Math 1A - Fall 2005 - Borcherds - Final

# Math 1A - Fall 2005 - Borcherds - Final - 8832832885 15:85...

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Unformatted text preview: 8832832885 15:85 5185422415 LICE MiﬂiIN LIERﬂiR‘r" F'iﬂiGE 81.-"'81 Math 1A Final 2005—12—15 sine-3:00pm R. annchetcls You are allowed 1 sheet of notes. Calculators are not allowed. Each question is worth 3 marks, which will only be given for a clear and correct answer in simpliﬁed form. Draw the graph of the function y = lcos(:r)| for —rr 5 :13 :1" 7r. Evaluate the limit liin.._.e ﬁgg; Prove that 9:4 -|— 1 = 3:3 has at least one real root. Diﬁerentiate new/(1r: + 1). Find the derivative of the function y = cos(cos(cos(m))). Find sly/dd: if 3:23; + day? = 23:. Find the derivative D57 sin(2rr:). (D means d/da') If f(1) = 10 and f’ (:r) 3 —1 for all tr, what is the smallest possible value of f(5)'? Find li.mm_.+m eel/s. Sketch the curve :1; = :l: lrl[:r)2 for a :2» U. 11. Find two numbers Whose difference is 10 and whose product is a minimum. 12. Use one iteration of Newton’s method applied to the initial. approximation a1 = 2 to estimate 91/3. 13. Find a function f such that f’(:t) = 333 and the line .1: +12; = D is tangent to the graph of f. 14. Find 3” given that f”(.t) = sin(.1:), ﬁll) = 1, fan = 0. 15. Estimate the area. under the graph of f (rt) ——- 3:2 from a: = 1 to rs = 4 using three rectangles and left endpoints._Sl~:etch the graph and rectangles. ls. Ii ff f(e)de = 12 and ff f(:t)drr = 14 13.11de f(:r)d:c. 1?. Evaluate the integral fogﬂ + i/Q — 3:3)ds: by interpreting it as an area. 18. Prove that 1/e g fol e-ﬁdn g. l. 19. Find the derivative of the function 9(a) = I; e‘tzdt. 20. Find the derivative of y = fjini“) tan(t)dt. os(.'.t) 21. Evaluate the integral f_‘11(.l:3 + 2x + 1)d:r. 22. Evaluate the integral. few/4 sec((9) tan(9)d9. 23. Evaluate the indeﬁnite integral. f (1 + y2)mydy. 24. Evaluate the indeﬁnite integral f tan(.r) ln(cos(m))d:t. 3 25. Evaluate the deﬁnite integral f: HTELdi‘. 26. By comparing areas, show that 1-1- 1/2 +1/3 + - . - + 1/(77. — 1) 2: Info) if n 2 2. 27. Find the area enclosed by the curves :1; = 332, y = 2/ (3:2 + 1). 28. Find the volume of the region obtained by rotating the region bounded by the curves y = v/a: m 1, y z 0, :13 = 2, :1: = 10, about the :tvaxis. 29. Use the method of cylindrical shells to ﬁnd the volume generated by rotating the region bounded by y = 3:2, 3; = 0, at = 1 about the yvalds. 30. Find the average value of sin(:r.)2 on [0, 271*]. |_l. EDSDPOTJF‘F’TPFON!‘ ...
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