Math 1A - Fall 2005 - Borcherds - Midterm 2

Math 1A - Fall 2005 - Borcherds - Midterm 2 - UHIZHIZUUb...

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Unformatted text preview: UHIZHIZUUb lb:U4 blflb4224lb UUH MHLN LlfiHHHV Pfibt Math 1A Midterm 2 sees—11-3 11:00-12:30pm. R. Benscth-wdfi 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. 1. For what values of a: does the graph of fin) = :r — 2sin(:r) have a horizontal tangent? 2. Find an. equation of the tangent line to the curve 3,; = (1 + 3:13)1U at the point (0,1). 3. Find dy/ds: by implicit differentiation if fl“ = l ~4- 1:23;. ‘4. Find '1 ”’, where y = :r/(Qa: - 1). 5. Differentiate ln(1n(ln(ln(:r))j). 6. Find the derivative of Sinh(m),tanh(a2). 7. Use diflerentials or a linear approximation to estimate m 8. Find the absolute maximUm and absolute minimum values of flat) 2 3:3 — 3m + l. on the interval [0, 3] 9. Find all critical numbers of the function f (m) = 3.1/3 — neg/3. 10. Verify that f(fl:‘) : m2 — 4m+ 1 satisfies the three hypotheses of Rolle’s theorem on the interval [0,4]. Then find all numbers c that satisfy the conclusion of Rolle’s theorem. 11. Find the intervals on which f is increasing or decreasing and all local maximum and minimum values of f(s:) = :rge“. ' 12. Find the limit lim.,,_,.0{em — 1 — $)/LL'2. 13. Find the limit limmmo sin(:z:)/[sinh(:r) + 1). In‘ questions 14 and 15 your sketch should show the domain of the function, local. maxima and minima, where the function is increasing or decreasing, any zeros of the function, the behavior for large values of lwl, and the behavior near a: = D. You need not show convexity or points of inflection. 14‘ Sketch. the curve” 3,: = sin(:c)/(l + cos(zt)). 15. Sketch the curve 3; m 321/!" for it 2:- D. Ulffll ...
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