Question

please just answer c, d, and e

Puges together UL Writing except your writing exce For the following questions, let P(t) =- 1+e? 1. Derivatives: a. Using At = 30.5 in the definition of the derivative, find both an underestimate and an overestimate for P'(2). What is a bound on the error for these approximations? b. On a full-page graph, clearly label P'(2), your underestimate, your overestimate, the error for each approximation, and the error bound. (Use an appropriate scale to best represent your answers!) c. Find an underestimate and overestimate to approximate P'(2) accurate to within 0.001. d. Use differentiation rules to compute the exact value of P (2). e. What are the exact errors for your approximations in Parts a and c? Are the errors smaller than the corresponding error bounds?

Answer 1

C PLt +et Usina he definition of the clenivctive we can either ppreximate p'it aJ . PLt+Dt p(t OY appimaute plt f'Lt Plt PLt-at) - P(2) Pl2.5) PL2+0.5) o. S 5 1-fe - P(2-0.5) P(2) _P(2) - P(i.s) ) - O S 1e2 +e-s C1264 Unclerestimate for ple) OORG7 for pa) o1264 Over estimate Somewhere Nou, the ercut yalve of pte) u'es between o.0867 ond o.1264 0.264

2) in ei ther of our marimum error estimates equal to he clistane loetween o.08G ano 0 1264 ersor bound O 037 error bound hwo cleeimal plauces only, you went Unelerestimete fer pl2) O. o3 Overestimate for p2) O.13 CrYor bouned

saw eosier, for As we given At Uneler estimute for f'(2) P (2+0P2) for Plez) Over estimate Pl t P(2-ot) esrorbound Underesfimate Overestimute P (2) - Pl2-0t P(2t ot) () . errer bound P(2 -0t) - P(2 +At) 2-P (2) : . Led s ty different valuey of ot Over eshimate exTor bounad Unelerestimate O10go O.O07J o 1030 o.070 Ot046 o os o -00og error bouned for ot o.0l dhe see that we hich is than o.001 aueurate ithin o.001 bnderestimeute Goveresthmte for 2) cre Undereshimate O-I04G Overestimate Oos4 Underestimate 104 Upto three desimed Overestimate o.105 Ploe

d) PCt e-b cli'ferenhaton rde for funhon of the form; CA in our Ce 1+e- - et (1e-Co )-e 1 e. ercut volue of Plzs a.1050 f((2) (upto three de'mel plaory) = o-10s Cupto tao cdoi'mal ploery ) plc2)= 011

cej eroct error in port (a) . exror in ndderes timutc O 05O - 0.086子 o. 018 3 crror in dverestimale o .2G4 o loso O02 see that in both dhe coses, he exror is e tess thon he bound o.0337 ersor Upto toe desimal plauez . OO2 error in unclerestirede O C2 O 13 OI error in overes innete tess me than Cre see that he errors e o.o4 error oOunal Now, exat error in port ce) Upto 4 euoul pluces . exror in underestimate ニ ○. lo5o ~ 0.1046 = 0- 000 O.IO54 o1oso exror in overeshimate hon the erTor see that dhe errors ort less twe O oo bound 3 oleimal p(oeez . Upto 0 00 exror in underestimate IOS005 errar in overeshimate itn ne error Care ne Crror wc see that bouno