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HW04-solutions

# HW04-solutions - husain(aih243 HW04 Gilbert(56215 This...

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husain (aih243) – HW04 – Gilbert – (56215) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Sketch the graph of a function g for which g (0) = 1 , g (0) = 2 , while g (1) = 0 , g (2) = 4 . 1. 1 2 1 2 2. 1 2 1 2 cor- rect 3. 1 2 1 2 4. 1 2 1 2 5. 1 2 1 2 6. 1 2 1 2 Explanation:

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husain (aih243) – HW04 – Gilbert – (56215) 2 One possible graph can be eliminated im- mediately because its y -intercept does not sat- isfy the condition g (0) = 1. On the other hand, the slope both at x = 0 and x = 2 of the remaining graphs can be estimated us- ing the grid provided. By this method we thus see that the only graph satisfying all the conditions is 1 2 1 2 002 10.0 points For which of the following functions f and corresponding numbers a is the limit lim h 0 (1 + h ) 9 1 h the value of f ( a )? 1. f ( x ) = x 9 , a = 9 2. f ( x ) = ( x 1) 9 , a = 1 3. f ( x ) = ( x + 1) 9 , a = 9 4. f ( x ) = x 9 , a = 0 5. f ( x ) = ( x + 1) 9 , a = 1 6. f ( x ) = x 9 , a = 1 correct Explanation: By definition f ( a ) = lim h 0 f ( a + h ) f ( a ) h . When f ( a + h ) f ( a ) h = (1 + h ) 9 1 h , therefore, inspection shows that f ( x ) = x 9 , a = 1 . 003 10.0 points If f is a function having -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 4 2 4 2 4 2 4 as its graph, which of the following is the graph of the derivative f of f ? 1. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 4 2 4 2 4 2 4 2. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 4 2 4 2 4 2 4
husain (aih243) – HW04 – Gilbert – (56215) 3 3. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 4 2 4 2 4 2 4 correct 4. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 4 2 4 2 4 2 4 5. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 4 2 4 2 4 2 4 Explanation: The tangent line to the graph of f is hor- izontal at ( 2 , f ( 2)) and (3 , f (3)), so the graph of f must have x -intercepts at x = 2 and x = 3; in particular, the graph of f can- not be a straight line. This eliminates three of the possible graphs. By observing the slope of the graph of f near x = 2 and x = 3 we see that the graph of f is -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 4 2 4 2 4 2 4 004 10.0 points The figure below shows the graphs of three functions of time t : t One is the graph of the position function s of a car, one is its velocity v , and one is its acceleration a . Identify which graph goes with which function. 1. s : v : a : 2. s : v : a : 3. s : v : a : correct 4. s : v : a :

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husain (aih243) – HW04 – Gilbert – (56215) 4 5. s : v : a : 6. s : v : a : Explanation: Experience tells us that the car is (i) moving forwards when its velocity is positive, (ii) moving backwards when its velocity is negative, (iii) speeding up when its velocity is positive and increasing, i.e. , when both velocity and acceleration are positive, (iv) slowing down when its velocity is pos- itive but decreasing, i.e. , when its velocity is positive but its acceleration is negative.
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HW04-solutions - husain(aih243 HW04 Gilbert(56215 This...

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