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Unformatted text preview: Spring 2011 Math 151 Exam III Version A Solutions 1. D Since the numerator and denomina- tor both approach 0, use LHospitals Rule: lim x e x- cos x- 2 x x 2- 2 x = lim x e x + sin x- 2 2 x- 2 = 1 2 . 2. E 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 > , 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. B The original function f is decreasing when f is negative, which occurs when x ( a , c ) ( e , ) . 5. B 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 ....
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- Spring '06