Solutions to Final Exam Ma12, F05

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

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Unformatted text preview: me— Exam fbwmml — Pea/7,005 1. (35 pts.) Find 3x2)- for each y below. a. y=2x3 +5x2+7x+10 417 ; éﬁl—quo-I- 7 512a b. y =(x4 +5x3 —x2 +9)sinx if! = (ma/nu -7-x) >42“ J K (x*+5x’-X1+7>W7€ _ 5x3 +8x2 +x—2 x4+6x3 +4x—1 (i3 : K/F'Kvlal' /v6?< 4' ‘)<xq‘.LC‘7c3+ 77(4) .. JX 6K3 +5K1+wc—’l)(4v<5+/87<1 +§>j (7611 (73+ 7‘K‘ ))L c.y d. y = sin(\/3x2 +5x+7) -—I 3‘12 " w» J2 (9XL*4K*7)/L (‘K+’;) x 6. 5x23)3 + 2x+xsin(y) = 2005 } c1 - J [OX/y 4' Ext/71 if + Z 4- WG‘j)+ XWQJ) 2:: '7' 0 (ix 133659" 4' wow?) : WIN/95+ 7— *Mm) ckj _ ’((01<:/~73*'1+M’j> 962C (lngM1+--swv‘7> 2. (14 pts.) For each of the following indicate whether the statement is true or false. Assume that all derivatives exist. No work is necessary. a. If f is differentiable at x, then f is continuous at r False (circle one) b. If f is continuous at x, then f is differentiable at x. True ocircle one) c. For any function f, lim f(x) = f(c). True orircle one) (MAW 4W“ “5“” 14C L (S Cbn’lqvww @ 709 d. i—(fg) = . True or circle one) e. —( f + g) = — +— or False (circle one) f. i i =1 True ocircle one) dx g dx dx g. %( f ( g(x)) = f '(g(x)) g '(x). Tr or False (circle one) ’ 3. (32 pts.) Compute the integrals below. 5 4 2 ,1 a.I§-_3x;_§x_-1gx ;8 73-37813 - x (4,0 x b. Jam/5x2 +9dx : “((1 Au [0 c. [(x+1)2J2x—1dx K : 2‘4 P f x: (our 0/7. / S “I/ma‘k :% all-L Au : LX_SQ&1}(M+q) 7. a ( 5 5’1 3/1 #1 u : i u. ’4' Ar q k ‘ 7 Q + é “ + at u )‘l’ 3’ 7/1 I “9,?” 3,1 4. (10 pts.) When a foreign object lodged in the trachea forces a person to cough, the body apparently maximizes the force on this object by maximizing the velocity of the air stream of the cough. In particular, the body does this by contracting the trachea. Suppose that the air stream velocity, v(r), is related to the radius of the trachea, r, by v(r)=(n-—r)r2, SrSn where n is the normal radius of the trachea. < n 7 o > Find the value of the radius (in the interval [n/2, n]) which maximizes the velocity of the air stream. Explain your work/reasoning. Note: It is acceptable to have your optimal value of r expressed in terms of the normal radius value, n. '2 r90 OV‘ V‘: 32 I 3 W (N [51/1th / \/ Cow’Hun» 0k CM/L/ In) J" moﬁr" 1/111... wxlwm 4‘. V‘: or VD‘NJ.(S>} 5. (10 pts.) Koch’s Snowﬂake, named after the Swedish mathematician Helge von Koch (1870-1924) is obtained as follows. Given an equilateral triangle, place equilateral triangles on the middle third of each side. Repeat this process. The first four Koch Snowﬂakes are shown below: If the first snowﬂake (the equilateral triangle on the left) has sides of length one, then it can be shown that n—l p(n) = perimeter of 11th snowﬂake = I n—l a(n) = area of nm snowﬂake = -‘/—§+ M 1-[:4_] 4 20 9 Here, ﬁnally, are your questions: a. What can you say about the snowﬂake perimeter as you continue to repeat this process? That is, tell me about 332M”)? / 4 ““ <1 ﬂ.» : 00 (45 Ill-90’ b. What can you say about the snowﬂake area as you continue to repeat this process? That is, tell me about lim a(n)? n—wo 2/1»; Eat afﬂeg‘fq) (45%", new 4‘ :Eiriqét) :’-\F3 + w 5’ ...
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