MATH
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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