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mt01s06 - MIDTERM 1 MATH 19 A Instructor Frank Béiuerle...

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Unformatted text preview: MIDTERM 1 MATH 19 A 5/5/2006 Instructor: Frank Béiuerle, Ph.D. Your Name: _______________________________ Your TA: _—_——_——————.————_—_——.——_——_—.—_—._———.—— Max Your score Problem 1:“ 15 Problem 2: ' 10 Problem 3: 10 Problem 4: 20 Problem 5: 25 Problem 6: 10 Problem 7: 10 TOTAL: 100 Good Luck! 1. (15 points) (a) Give an informal description of what it means that ” f (at) is continuous at J: = a”. (b) Give a formal definition (using linis) of ” f (:3) is differentiable at :c = a”. (c) Use the limit definition of the derivative to compute the derivative of f (:5) = $2 + 1 at 5:: = 2. (d) Use the appropriate diiferentiation rules to compute the same derivative again to verify your answer in (c). i’. (10 points) Assume 5(t) = t3 - 121E + 5 is the position of an object in a one-dimensional system. (a) What is the velocity of the object at time t = 0? (b) What is the speed of the object at time t = 0? Explain the difference to your answer in (a). (c) At what time(s) is the object at rest? 3. (10 points) Use the given graph of f (cc) to carefully sketch the graph of f’(a:) in a new coordinate system. ’19 ‘6 Hint: Find/estimate f’(-1),f’(l),f’(0) and think about what it means for f[:c) when f'($) > U or when f’(:L') < O. sin 3: 4. (20 points) Compute the following limits. Justify steps. You may use that iim0 = 0 3—4» a: sin2 a: 1. (a) :c-l—I—Ii-lo 2;; l _ 3 (b) lim _+_$i =—*°° 1 +z2 , tanh (c) blinoT (d) 1i1n s_1n_t t—->0+ \fl 5. (25 points) Compute the requested derivatives of the functions: (NO NEED TO SIM- PLIFY) dy (8—) Ex’ for y — x/5 (b) y' for y =x2+cosm+secx—3 (c) f’(x) for M) = —f§ df x (d) dx for f(:1:)—2:2 tan: d?! 5 4 9 2 a (e) Efory=(z ——:L' +3—2)(5:r: ’13:: —-:1: +3—12) 1' 6. (10 points) Compute an equation of the tangent line to the curve 3; = at the point 8 Pl-. (’2) 3+1 7. (10 points) Find all real number values for a so that the function f(:c) = ax? —— (12.17 + 1 has a horizontal tangent line at :c = 1. ...
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