cal2 - choi(kc24547 – HW02 – BERG –(56525 1 This...

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Unformatted text preview: choi (kc24547) – HW02 – BERG – (56525) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Find all functions g such that g ′ ( x ) = 3 x 2 + x + 2 √ x . 1. g ( x ) = √ x parenleftbigg 3 5 x 2 + 1 3 x + 2 parenrightbigg + C 2. g ( x ) = 2 √ x parenleftbigg 3 5 x 2 + 1 3 x + 2 parenrightbigg + C cor- rect 3. g ( x ) = 2 √ x ( 3 x 2 + x − 2 ) + C 4. g ( x ) = 2 √ x parenleftbigg 3 5 x 2 + 1 3 x − 2 parenrightbigg + C 5. g ( x ) = 2 √ x ( 3 x 2 + x + 2 ) + C 6. g ( x ) = √ x ( 3 x 2 + x + 2 ) + C Explanation: After division g ′ ( x ) = 3 x 3 / 2 + x 1 / 2 + 2 x − 1 / 2 , so we can now find an antiderivative of each term separately. But d dx parenleftbigg ax r r parenrightbigg = ax r − 1 for all a and all r negationslash = 0. Thus 6 5 x 5 / 2 + 2 3 x 3 / 2 + 4 x 1 / 2 = 2 √ x parenleftbigg 3 5 x 2 + 1 3 x + 2 parenrightbigg is an antiderivative of g ′ . Consequently, g ( x ) = 2 √ x parenleftbigg 3 5 x 2 + 1 3 x + 2 parenrightbigg + C with C an arbitrary constant. 002 10.0 points Determine f ( t ) when f ′′ ( t ) = 6( t + 1) and f ′ (1) = 2 , f (1) = 4 . 1. f ( t ) = t 3 − 3 t 2 + 7 t − 1 2. f ( t ) = t 3 + 3 t 2 − 7 t + 7 correct 3. f ( t ) = 3 t 3 − 6 t 2 + 7 t + 0 4. f ( t ) = 3 t 3 + 3 t 2 − 7 t + 5 5. f ( t ) = t 3 − 6 t 2 + 7 t + 2 6. f ( t ) = 3 t 3 + 6 t 2 − 7 t + 2 Explanation: The most general anti-derivative of f ′′ has the form f ′ ( t ) = 3 t 2 + 6 t + C where C is an arbitrary constant. But if f ′ (1) = 2, then f ′ (1) = 3 + 6 + C = 2 , i.e., C = − 7 . From this it follows that f ′ ( t ) = 3 t 2 + 6 t − 7 . The most general anti-derivative of f is thus f ( t ) = t 3 + 3 t 2 − 7 t + D , where D is an arbitrary constant. But if f (1) = 4, then f (1) = 1 + 3 − 7 + D = 4 , i.e., D = 7 . Consequently, f ( t ) = t 3 + 3 t 2 − 7 t + 7 . choi (kc24547) – HW02 – BERG – (56525)...
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cal2 - choi(kc24547 – HW02 – BERG –(56525 1 This...

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