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2011a_x3b_sols - Spring 2011 Math 151 Exam III Version B...

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Spring 2011 Math 151 Exam III Version B Solutions 1. E Since the numerator and denomina- tor both approach 0, use L’Hospital’s Rule: lim x 0 e x - cos x - 2 x x 2 - 2 x = lim x 0 e x + sin x - 2 2 x - 2 = 1 2 . 2. A Use properties of logarithms: ln( x 2 + x ) = ln( x +4) , so x 2 + x = x +4 , x 2 - 4 = 0 , which yields x = 2 or x = - 2 . Since the domain of the left-hand side of the original equation is x > 0 , the only solution is x = 2 . 3. A f positive means f is increasing, and f decreasing means f ′′ is negative, so f is con- cave down. The only graph increasing and concave down is graph A . 4. C The original function f is decreasing when f is negative, which occurs when x ( a , c ) ( e ) . 5. E f has a critical value when f = 0 , namely when x = a, c, e . Using the signs of the derivative f , we find that f is increasing for x ( -∞ , a ) ( c, e ) and decreasing for x ( a, c ) ( e, ) . Therefore, f has a local minimum only when x = c . 6. D Let y = log 4 parenleftbigg 1 8 parenrightbigg . Then 4 y = 1 8 , or 2 2 y = 2 3 , which means 2 y = - 3 and y = - 3 2 .
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