201-103-dw-winter2011

201-103-dw-winter2011 - Dawson College Department of...

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Unformatted text preview: Dawson College Department of Mathematics — Winter 2011 Instructors : I. Rajput, A. Jimenez, M. Marchant, M. Ishii Final Examination Tuesday May 17th 2011 Calculus I (201-103—DW) STUDENT NAME: STUDENT 1]): ' No graphing I programmable calculators allowed. There are 16 questions in total worth 100 marks. Show all your work in the space provided and circle your ﬁnal answers. [9] 1. Find each of the following limits (show all your work): . 3x2 + 6x — 24 ,3 o" (a) 11m —,)—-— -- ‘— x-——>2 x“ _ 4 O :: Jim 30992-761410 7‘41 (ac—17009.) :. ~€im 306147 '—_—____—l—-‘ 2292. 1+2. mm x = 9— x—>O‘\11—3x-—l O . 5—x—2x3 _..ao (c) 11m —-§—— — --—-- x—>°0 32: +17 00 .——-"—-" 7”" 313 [3] 2.. Consider the following piece-wise deﬁned function: ﬁb+x—3ﬂ y x22 f (x) = 3x — 8 x <' 2 Find the limits: wlmtﬂﬂ = 31+Z—‘L : <::> x->2+ ‘ I (wﬁmfh): G- 3 = <::> x—>2_ [6] 3. Consider the piece-wise deﬁned function: x —x¥D ifxi_3 x+3 f(x)= —5 ifx=—3 a) State the deﬁnition for a function to be continuous at a point. “9”” “W” W60 :c-M b) Use the deﬁnition from part a) to show is discontinuous at x = -3 . [6] 4. State the limit deﬁnition of the derivative and use ONLY the limit 3 deﬁnition to evaluate f ' (x) for = 5x2 + 7x41. f5): WM [11°57 *‘ {76> ' l1~>o [1 : WM 23(114432 + i}(x+k7 — \ "(22."1+ 77"" D h‘PO in ’4" i ? 1 3’ 2" - L117"; 276 + 374* til“ + ¥I+€Lh~lg£l¥x+lg A :W’“ A[%x+§la+¥) h~>o A _ Jim ~ hao 81 + %h + ;L .— .- x [6] 5. Find the equation of the tangent line to the graph of = 7 3 — x at the point where x = 1. m: 1’0) V1 'V2 5770?— (3r“3x> [1) - Zia/\$336) (-3) (?-3x> _ " - __ __ _{_/ u 7i [0.“ 2% >- J- “a X "Eb = (I): __._.' f 2 J = __/_r, g, 1 la. f 10-.- :2 2x+1 [5] 6. For the function f(x) : 1 x ....... ﬁnd f"(x) and simplify your answer. (76—02" : --3 (1-0—2 -3 H [5] 7. Compute the value of f’(x) when x =0 given that f(x) = (e—zx + 3)3 + sin(6x). 7W:- 8 (e'zx+37z(€zx>(~2> +écoswx> W: [7] 8. Suppose the monthly revenue and cost functions (in dollars) for x units of a 2 x commodity sold and produced are R(x) = 400x — 56 and C(x) = 5000 '+ 70x. a) Find the Proﬁt function P(x) . PK?) =— KCx) ‘“ C(76) Z. 1)) Find the marginal proﬁt function and use it to estimate the proﬁt from selling the 42“ unit. FYI): i322 +330 = :29. +330 10 I0 740: ‘4' : ‘ l’ 75 +330 325.90 tit/M c) Calculate the actual proﬁt from selling the 42nd unit. H423» mm =- 323.39\$ [5] 9. Find the x—value(s) where the graph of f (x) = tangent line. (Z76 +f> «,1 Co: + 5 — 3:: +2 (Zx—HS/z- Z 31. + 5 : O {ZIHY’Z 31’ z - 5 V [90+ ‘F-(x [5 not gig-PK'MC‘ 0L+ 3-9/3 SIM-CL 21H 7 0 7-4 '5 KM 7: bumcitvx [6] 10. Consider the relation 1/x + y =1+ x2522 . (:1) Find the derivative y'. - 1/1 E’Kﬁﬁﬂg’): 25223:? + 21f (7(7‘5317L (jib/3g 41%;“ + 41011 [/L ‘ (ﬂan) tr“ “ 4ngg’ 2* [5] 11. Use logarithmic differentiation to find the derivative f'(x) given that f(x) : (cos 290‘“. .1 f: [575szan c/ 401;: “(M ((05820 [4x 401‘} 1 (Jam) ' Wt (max) I . g z m (—sinx>-2 + Jmfost)_L / (j, (0527C x I g; 001 Hwy 76 D/q[(0’f?x>\] 05 [ Zx I 6’0st 40m ; (@5276) [~«Z-ﬂh7C/9ih270 _F Lymaoszyaj 7C 4M ;(Coszx> W " Z4142: (hi/1270] 7C '[5] 12. Consider the function f (x) = 1n{ exxlx?‘ +1] (x—2)3 (a) Fully silnplify f (x) using the properties of logarithms. 76/70: We”) +1! swam) — 3 «lac—z) \l' X + 4:1(12-60 - ; snot—2,) 1 Z (b) Find f'(x) and sinlplify your answer. \ ._ ZJC TNT?” / + L , .~— 3 (i) 21wa :02 [121 13. Consider the function f (x) = x3 — 6x (a) Find the x and y intercepts. 3181* =0 :C-ud 1C¢Z"(’7:O J J, {=0 7C: i550 (fwd Mom: (44/, age) {and MM (l.‘H,‘\$‘.C;(o> (d) Sketch a graph of f(x) with important points labeled. [5] 14. The productivity P of a worker after t hours on the job was modeled by PG) 2 24f + 31‘2 — 1‘3, Where 0 S l‘ _<_ 8 and P denoted the number of units produced per hour. What is the maximum productivity, and after how many hours is productivity maximized? PW): 24 + (0+, at; = o [7] 15. A rectangular box has a square base. The volume of the box is 256 cubic centimeters. The material for the base and the top of the box cost \$0.10 per square centimeter and the material for the sides cost \$0.05 per square centimeter. Find the dimensions of the box that minimize the cost of producing the box. Vii—.1. 1 I 3‘ V? 299": 12(3— [8] 16. Find the anﬁderivatives: (a) [ «5(2)? —7) abc 5Q 95% —— 41"” > (/1 q/ 7‘ Zizé —- \$31.2... H, ‘1 3 u: 5:3»sz du= IOJIC’Z d>C 5 1L“. ' du=~a(5‘x~Dc{:c ...
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