p1s - MATH 2144 - FALL 2011 - PRACTICE EXAM I NAME:...

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MATH 2144 - FALL 2011 - PRACTICE EXAM I NAME: SOLUTIONS The exam has 13 problems. The total number of points is 100. Be sure to follow the instructions for each problem. Draw a box around each of your final answers. Write neatly and legibly. Unreadable answers are wrong.
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1. (8 points) After each function below state all the classes of functions to which that function belongs out of this list: power, root, polynomial, rational, algebraic, trigonometric, exponential, logarithmic . On each part of the problem you will receive one point for each correct answer and lose one point for each incorrect answer that you state (but not get less than zero). (a) 3 x power, root, algebraic (b) 3 x exponential (c) ln x logarithmic (d) x + 1 x 2 + x + 2 rational, algebraic (e) x + 1 x 2 + 1 ! 3 algebraic 2. (4 points) State the domain of the function f ( x ) = q ( x - 2)( x - 5) using interval notation. Solution : For the square root to be defined we need ( x - 2)( x - 5) 0. There are two ways this can happen: both factors non-negative or both factors non-positive. In the first case we have x - 2 0 and x - 5 0, so x 2 and x 5. The only way we can have a number that is at least 2 and simultaneously at least 5 is for it to be at least 5, so we have x 5 for this case. In interval notation this is the set of numbers [5 , ). In the second case we have x - 2 0 and x - 5 0, so x 2 and x 5. The only way we can have a number that is at most 2 and simultaneously at most 5 is for it to be at most 2, so we have x 2 for this case. In interval notation this is the set of numbers ( -∞ , 2]. So we must have either x in ( -∞ , 2] or x in [5 , ). This means that the domain is the union of these two sets, which is written ( -∞ , 2] [5 , ) .
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3. (6 points) H ( x ) = (ln( x 2 + 1)) 3 + 4. Find f ( x ), g ( x ), and h ( x ) such that H = f g h . Do not use the identity function i ( x ) = x . Solution : Recall that ( f g h )( x ) = f ( g ( h ( x ))), so one first computes h ( x ), then feeds it into g to get g ( h ( x )), then feeds that into f to get f ( g ( h ( x ))). To identify f , g , and h think about how H ( x ) is built up from x . x x 2 + 1 ln( x 2 + 1) (ln( x 2 + 1)) 3 + 4 Think of this as an assembly line in which something is produced from the input x by a sequence of three machines. The first machine h takes its input x , squares it and adds 1, so it can be described as h ( x ) = x 2 + 1. The second machine g takes the natural logarithm of its input x 2 + 1 to get ln( x 2 + 1). Note that the only thing it does with its input is to take its natural logarithm, so its job can be described by g ( x ) = ln x . The third machine
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This note was uploaded on 10/12/2011 for the course MATH 2144 taught by Professor Pagano during the Fall '08 term at Oklahoma State.

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p1s - MATH 2144 - FALL 2011 - PRACTICE EXAM I NAME:...

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