ~ :€ 90"/4 ~• ~ Sat 7:10:09 PM + Sketch the curve and t he approxima ting rectangles for R 6 . y y y 20 15 I I 10 I _:.;...:.c._ o-1 ...;::..+--=.=.ii..=...::,_.......,o::.::,._,.,._~ 0.5 - 0.5 y 1· 1.5 _._.._ X 2 X (]) o-1 - 0.5 0.5 1 1.5 2 (j) 0 - 1 -0.5 0 ·- 0 webassign.net/web/Stu dent/Assignment-Responses/last?dep=27003985 i Q. 0.5 1 1.5 2 (j) • \.IIIIU••• - r~ ••• C (- Resources for 4.1: Fal21 MATH .._, ➔ i . webass1gn.n e x •--i I L, 4 1 HW·. Areas and Distance X . /I t?dep=270 03985 as . t/W bjStudent/A ssignment- Responses ~- _ _ e y X (b) Repeat part (a) using a le~ endpoints. L3 = 'l...--------' Sk etch the curve and the approximat ing rectang les for L 3 . y y Chrome • • ➔ File Edit View History () Resources for 4.1: Fal21 MATH x Bookmarks Profiles Tab I () 4.1 HW: Areas and Distance Window x 'ii\ 90% ~ • ~ Help 4.1 HW: Areas and Distance - r x Sat 7:10:27 PM (b) Repeat part ( a) using a le~ e ndpoints. L3 = L.I_ __ ...J L5 = L.I_ _ __, Sketch t he curve and the approximating rectangles for L 3 . y y y 5 0-1 y 2 X X X 1 G) 0 - 1 1 0 ·0 + i webassign.net /web/Student/A ssignment- Responses/1ast?dep=270039 85 C Q_ 2 G) 0 - 1 1 2 Ci) •• • ~ ➔ .-, U C Resources for 4 .1: Fal21 MAT H i x J () 4 .1 HW: Areas and Distance X '1\ 4 .1 HW: Areas and Distance - x + webassign.netjweb/Student/Assignment-Responses/last?dep=27 003985 X o-1 1 2 (i) X X o-1 1 2 1 (j) 0- 1 y X o- 1 1 2 (j) Sketch the curve and the approximating rectangles for L 6 . y 20 20 15 a ~ .. y y 20 2 (j) Chrome • ➔ File Edit View History Bookmarks Tab Profiles () Resources for 4.1: Fal21 MATH X / () 4.1 HW: Areas and Distance C i Window x Q Help ;t 90% Rl• ~ Sat 7:10:40 PM + ~ 4.1 HW: Areas and Distance - r x 1 2 Q) Sketch the curve and the approximating rectan gles for L . 6 y y y 20 15 10 _f]~~~IC!j~~:f:_!~!_ ,l~ ~ ~'..,_j_ X 2 1.5 0 .5 -0.5 1 0-1 y 0 0-1 - 0.5 0 .5 1 1.5 0 ·Cl webassign.net/web/Student/Assign ment-Responses/last?dep=270039 85 0-1 Q. 2 Ci) ~ II • • Chrome • File Edit View I S r ¢; 11w History ( ; Resources for 4.1: Fal21 MATf-- X Bookmarks J () Profiles Tab 4.1 HW: Areas and Distance Window x ,..._ Sat 7:l C Help 4.1 HW: Areas and Distance - X + Iii webassign.net/web/Student/Assignment-Responses/1ast?dep=27003985 l . o-1 I X - 0.5 0.5 1 1.5 2 0.5 1 1.5 2 Ci) o- 1 y o- 1 (c) - 0.5 (j) Repeat part (a) usin g a m idpoi nts. M3 = M6 = ' - I- - - - - - - ' - I- - - - - - - Sketch the curve and the approximating rectang les fo r M . 3 -- X - 0.5 0.5 1 1.5 2 (j) 0- 1 - 0.5 1~ 1· L •'!'- . I ..._ 0.5 '111111 t, _..J .__ 1 ..., --~ 1.5 Window • n • Q Resources for 4.1: Fal21 MATH x i (c) / 0 4.1 HW: Areas and Distance x Q Help ','JI 4 .1 HW: Areas and Distance - r x ~ 90% ~ • ~ Sat 7:10:49 PM Repeat part (a) using a m idpoints. -I ::=======::: M6 •~'---- Sketch the curve and the approximating rectangles for M 3 . y y y 20 15 o- 1 y 2 X X X 1 G) 0 -1 1 0 ·0 + webassign.net/web/Student/Assignment-Responses/1ast?dep=27003985 M3 Q. ' 2 0 0 -1 1 2 G) a • f- • Chrome • ➔ File Edit History View •:) Resources for 4.1: Fal21 MATh C x Bookmarks I () Tab Profiles 4.1 HW: Areas and Distanc e