Math 1A - Fall 2001 - Vojta - Final

Math 1A - Fall 2001 - Vojta - Final - THU 10:22 FAX 6434330...

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Unformatted text preview: 04/25/2002 THU 10:22 FAX 6434330 MOFFITT LIBRARY 001 Ell/gilt; Math 1A Final Examination 3 hours 1. (18 points) Find the following limits: - :c 1/3: (a). me +w) tan2 :6 — 3 . l' (b) 3—1;;- 7tan2 () 1. 3$+2 C's—1131mm 2. (18 points) Find the following derivatives: at: -|— sin a: a! 1n cosh $2 (b)‘ Fig? (cw + 2 ) (18 points) (a). Find the slepe of the curve ya = m4 + By — 9 at the point (1, 2). (b). Find y” at (1,2). (12 points) Use differentials to estimate tan(g + .05). (12 points) State fully and carefully the Mean Value Theorem. (b). Let 0 < a < b. Show that there exists a number c such that a < c < b and cln 2 b — a. 6. (20 points) A cylindrical gob of goo is undergoing a transformation in which its height is decreasing at the rate of 1 cm/ sec, while its volume is decreasing at the rate of 271' cm3 / sec. (It retains its cylindrical shape while all this is happening.) If, at a given instant, its volume is 2411' cm3 and its height is 6 cm, determine whether its radius is increasing or decreasing at that instant, and at what rate. 7. (15 points) Determine the maximum and minimum values of the function f (w) = a: — 33: on the interval [—1, 27], and find all points (ac—values) where they occur. 8. (12 points) Find all asymptotes (horizontal, vertical, and slant) of the function 2/3 373-1 332—1. ft?) : 9. (15 points) A cyclist traveling at 30 ft/sec decelerates at a constant 3 ft /secz. HOW many feet does he travel before ceming to a complete stop? 10. (10 points) If Newton’s method is used to solve x3+m2+2 = 0 with an initial approximation 221 = —2, What is the second approximation, :52? 04/25/2002 THU 10:23 FAX 6434330 11. 12. 13. MOFFITT LIBRARY 002 P. Vojta Math 1A Final Examination Fall 2001 (35 points) Find the following integrals: (a). f1 tan dar; —1 d3: 0)) 9m2 + 1 3:4 + 1 (c). f 333 day (d). sin 932k!”S $2 due 2 (e). f asxé/m—ldm 1 2 (20 points) Find the integral / 3:2 d3: using the definition as a limit of Riemann sums. 1 You may use any (valid) choice of elf, and may also use any of the formulas i1=n i=1 i=1 (20 points) (a). The region bounded by the curves 1; = 2mg and y = 3:13 — 3:2 is rotated about the y-axis. Find the volume of the resulting solid. 71 . nn 1277, 1 n ,3 1 2 232: (+ ’1‘) Z? ' 73:1 11:1 (b). The same region is rotated about the line 3; = ——1. Write an integral expressing the volume of the resulting solid. (DO NOT evaluate it.) ...
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