CalcHW10Answers - Villarreal, Natalie Homework 10 Due: Oct...

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Villarreal, Natalie – Homework 10 – Due: Oct 30 2007, 3:00 am – Inst: Louiza Fouli 1 This print-out should have 21 questions. Multiple-choice questions may continue on the next column or page – fnd all choices be±ore answering. The due time is Central time. 001 (part 1 o± 1) 10 points The derivative o± a ±unction f is given by f 0 ( x ) = ( x 2 + 3 x - 10) g ( x ) ±or some unspecifed ±unction g such that g ( x ) > 0 ±or all x . At which point(s) does f have a local minimum? 1. local minimum at x = - 5 2. local minimum at x = - 2 3. local minimum at x = 5 4. local minimum at x = - 5 , 2 5. local minimum at x = 2 correct Explanation: At a local minimum o± f , the derivative f 0 ( x ) will be zero, i.e. , ( x - 2)( x + 5) g ( x ) = 0 . Thus the critical points o± f occur only at x = - 5 , 2. To classi±y these critical points we use the First Derivative test; this means looking at the sign o± f 0 ( x ). But we know that g ( x ) > 0 ±or all x , so we have only to look at the sign o± the product ( x - 2)( x + 5) o± the other two ±actors in f 0 ( x ). Now the sign chart - 5 2 + + - ±or ( x - 2)( x + 5) shows that the graph o± f is increasing on ( -∞ , - 5), decreasing on ( - 5 , 2), and increasing on (2 , ). Conse- quently, f has a local minimum at x = 2 , . keywords: local minimum, First Derivative Test critical points, sign chart, conceptual, 002 (part 1 o± 1) 10 points Let f be the ±unction defned by f ( x ) = 5 - x 2 / 3 . Consider the ±ollowing properties: A. derivative exists ±or all x ; B. concave down on ( -∞ , 0) (0 , ); C. has local minimum at x = 0; Which does f have? 1. C only 2. A only 3. B and C only 4. A and C only 5. B only 6. A and B only 7. All o± them 8. None o± them correct Explanation: The graph o± f is 2 4 - 2 - 4 2 4 On the other hand, a±ter di²erentiation, f 0 ( x ) = - 2 3 x 1 / 3 , f 00 ( x ) = 2 9 x 4 / 3 .
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Villarreal, Natalie – Homework 10 – Due: Oct 30 2007, 3:00 am – Inst: Louiza Fouli 2 Consequently, A. not have: ( f 0 ( x ) = - (2 / 3) x - 1 / 3 , x 6 = 0; B. not have: ( f 00 ( x ) > 0 , x 6 = 0); C. not have: (see graph). keywords: concavity, local maximum, True/False, graph 003 (part 1 of 1) 10 points Use the graph a b c of the derivative of f to locate the critical points x 0 at which f does not have a local minimum? 1. x 0 = c, a 2. x 0 = b 3. x 0 = a, b 4. x 0 = a 5. x 0 = c 6. x 0 = a, b, c 7. x 0 = b, c correct 8. none of a, b, c Explanation: Since the graph of f 0 ( x ) has no ‘holes’, the only critical points of f occur at the x - intercepts of the graph of f 0 , i.e. , at x 0 = a, b, and c . Now by the ±rst derivative test, f will have (i) a local maximum at x 0 if f 0 ( x ) changes from positive to negative as x passes through x 0 ; (ii) a local minimum at x 0 if f 0 ( x ) changes from negative to positive as x passes through x 0 ; (iii) neither a local maximum nor a local minimum at x 0 if f 0 ( x ) does not change sign as x passes through x 0 . Consequently, by looking at the sign of
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CalcHW10Answers - Villarreal, Natalie Homework 10 Due: Oct...

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