Homework #13 - jiwani(amj566 – Homework13 – Fouli...

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Unformatted text preview: jiwani (amj566) – Homework13 – Fouli – (58395) 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 Find the larger solution of the exponential equation 3 x 2 = 9 x +4 . 1. none of these 2. x =- 4 3. x =- 2 4. x = 2 5. x = 4 correct Explanation: By properties of exponents, 9 x +4 = 3 2 x +8 . Thus the given equation can be rewritten as 3 x 2 = 3 2 x +8 , which after equating exponents becomes x 2 = 2 x + 8 . This equation factors as ( x- 4)( x + 2) = 0 , and so its solutions are x = 4 ,- 2. Hence the larger solution of the given equation is x = 4 . 002 10.0 points Which function has 2 4- 2- 4 2 4- 2- 4 as its graph? 1. f ( x ) = 2 x − 1- 3 2. f ( x ) = 3 − x- 2 3. f ( x ) = 2- 2 − x − 1 4. f ( x ) = 2- 3 − x correct 5. f ( x ) = 3 x- 3 6. f ( x ) = 2 − x − 1- 2 Explanation: The given graph has the property that lim x →∞ f ( x ) = 2 . But lim x →∞ 2 − x = 0 = lim x →∞ 3 − x , while lim x →−∞ 2 x = 0 = lim x →−∞ 3 x , so f ( x ) must be one of 2- 3 − x , 2- 2 − x − 1 . On the other hand, the y-intercept of the given graph is at y = 1. Consequently, the graph is that of f ( x ) = 2- 3 − x . 003 10.0 points jiwani (amj566) – Homework13 – Fouli – (58395) 2 Find the value of lim x →∞ parenleftbigg 4 e 2 x- 3 e − 2 x 3 e 2 x + e − 2 x parenrightbigg . 1. limit = 1 4 2. limit =- 1 4 3. limit = 3 4 4. limit =- 3 4 5. limit = 4 3 correct 6. limit =- 4 3 Explanation: After division we see that 4 e 2 x- 3 e − 2 x 3 e 2 x + e − 2 x = 4- 3 e − 4 x 3 + e − 4 x . On the other hand, lim x →∞ e − ax = 0 for all a > 0. But then by properties of limits, lim x →∞ 4- 3 e − 4 x 3 + e − 4 x = 4 3 . Consequently, limit = 4 3 . 004 10.0 points Determine f ′ ( x ) when f ( x ) = e √ 4 x +3 . 1. f ′ ( x ) = 4 e √ 4 x +3 √ 4 x + 3 2. f ′ ( x ) = 2 e √ 4 x +3 √ 4 x + 3 3. f ′ ( x ) = 4 e √ 4 x +3 4. f ′ ( x ) = 2 e √ 4 x +3 √ 4 x + 3 correct 5. f ′ ( x ) = 1 2 e √ 4 x +3 √ 4 x + 3 Explanation: By the chain rule f ′ ( x ) = e √ 4 x +3 parenleftbigg d dx √ 4 x + 3 parenrightbigg = 2 e √ 4 x +3 √ 4 x + 3 . 005 10.0 points Find f ′ ( x ) when f ( x ) = e x cos( e x )- sin( e x ) . 1. f ′ ( x ) = e 2 x sin( e x ) 2. f ′ ( x ) = e x cos( e x ) 3. f ′ ( x ) = e x sin( e x ) 4. f ′ ( x ) = e 2 x cos( e x ) 5. f ′ ( x ) =- e 2 x sin( e x ) correct 6. f ′ ( x ) =- e x cos( e x ) 7. f ′ ( x ) =- e 2 x cos( e x ) 8. f ′ ( x ) =- e x sin( e x ) Explanation: By the Chain Rule, f ′ ( x ) = e x cos( e x )- e x cos( e x )- e 2 x sin( e x ) . jiwani (amj566) – Homework13 – Fouli – (58395) 3 Consequently, f ′ ( x ) =- e 2 x sin( e x ) ....
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Homework #13 - jiwani(amj566 – Homework13 – Fouli...

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