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Review 2-solutions - white(taw933 Review 2 ben-zvi(55600...

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white (taw933) – Review 2 – ben-zvi – (55600) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Which one of the following properties does f ( x ) = x - 2 x 2 + 12 have? 1. local min at x = - 6 2. local max at x = 6 correct 3. local max at x = 2 4. local max at x = - 6 5. local min at x = 2 6. local min at x = 6 Explanation: By the Quotient rule, f ( x ) = x 2 + 12 - 2 x ( x - 2) ( x 2 + 12) 2 = 12 + 4 x - x 2 ( x 2 + 12) 2 . The critical points of f occur when f ( x ) = 0, i.e. , at the solutions of f ( x ) = (2 + x )(6 - x ) ( x 2 + 12) 2 = 0 . Thus the critical points of f are x = - 2 and x = 6. To classify these critical points we use the First Derivative Test. But the sign of f de- pends only on the numerator, so it is enough, therefore, to look only at a sign chart for (2 + x )(6 - x ): - 2 6 - - + From this it follows that f is decreasing on ( -∞ , - 2), increasing on ( - 2 , 6), and de- creasing on (6 , ). Consequently, f has a local maximum at x = 6 . keywords: local maximum, local minimum, critical point, quotient rule, First Derivative Test, rational function 002 10.0 points Determine if the limit lim x → ∞ x + 3 x 2 - 4 x + 5 exists, and if it does, find its value. 1. limit = 3 5 2. limit = 5 3. limit = 0 correct 4. limit = 3 5. limit doesn’t exist 6. limit = - 1 4 Explanation: Dividing in the numerator and denominator by x 2 , the highest power, we see that x + 3 x 2 - 4 x + 5 = 1 x + 3 x 2 1 - 4 x + 5 x 2 . On the other hand, lim x → ∞ 1 x = lim x → ∞ 1 x 2 = 0 . By Properties of limits, therefore, the limit exists and limit = 0 .
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white (taw933) – Review 2 – ben-zvi – (55600) 2 003 10.0 points Which of the following functions has 4 3 -2 3 as its graph. 1. f ( x ) = 3 x + 4 x - 2 2. f ( x ) = 3 x - 4 x - 2 3. f ( x ) = 4 - 3 x x + 2 4. f ( x ) = 3 x + 4 x + 2 5. f ( x ) = 3 x - 4 x + 2 correct Explanation: The graph of any function f ( x ) = ax - b x + c has the properties (i) a vertical asymptote x = - c , (ii) a horizontal asymptote y = a , (iii) an x -intercept at x = b a . From the graph above, therefore, c = 2 , a = 3 , b a = 4 3 . Hence the graph is that of f ( x ) = 3 x - 4 x + 2 . 004 10.0 points Use calculus to decide which of the follow- ing is the graph of f ( x ) = 3 x 2 / 3 - 2 x . 1. x y 2. x y correct 3. x y 4. x y
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white (taw933) – Review 2 – ben-zvi – (55600) 3 5. x y Explanation: After di ff erentiation of f ( x ) = 3 x 2 / 3 - 2 x we see that f ( x ) = 2 x 1 / 3 - 2 = 2(1 - x 1 / 3 ) x 1 / 3 ; in particular, (i) f has critical points at x = 0 , 1. Di ff erentiating again we next see that f ( x ) = - 2 3 x 4 / 3 , from which it follows that (ii) f < 0 , so f concave down, on ( -∞ , 0) , and again (iii) f < 0 , so f concave down, on (0 , ) ; in particular, f has a local maximum at x = 1. Of the five graphs only x y has these properties. 005 10.0 points If a tank holds 2000 gallons of water, and the water can drain from the tank in 40 min- utes, then Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as V = 2000 1 - t 40 2 .
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