Review 2-solutions

# Review 2-solutions - white(taw933 Review 2 ben-zvi(55600 This print-out should have 18 questions Multiple-choice questions may continue on the next

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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 – fnd 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 6. local min at x Explanation: By the Quotient rule, f ± ( x x 2 - 2 x ( x - 2) ( x 2 +12) 2 = 12 + 4 x - x 2 ( x 2 2 . The critical points oF f occur when f ± ( x )=0, i.e. ,atthesolutionsoF f ± ( x (2 + x )(6 - x ) ( x 2 2 =0 . Thus the critical points oF f are x = - 2and x =6. To classiFy these critical points we use the ±irst 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 ): - 26 - - + ±rom 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 . keywords: local maximum, local minimum, critical point, quotient rule, ±irst 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, fnd 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 ,theh ighestpower,weseethat x x 2 - 4 x = 1 x + 3 x 2 1 - 4 x + 5 x 2 . On the other hand, lim x 1 x =l i m x 1 x 2 . 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 x 5. f ( x 3 x - 4 x 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 . 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
white (taw933) – Review 2 – ben-zvi – (55600) 3 5. x y Explanation: After diFerentiation 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. DiFerentiating 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 ±ve 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 =2 0 0 0 ± 1 - t 40 ² 2 .

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## This note was uploaded on 12/05/2011 for the course M 408k taught by Professor Schultz during the Spring '08 term at University of Texas at Austin.

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Review 2-solutions - white(taw933 Review 2 ben-zvi(55600 This print-out should have 18 questions Multiple-choice questions may continue on the next

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