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1 I NAME: E - SECTION: Page 1 MATH COMMON FINAL Spring 2005 General Instructions: 1. The exam consists of 10 pages, including this cover; the test is printed on both sides of the page, and contains 9 questions. Please verify that your copy of the exam has all 10 pages and 9 questions. 2. Write your full name and section on the space provided at the top of each odd numbered page. 3. Justify all your answers by showing your work. A final answer with no work to support it will get no credit, even if the answer is correct. 4. Clearly mark your final answer by boxing it or circling it. 5. Use the space provided for each problem. If you need more space for a question, raise your hand and request an extra page. Write the problem number and your name at the top. When you turn your exam in, make sure we staple the extra pages into the exam. 6. Each question indicates its point value. The exam consists of 200 points in 9 questions. 7. If you have any questions, raise your hand, and someone will come to answer your question. 8. GOOD LUCK! Question Grade Out of Question Grade Out of Total 200
2 Page 2 MATH Common Final - Spring Evaluate the following limits using any valid method. If the limit exists, you must give its exact value. If it does not exist, and equals 00 or -oo, then indicate this. If it simply does not exist, then state so explicitly. (10 points each, 30 points total) (a) lim x - 2 = x x2 U V10 x -> ' +X A 5x2 (b) :.3x2-2x+13 "`^' x->co x2- (3xZ-^x+ 13) l:r^ x-a o0 5 (c) 11m x + sin(x) U+ 0 = O i o l e- fe rm i V1 a 0 0 -O r VY1 Uce, L'43 PI-Fa-ls rv-l LX + Si 3 X,) I -+eosx U,nn i +Cos^c X-^n
3 NAME: SECTION: Page 3 2. For each of the following functions, find the derivative f'(x). You do not need to worry about algebraic simplifications, but there should be no derivative left in the final answer. (10 points each, 30 points total ) (a) f(x) =In (xe') of km i r, r tk t e dl& I dx (e- "' t Xel X) X- CA LA, '_ & x+x - 6X + Xe-x (b) f (x) _ / sin(x2) + x- v-x c,o s `X-Z (c) f (x) = tan x 3x2-1 zx 3x2-I^ - ^?^ -L CMV x Lax ) 2- 'x Cox -twx
4 Page 4 MATH 152 Common Final - Spring Evaluate each of the following integrals. For the definite integral, give the value. For indefinite integrals, give the most general antiderivative. (10 points each, 30 points total) / 9 (a) J (3x2 + ^ dx 4 x rl3 + 2x12 J 9 4 TA7 (b) fsin2(x) cos(x) dx k r1h add c' f silo s^, ^; o^ off. V. c(.- = Cos x d x 3 e ^ (c) 1 c + Lk dxx ^'..,rlth a b s ^^ 4 Iti '- x OW- = "dx
5 NAME: SECTION: Page 5 4. Suppose f is a differentiable function. (Total: 20 points) (a) Carefully state the formal definition of the derivative of f, the one involving limits. (10 points) I ^C% ( x o VVI X-) 1X0 (b) Apply that definition to show f'(5) = 23 if f (^) = 4-7x + 3x2. (10 points) Xo = S 7^-( K ) = zs7 ^Cx) -QCs) 4_7x +3X -44 3Xz -7 x-^ s ei X >5 ( %+S) % --s) 3x+ 8 = ^
6 Page 6 MATH Common Final - Spring Related Rates A square metal plate is being heated. Suppose that the area of the plate is expanding at the rate of 2 square centimeters per minute. Determine how fast the length of the edge is increasing when the area is 16 square centimeters. (20 points) A re, a. = A Gx) = XZ d A I Lp C-W12 so x - 4 O(A C^ja) VI C K, da 0(t d t- vq^, old j - ^o I v-e ^v r ḏ lx Ok t i cv1 4 w.
7 NAME: SECTION: Page 7 6. Optimization A rectangular field is to be bounded by a fence on three sides and by an existing straight wall on the fourth side. Find the dimensions of the field with maximum area that can be enclosed with 1000 feet of fence. (20 points) )q P P9 V_ i7d -b be fro cqd vin ouc 'I ry'1,1 A s LAL!o j ^G`F 17 o c o n s _rca oz + o o C -h i-u fe i r-, f-0 74 irn x o (^ A Fi v, a- C r +-, rp+.s) cv- ^.^ % a otx A A = t oad x --oz x Ivco x o = ooo x IooO 4x 25 0 = X vrl A_ x d v t,^sv 5o e s x= S O 1 ' S f o cc (a ^^ 91 oio v i F:7i V, c q ^^ _ food ax = iooo -goo =Soo so x = ^L, So y Sc^ o 5)v e.s max r ^0.^
8 Page 8 MATH Common Final Spring Find the slope of the tangent line at (n, 0) to the curve determined by the equation sin(x + y) = y2cos(x) (15 points) r e e. d ^^ C0S(1K4-d)( 1"4-^_91) -= P1.^ r ^ x = rr^ Y -0 bt-ore sd /Vi n C 0 2 / CoS% aj ^ / cosx 2Sin x GO Srr'(I/ ) ifs -e -^a. TF T- F 1 0 s I o pe- c^-f- ^o'ir^+- slope rm Y _ Y o m(x-xc)) C-I)c Y,--T) -S Tv^ V- tl-^ +avi9e0,4 Livre
9 NAME: SECTION: Page 9 8. This problem asks you to state an important theoretical result and then to apply it. (Total: 15 points) (a) Carefully state a version of the Fundamental Theorem of Calculus which is sufficient to compute the exact value of the definite integral f 31(20 + 6t - 12t2)dt. (5 points) ^Cx)dx = fi(b)-fco-^ wk C V ( x) is `fie a n d ev-`, vj^ u-e o^ ^cx ) (b) Show how the theoretical result which you stated in part (a) can be used to compute the exact value of f 31 (20 + 6t - 12t2 ) dt. (10 points) 0 4- fi(x ) - 4x + c ^-- r + (- ^) + 3C-^)^ -- 4C- i^3 + C F(:;^) - F(-1 )
10 - Cox -ia Page 10 MATH 152 Common Final Spring Suppose f is a differentiable function. The graph of the derivative of f over the interval [-7, 5] is shown below, and a brief table of rounded values of f'. (Total: 20 points) (a) On what interval(s) is f increasing? Explain the evidence. (5 points ) L w k e_.v- o f p^ wk-i c± T f'(x) S ovl e (` 0 to v O& C V^^c ^ : ^ )^ C L `- 1 (b) On what interval (s) is f concave up? Describe the relevant evidence. (5 points) w ^ C C 9 1 l > v / r J 7 so ^1^^E of ^'/ i s pos rf-!v-^2 'u`-l^cgr wl ax - C QQ,c A-v e CIOt'N (c) Does f attain a local maximum somewhere in the open interval (-7, 5)? Describe the relevant evidence. (5 points) T _ O a.-+ - S J 3e_ s I o P e ^` -yu c t +- i S $ 0 l_` y g i ve I oca._p MooX (d) Suppose f (1) = -4. Write an equation of the line tangent to the graph of f at the point (1, -4). (5 points) G,( g_p tc. L M slop - la C X -I
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