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Unformatted text preview: MATH 140 FINAL EXAM
TIME: MAY 16, 1.30—3.30, 2005 Write your name, TA and section, and the question number on each of 6
answer sheets. Answer each question on ONE answer sheet, using the front
for part (a) and the back for part (b) (for questions 1 —— 4). Show all your
work, using complete English sentences. Cross out any material that you do
not wish graded. Calculators are NOT allowed. Good luck! 1(3) (30 pts) Compute the following (if they exist) . . 51:1/3—1 .. . sin(\/33) . :3 $2
(2) :16qu 90—1 (M) $1320 at ’ (m) fol14% {ac—2—x2—4} 1(b) (10 pts) Consider the equation 1+x = tan(x). (i) Explain why there is a root in the interval 0 < at < 7r/ 2. (ii) Explain why there is only one root in the interval 0 < 06 < 7r / 2. 2(a) (30 pts) Find the ﬁrst derivative of the following functions = 1:0;(235 , (a) h(t) = 16—9 ) (m) 9(8) :1n(1+3+82+83)‘ (2') m a 2(1)) (10 pts) At time 0 < t < 1 (hours) a car has velocity
11(t) = 10 — 90 t2 mph. What is the total distance travelled over the one hour (i.e. the increase
in mileage shown on the speedometer). w/Z‘xm/ K/Y‘ // 2 3(a) (10 pts) Find the critical points of the function 2 f (96) H 1 + $4
and explain which one gives the maximum on 0 < a: < oo. 3(b) (30 pts) Sketch the graph of g(x) = ln(1 + x2) . 4(a) (30 pts) Determine (2') / sin(2m)dcc, (a) Alarms, (iii) 55/) 1 dt 1+1t2 4(b) (10 pts) Find points a, b so that 3 1 b
/1(1+:c2)43alx+[1 (1+w2)43dx=/ (1+x2)43da:. a 5 (20 pts) A spherical balloon has volume inﬂated at the rate of 1 cm3/sec.
If the balloon’s radius is 10 cm how fast is its surface area increasing. 6 (20 pts) Find the quadratic expression for the ellipse with focii at points
(1,0), (“1,0) and containing the point (0, x/g). WMﬂﬂ ...
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