Window x "'i • Help ~ 4.1 HW: Areas and Distance • f x ~ 90% ~], ~ Sat 7: 10:53 PM + i webassign .net/web/ Student/A ssignment - Responses/last?dep=27OO3985 X o-1 1 X X 2 CD o-1 2 Ci) 1 2 1 0 0- 1 y 0-1 Sketch the curve and the approxima ting rectangles for M 6 . y y y 2 (j) Q. ~ • ~ • ( ) Resources for 4.1: Fal21 MATH X / ( ) 4.1 HW: Areas and Distance • ➔ C r x "'i 4.1 HW: Areas and Distance - r x f 90% ~ • ~ Sat 7: 10:59 PM u * w - I I Sketch the curve and the approximating rectangles for M 6 . I y y y I X o- 1 0.5 -0.5 y 1 1.5 2 (i) 0 - 1 -0.5 0.5 1 1.5 0 ·0 + webassign.net/web/Student/Assignm ent- Responses/last?dep=27003985 i Q. 2 (i) 0 -1 -0.5 0.5 1 1.5 2 (i) • ~ • • ➔ () Resources for 4.1: Fal21 MATH x C J () 4.1 HW: Areas and Distance x \AA. 4.1 HW: Areas and Distance - x + ii webassign.net/web/Student/Assignment-Responses/last?dep=27003985 X o-1 - 0.5 0.5 1 0.5 1 1.5 2 X Ci) o-1 0.5 - 0.5 y X 0-1 - 0.5 1.5 2 (j) ~ ( d) From you r sketches In parts (a) - (c), which estimate appears to be the most accurate? Q M Q R 0 L6 6 6 1 1.5 2 (j) 0- 1 - 0.5 0.5 1.5 z Flle Chroma ••• C t:cllt view IO () Resources for 4.1: Fal21 MATH x ➔ i 4.1 HW: Areas and Distance x VJ,. 4 .1 HW: Areas and Distance - F X + webassign.net/w eb/Student/Assi gnment-Respon ses/last?dep= 27003985 5. [-/2.22 Points) I MY NOTES SCALC9 4. 1.016. DETAILS h Use this definition with right endpoints to find an expression for the area under the grap ftx) 6. [-/2.22 Points) =~ 9x x-+ 7 , Of f I[ ASK YOUR TE ACHER as a llmlt • Do not evaluate the llmlt. 1:Sx:S3 MY NO TE S SCALC9 4.1.017. DETAILS 11 ASK YOUR TEACHER the limit. use thls definition with right endpoints fo find an expression for the area under the graph off as a limit. Do not evaluate 2 ftx) = 9 + sln (x), 7. [- /2.22 Points] 0 :S x :S n DETAIL S SCALC9 4.1.018. ~ MY NOTES the llmlt. Use this definit ion with right endpoints to find an express ion for the a rea under the graph of f as a limit. Do not evaluate f(x) = x2 + ~, 7 s x s 9 11 ASK YOUR TEACHER S • • Chrome • () File Edit View History Resources for 4.1 : Fal21 M ATH i Bookmarks j () X Profiles Tab 4.1 HW: Areas and Distance Window x "'Ji. Help • 4.1 HW: Areas and Distance - F x (:;;_:J ;t 90% lii]• • Sat 7:i i :24 PM DETAILS SCALC9 4.1 .018. 0 ,_ + 0 webassign.net/web/ Student/Assignment-Responses/last?dep=27003985 7. (- /2.2 2 Points] Q. * • CD M Y NOTES 11 ASK YOUR TEACHER 11 PRACTI CE ANOTH ER Use this definition with right endpoints to flnd an expression for the area under the g raph off as a lim it. D o not eva luat e the limit . f(x) 1 = x2 + 8. (-/2.22 Poin ts] ~ , DETAILS 7 s x s 9 SCALC9 4.1 .020. M Y N OTES 11 ASK YOUR TE ACH ER Det er mine a region whose area Is equal t o the g i ven llm lt . Do not evaluate t he limit. ~ _:_(.!_) 8 llm \ n-00 0 x9 0 ~ I = 1 n n on ( 1 , 2) x 9 on [0, 1) 0 x 8 on 0 x 7 (0, 1) on (0, 1) 0 x 8 on (1, 2) - - - - - - -- - - - . . . . j . _ ~ - ~ - - - - - - - - - -........-- - - - -- - : : - - - - - - - - --=---"'M-Y_N_o_TE_s__ [ ASK YOUR TEACHER :\\UL~~ ~ ~ . I\ PRACTI CE ANOTHER
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