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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 (201103—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 30992761410
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 piecewise 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 piecewise 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 :cM 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/$336) (3)
(?3x>
_ "  __ __ _{_/
u 7i [0.“ 2% > J “a X "Eb
= (I): __._.'
f 2
J = __/_r, g,
1 la. f
10. :2 2x+1 [5] 6. For the function f(x) : 1
x ....... ﬁnd f"(x) and simplify your answer. (76—02"
: 3 (10—2
3 H [5] 7. Compute the value of f’(x) when x =0 given that
f(x) = (e—zx + 3)3 + sin(6x). 7W: 8 (e'zx+37z(€zx>(~2> +écoswx> W: [7] 8. Suppose the monthly revenue and cost functions (in dollars) for x units of a
2 x
commodity sold and produced are R(x) = 400x — 56 and C(x) = 5000 '+ 70x. a) Find the Proﬁt function P(x) . PK?) =— KCx) ‘“ C(76) Z. 1)) Find the marginal proﬁt function and use it to estimate the proﬁt from selling the 42“ unit.
FYI): i322 +330 = :29. +330
10 I0 740: ‘4' : ‘
l’ 75 +330 325.90 tit/M c) Calculate the actual proﬁt from selling the 42nd unit. H423» mm = 323.39$ [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 gigPK'MC‘ 0L+
39/3 SIMCL
21H 7 0
74 '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) [~«Zﬂh7C/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(1260  ; 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
:Cud 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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 Calculus

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