exam-final-mat1720-b

Studocu is not sponsored or endorsed by any college or university Exam Final MAT1720 B Calcul différentiel et intégral I (University of Ottawa) Studocu is not sponsored or endorsed by any college or university Exam Final MAT1720 B Calcul différentiel et intégral I (University of Ottawa) Downloaded by Bintoup (binou.24.05@gmail.com) lOMoARcPSD|31381298

MAT1320B Final Exam 8 December 2017 Calculus I Elizabeth Maltais LAST NAME: FIRST NAME: STUDENT NUMBER: • This is a three hour exam. • No calculators are permitted. No notes, books, papers of any kind, or any other aids. Scrap paper will be provided on request. • Print your name and student number on this page. • Verify that your copy has all 15 pages (including this one). • Writeyoursolutionsinthespaceprovided(usethebacksofthepagesifnecessary). Youmust show all of your work. • Cellular phones, unauthorized electronic devices or course notes are not allowed during this exam. Phonesanddevicesmustbeturnedo ff andputawayinyourbag. Donotkeepthemin yourpossession,suchasinyourpockets. Ifcaughtwithsuchadeviceordocument,thefollow- ing may occur: you will be asked to leave immediately the exam, academic fraud allegations will be filed which may result in you obtaining a 0 (zero) for the exam. By signing below, you acknowledge that you have ensured that you are complying with the above statement. • Sign below to acknowledge that you have read these instructions. SIGNATURE: • Do not write below this line. 1 2 3 4 5 6 7 8 9 10 11 12 13 total /3 /6 /3 /3 /6 /6 /4 /6 /6 /8 /4 /5 /+4 /60 Downloaded by Bintoup (binou.24.05@gmail.com) lOMoARcPSD|31381298

MAT1320B Calculus I Final Exam 8 December 2017 page 2 of 15 1. Find the derivative directly from the definition for the function f ( x ) = 1 x +2 . You must use [3] the definition, not some other method. 2. Find the derivative of each function. [6] a) f ( x )=sin(ln( x 2 )) b) g ( t )= t ln( t ) fyy = qkjmo fKth)g# =fmsoFt¥yk+Fx =lfjmo k¥+2 - ¥2 = him a- = :* .tn#t*ynfyEIgITItt*=ghjmoyt2-x-hx+ht2)KtZ)(h1f'K1=cos(lnk4)(xtz)Kx ) ln(gHD=ln( that ) ⇒ lnlgttt )=lnHlntH ⇒ gttjg 'H= ttbnttltlntttt ⇒ gtttgtttfflnttttbntttf ] ⇒ gytktfnty.ph#y Trouvez la derivee de f(x) utilisant la definition de la derivee: (N'employez pas les regles de derivation). Trouvez la derivee de chacune des fonctions suivantes: Downloaded by Bintoup (binou.24.05@gmail.com) lOMoARcPSD|31381298

MAT1320B Calculus I Final Exam 8 December 2017 page 3 of 15 3. Estimate the value of f (0 . 1) using a linearization of f ( x )=tan( x )+1. Choose the point a of [3] the linearization appropriately. 4. Show that d d x Z x 2 3 1 / 2 1+ t d t + Z 2 tan - 1 x tan( t )d t ! is zero. [3] 4×1 = Ha ) + f ' (a) ( × - a) Let a=O . fkktank ) +1 Hot tank +1 = 0+1=1 f ' k ) = seek ) f 'H=sec44= ¥ ,p= fz =L : . LK ) = It 1 ( x - o ) = Hx % f ( 0 . 1) ~~L(O . 1) = It 0.1 By FTC 1 , a¥(B¥tat + t.IN?.nafttdt ) =Y÷×zk4 ' + - da , fstantxtanttldt =tgGy÷ - ( tankan ' ' KD . Clan ' ' KH ) =*× . - Handy .tt ) ) =X = X - I Fxz ltxz = 0 Estimez une valeur de utilisant une linearisation de Faites un choix approprie' du point a de linearisation. Montrez que = 0 Downloaded by Bintoup (binou.24.05@gmail.com) lOMoARcPSD|31381298

MAT1320B Calculus I Final Exam 8 December 2017 page 4 of 15 5. Consider the curve defined by y + xe y = x 2 . [6] a) Give d y d x in terms of x and y . b) Findallvaluesof x suchthatthepoint( x, 0)isonthecurvedefinedbytheaboveequation. For each of these give the slope of the tangent line to the curve at that point. ytxey =×2 ⇒ da¥t1eY+xeY.dd¥ =2x ⇒ day + xeY.ge#=2x-eY ⇒ aduxtdtxey ) =2x - EY ⇒ day = 2×-1 ltxet y+×eY=×2 at ( × ,gy=o ⇒ 0t×e°=×2 ⇒ ×=×2 =) O=×2 - × ⇒ o=x( x-D ¢ x X=O 11=1 at ( 0,01 , slope = data = 210164=-1 ate ,o ) , s1°pe=ad¥=2yfpe÷=2÷=z Considerez la courbe definie par: Trouvez en fonction de x et y. Trouvez la valeur de x telle que le point est sur la courbe d'equation donnee ci-dessus. Pour chaque valeur, trouvez la pente de la tangente a' la courbe en ce point. Downloaded by Bintoup (binou.24.05@gmail.com) lOMoARcPSD|31381298