Review 2-solutions-1

Review 2-solutions-1 - white (taw933) – Review 2 –...

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Unformatted text preview: 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 . 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 white (taw933) – Review 2 – ben-zvi – (55600) 3 5. x y Explanation: After differentiation 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....
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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.

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Review 2-solutions-1 - white (taw933) – Review 2 –...

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