practice1 - Consider the function f ( x ) = x 3-2 x 2 +2...

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MAT 21A Fu Liu Practice Exam 1 1 (28 pts.) Determine whether or not the following limits exist, and calculate them. If the limit does not exist as a number, state whether or not it can be written as or −∞ . (a) lim x 3 x - 3 x 2 - 2 x - 3 (b) lim t 0 t +1 - 1 t (c) lim x 1 + 2 - x 2 1 - x (d) lim x →∞ sin x
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2 (12 pts.) Directly from the deFnition of a limit, show that lim x →∞ 1 e x = 0 . 3 (8 pts.) Using the sandwich theorem, show that lim x →∞ 1 - cos x e x = 0 . (You may use the result from problem 2 even if you did not do it.) 2
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4 (10 pts.) Find lim x →∞ 1 - cos(1 /x ) (1 / 2 x ) . (Hint: use the double angle formula) 5 (10 pts.) For each question, answer only “true” or “false”. No justi±cation is needed. You will receive 5 points for a correct answer, 2 points for no answer, and 0 points for an incorrect answer. (a) Suppose f ( x ) is continuous on [0 , 1] . If f (0) = 3 and f (1) = 2 , then f ( x ) has at least one root in (0 , 1) . (b) A step function cannot be the derivative of any function. 3
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6 (20 pts.)
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Unformatted text preview: Consider the function f ( x ) = x 3-2 x 2 +2 x-4 x 2-3 x +2 : x n = 1 , 2 4 : x = 1 , 2 (a) Find all horizontal/oblique asymptotes to the graph of f ( x ) . (b) Find vertical asymptotes and points of discontinuity of f ( x ) . For each discontinuity, say whether or not it is removable. (c) Use the information obtained above, graph f ( x ) . (Hint: factor the denominator rst) 4 7 (12 pts.) At t seconds after liftoF, the height of a rocket is 4 t 2 feet. (a) How fast is the rocket climbing t seconds after liftoF? (Compute this from our denitions, without using derivative laws) (b) If you graph the position of the rocket as a function of time, what is the equation of the tangent line to the graph at time t = 10 ? (The equation should be in the form of y = g ( t ) . ) 5...
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practice1 - Consider the function f ( x ) = x 3-2 x 2 +2...

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