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Unformatted text preview: Math 20A First Midterm Exam. October 22, 2002 VERSION 1 Instructions: Fifty-ﬁve minutes. No books or notes; graphing calculators without symbolic manipulation programs are permitted. Do all 6 problems in your blue book. Show all work; unsubstantiated answers will not receive credit. Turn in your exam sheet with your blue book. 1. (20 points) The function g (x) deﬁned on the interval (0, 2) satisﬁes g (1) = 2 and g (1) = −1. (a) Find an equation for the line tangent to the graph y = g (x) at x = 1. (b) Find the value of lim g (x) − 2 . x→1 x − 1 √ 2. (20 points) Find an exact value of lim
x→0 2−x− x √ 2 and justify your answer. 1 1 − 1+t 5 3. (20 points) A car drives down a road and is at distance (in miles) d(t) = 60 from its destination after t hours. (a) What is the average velocity while traveling between t = 2 and t = 3? (b) Express its instantaneous velocity at time t = 2 as a limit. (c) Compute the limit. Be sure to indicate the units in (a), (b), and (c) above. 4. (20 points) Prove that there is at least one negative real number x satisfying the equation x3 − x + 1 = 0. (You must use a theorem, not just a graph.) x+5 . Justify 5. (20 points) Find all the horizontal asymptote(s) of the curve y = √ 9x2 + 5 your answer. 6. (20 points) Let a > 0 and consider 3 − ax2 , x < 1; ax + 2, x ≥ 1. f (x) = Show that there is a unique value of a such that f is continuous at every real number. (A correct value of a will not be suﬃcient; you must justify your answer.) ...
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This note was uploaded on 11/09/2010 for the course MATH 20A 20A taught by Professor Yacobi during the Fall '08 term at UCSD.
- Fall '08