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Unformatted text preview: Version 100 – EXAM 1 – Radin – (58280) 1 This printout should have 20 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table t (min) 5 10 15 20 25 30 V (gal) 720 480 289 190 23 show the volume, V ( t ), of water remaining in the tank (in gallons) after t minutes. If P is the point (15 , V (15)) on the graph of V as a function of time t , find the slope of the secant line PQ when Q = (25 , V (25)). 1. slope = − 53 . 2 2. slope = − 19 . 1 3. slope = − 43 . 1 4. slope = − 9 . 9 5. slope = − 26 . 6 correct Explanation: When P = (15 , V (15)) , Q = (25 , V (25)) the slope of the secant line PQ is given by rise run = V (25) − V (15) 25 − 15 . From the table of values, therefore, we see that slope = 23 − 289 25 − 15 = − 26 . 6 . 002 10.0 points Below is the graph of a function f . 2 4 6 − 2 − 4 − 6 2 4 6 8 − 2 − 4 Use the graph to determine lim x → 4 f ( x ) . 1. limit does not exist correct 2. limit = 9 3. limit = 8 4. limit = 5 5. limit = 7 Explanation: From the graph it is clear the f has a left hand limit at x = 4 which is equal to 9; and a right hand limit which is equal to 1. Since the two numbers do not coincide, the limit does not exist . 003 10.0 points Determine lim x → x − 1 x 2 ( x + 5) . 1. limit = 1 2. limit = − 1 5 3. none of the other answers 4. limit = 0 5. limit = ∞ Version 100 – EXAM 1 – Radin – (58280) 2 6. limit = −∞ correct Explanation: Now lim x → x − 1 = − 1 . On the other hand, x 2 ( x + 5) > 0 for all small x , both positive and negative, while lim x → x 2 ( x + 5) = 0 . Consequently, limit = −∞ . keywords: evaluate limit, rational function 004 10.0 points Let F be the function defined by F ( x ) = x 2 − 49  x − 7  . Determine if lim x → 7 F ( x ) exists, and if it does, find its value. 1. limit does not exist 2. limit = − 14 correct 3. limit = − 7 4. limit = 7 5. limit = 14 Explanation: After factorization, x 2 − 49  x − 7  = ( x + 7)( x − 7)  x − 7  . But, for x < 7,  x − 7  = − ( x − 7) . Thus F ( x ) = − ( x + 7) , x < 7 , By properties of limits, therefore, the limit exists and lim x → 7 F ( x ) = − 14 . 005 10.0 points Determine lim x → 8 √ x + 1 − 3 x − 8 . 1. limit = 1 3 2. limit = 6 3. limit = 1 6 correct 4. limit doesn’t exist 5. limit = 3 Explanation: After rationalizing the numerator we see that √ x + 1 − 3 = ( x + 1) − 9 √ x + 1 + 3 = x − 8 √ x + 1 + 3 ....
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 Spring '08
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 Derivative, Limit, lim F

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