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M408CF08Exam1Sol - Exam 1 Math 408C Name Fall 2008 TA...

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Unformatted text preview: Exam 1 Math 408C Name _ Fall 2008 TA Discussion Time: TTH You must show sufficient work in order to receive full credit for a problem. Do you r work on the paper provided. Write your name on this sheet and turn it in with your work. Please write legibly and label the problems clearly. Circle your ansWers when appropriate. N0 calculators allowed. 1. Calculate the limits below. if they exist. In the case that a limit doesn‘t exist. indicate whether the limit approaches (‘0. —:>cr. or neither. You must show sufficient work to get credit for a, correct answer. 1 f .~ \ (a) (10 points) lim < A ‘ ’1‘—>-1 17+1 $2*.r—2/ (b) (10 points) lim lhl (1. — 1) h—>O’ h. V3 ~15 — l (c) (l0 points) lim—-— ’t—r'Z f—‘_) 2. (10 points) Find a value of C such that the tunction f below is continuous every- where. 1’2 — 9 ftr) = t“ “ 3 \/C:l‘. + 3, .l‘ 23 3. (10 points) Use the definition of (lEI‘ZV‘c'lllVQ to calculate f’(0) for f(;r) : 1‘ < 3 l (I —l— l)? (You must use the definition of derivative as the limit of the difference quotient. not the differentiation rules. in order to get credit for this problem.) 4. (10 points) An object moves along the yeaxis. its position at time 1‘ given by t W) = \th—fi (a) Find the velocity of the object at iinie f. Simplify your answer to a. reasonable for 0 S t, where y is measured in meters and f in seconds. degree. (b) Find all times (if any exist) when the object changes direction. If the object never changes direction. determine whether it always moves upward or always moves downward along the axis. 5. (10 points) Find all values of .I' in the interval [0.2a] where the tangent to the curve 3; = cos(2w) + V31 is horizontal. d2y 6. (10 points) Find I 9 if y : tan‘2(3;r). (1417‘ [More on the back] 7. (10 points) Find the equation of the "(urgent to the curve I : sin(I — g) 4.131;") at the point (151). 201‘ 2r — :3 Where f is measured in degrees (deicius and f in hours. Find the rate at which the 8. (10 points)The temperature of an object at timet is given by f(t) : 100 — temperature is Changing Ett the momeiu when the temperature of the object is 60 degrees Celcius. (l — eoxz‘)‘2 va. ¢ 0 Bonus: (5 points) Let f(I) : I U. ‘1‘ = U Show that f is (iiffei‘eiitia}_‘ile for all I, and hnd f’(0). Determine whether or not f’ is continuous at I = 0. l b , |, 003(03—\ // _ - 2—‘ f \ 50 w Jaclwmw‘w" WW3 g,‘:.—(\fi") l0:’¥+-Z" g gm): )00' “El: 2t'3 F‘rSAr IRA”; ,J/LLWWW SH)=60 [00“;C -‘0 avg 40:33: rété’i bgotvnw =1!” éOtC’w t:2_ 'th); _. 2% (Zt’3)L ) -tlot ...
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