Solutions to Exam 2 Ma123(Green version), F07

# Calculus (With Analytic Geometry)(8th edition)

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Unformatted text preview: Math 123 CalculusI Name: jobmo/Uﬁ - ﬁrm Exam I Fall 2007 RWJ No calculators allowed, one page of notes allowed Instructions: 0 Read questions carefully 0 Help me award you (partial) credit by showing your work (except on problems 4 and 5) 0 Note point values on questions — 105 points are listed, 1 00 points is possible 0 Check your answers you ﬁnish early 0 The exam ends promptly at 2:5; 0 Good luck! 1 / 2 8 2 / l 5 3 / 2 O 4 / l O 5 . / 8 6 / 9 7 . / l 5 /105 1. (28 pts.) Compute the derivatives of the following. Don ’t waste time simplz‘ﬁ/ing. a. 5x7 +3x3 —6x—10 95%cJ' 7762— e b- (5x6 + 2x4 —7x2 +x+3)sinx (30x5+3w3—I+x+|) 5.4;): 4_ [5Ké*2x¢'7x1+‘7c+3) (o- x W 2x3 +x—5 (5W5 7*” Haw 7%) » (whawh wwzaam) MW ( twg‘t'wc—é')?‘ C. d. tan[(x2 +5x+3)“2] —// klfo'z‘ +§x¢3)’"3: é(7c"k2;,r+g) 3' [27‘ +5) 2. (15 pts.) The position of an object at time x is given by f(x)=\/x+3 a. Find the position of the object at time x =1. Also ﬁnd the position of the object at time x = 6. PM = J7 =69 r9 [6) :ﬁ :1 b. What is the average velocity of the object over the time interval [1,6]? Q/é)-,P/)) 3-1 = L K— 1 2:7 ; c. Give an expression for the average velocity over the time interval [x, x + h]. XIXM) -Plx) _. \i'xrkk} - Jx+3 In .. ________________. (1. Using your answer from part c (and not any “shortcut” methods) determine the instantaneous velocity of the object at time x. [70,“) , (70,} m 447“) : (7;+Lar3)~—(7u-3) 14 m +‘1m3 la [ 79:46 hi x43) 1 ________________._. (m 4"] war?) ll glxkh) - IF/w) A M; I _,__l___ =ﬂf'zt2’3 441 J; J ,7 Jx 3. (20 pts.) Find Don’t waste time simplifying. x x2 sinx a. = y x3+2x+3 ._ (zﬁﬁnk XIMW)(K3+7—xr3) ’— (767-9‘; ,C')(3?(1+_7_ [ngzx f 3)1 b. y = (8x3 + 6x+ 5)3(x2 +3x+2)10 -, 5/3K34’éxi5)7(2+w‘+ 4)('X1vf'37:+1) ” + (5X9*4°<*'5)9 /0 {KW-37:41,)? ﬁnd» 3) 92 will be in terms of both x and y) c. 2xy2 = sin(xy) (your expression for d x 1. J - jig—*7] = J; Raw») zflfgﬁ)»'xj£,71 -: 410—037)'j-:[K0) .J ,L (71., x27 : Mbwy) . [LU/7+ 'K ify (kc? ' qCMJW») , gm/KO)’ 7"‘7 42¢ m é: 1x ’ 4X7'7Cm’hp7) 4. (10 pts.) For each of the following indicate whether the statement is true or false (no work necessary): a. If f is not continuous at x, then f is not differentiable at r False (circle one)? b. If f is continuous at x, then f is differentiable at x. True [email protected] one)? X-)C c. For any function f, lim f (x) = f (c). True [email protected]/(circle one)? (1. If f and g are differentiable, then (f + g) '= f '+ g ‘ @r False (circle one)? e. If f and g are differentiable, then (fg)‘ = f ' g'. True or @ircle one)? 5. (8 pts.) Fill-in the blank: a. If lim f (x) = f (c), then we say that f is (VP 7710004)} at c. X—>C b. If f '(x) exists, then we say that f is p/FFW W“:— at x. 6. (9 pts.) Suppose f is a continuous function on [2,8]. Also suppose f(2)=-3 f(8)=7 ff. a. f necessarily has a zero (or root) between what two x values? 4—H—‘i— 2 a2 7— 5 b. If the bisection method is used, at what x value do we now evaluate f? 2 + S’ ' " 7. (a . Suppose f is positive at the x value you gave in p . ere do you now evaluate j? ‘33. \ (K . L pltp ,a O? l. C'lv p W W 7. (15 pts.) For the following limit problems: 0 If the limit exists, give me the (ﬁnite) value of the limit 0 If a limit does not exist, answer using 00 or —00 when appropriate, otherwise write “Does Not Exist” \ b limx2+4x+l _ (Daur 710*] _ ,5 £72.) . x-—>1 x+2 3 ...
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