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HW3-solutions

# HW3-solutions - merino(aem2588 HW3 chen(55405 This...

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merino (aem2588) – HW3 – chen – (55405) 1 This print-out should have 19 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0points Apply the Product Rule twice to determine the derivative of f when f ( x ) = ( x 1)( x 2)( x 3) . 1. f ( x ) = 3 x 2 + 4 x 5 2. f ( x ) = ( x 2)( x 3) 3. f ( x ) = 3 x 2 + 12 x + 11 4. f ( x ) = ( x 1)( x 2) 5. f ( x ) = 3 x 2 12 x + 11 correct 6. f ( x ) = 3 x 2 4 x 5 Explanation: Applying the Product Rule to f once we see that f ( x ) = ( x 2)( x 3)+( x 1) d dx ( x 2)( x 3) . Applying the Product a second time, but now to the second term, we see that d dx ( x 2)( x 3) = ( x 3)+( x 2) = 2 x 5 . Thus f ( x ) = ( x 2)( x 3) + ( x 1)(2 x 5) = ( x 2 5 x + 6) + (2 x 2 7 x + 5) . Consequently, f ( x ) = 3 x 2 12 x + 11 . 002 10.0points Find the third-degree polynomial Q such that Q (1) = 0 , Q (1) = 3 , Q ′′ (1) = 10 , Q ′′′ (1) = 18 . 1. Q ( x ) = 3 x 3 + 4 x 2 2 x 1 2. Q ( x ) = 4 x 3 3 x 2 + 2 x 1 3. Q ( x ) = 4 x 3 + 3 x 2 2 x 1 4. Q ( x ) = 3 x 3 4 x 2 + 2 x 1 correct 5. Q ( x ) = 2 x 3 4 x 2 + 3 x 1 Explanation: A degree 3 polynomial can be written as Q ( x ) = ax 3 + bx 2 + cx + d where the values of the coefficients a, b, c and d are determined by the values of Q and its derivatives at x = 1. Now Q ( x ) = 3 ax 2 + 2 bx + c while Q ′′ ( x ) = 6 ax + 2 b, Q ′′′ ( x ) = 6 a . From the given values Q ′′′ (1) = 18 , Q ′′ (1) = 10 , Q (1) = 3 , therefore, we see that Q ′′′ (1) = 6 a = 18 = a = 3 , while Q ′′ (1) = 6 a + 2 b = 10 = b = 4 , and Q (1) = 3 a + 2 b + c = 3 = c = 2 . Finally, Q (1) = a + b + c + d = 0 = d = 1 .

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merino (aem2588) – HW3 – chen – (55405) 2 Consequently, Q ( x ) = 3 x 3 4 x 2 + 2 x 1 . 003 10.0points The graph of a differentiable function g is shown in 1 2 3 1 2 3 Arrange the values of 0 , g (0) , g (1) , g (2) , g (3) in increasing order. 1. g (1) < 0 < g (3) < g (0) < g (2) cor- rect 2. g (2) < 0 < g (3) < g (1) < g (0) 3. g (1) < 0 < g (2) < g (3) < g (0) 4. g (1) < 0 < g (0) < g (2) < g (3) 5. g (2) < 0 < g (1) < g (3) < g (0) 6. g (2) < 0 < g (1) < g (0) < g (3) Explanation: The sign of the derivative, g ( a ), of g at x = a is determined by whether the tangent at P ( a, g ( a )) is sloping down to the right (in which case g ( a ) < 0) or whether it is sloping up to the right (in which case g ( a ) > 0). On the other hand, the size of g ( a ) is determined by the value of the slope of this tangent: the inequality g ( a ) < g ( b ) will hold when the tangent at Q ( b, g ( b )) has greater slope than the tangent at P ( a, g ( a )).
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HW3-solutions - merino(aem2588 HW3 chen(55405 This...

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