Solutions Homework 8

# Solutions Homework 8 - MATH 106 CALCULUS I FOR BIO SOC SCI...

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INSTRUCTOR: JOS ´ E MANUEL G ´ OMEZ Solutions Homework 8. Please show all your work. (1) (5 Points) Consider the function f ( x ) = 2 x + 2. Use a suitable the linear approxi- mation on the function f ( x ) to estimate 4 . 2. Solution: Note that 4 . 2 = f (1 . 1). Thus we can use the linear approximation of f ( x ) at x = 1 to approximate f (1 . 1). By deFnition the linear approximation of f ( x ) at x = 1 is L ( x ) = f (1) + f (1)( x 1) . In this case f (1) = 4 = 2. Also f ( x ) = 1 2 (2 x + 2) 1 / 2 (2) = 1 2 x + 2 Therefore f (1) = 1 4 = 1 2 . It follows that the linear approximation of f ( x ) at x = 1 is L ( x ) = 2 + 1 2 ( x 1) = 3 2 + x 2 . We can estimate f (1 . 1) with L (1 . 1) to obtain 4 . 2 = f (1 . 1) = L (1 . 1) = 3 2 + 1 . 1 2 = 2 . 05 (2) (10 Points) Suppose that w ( x ) = x 2 x 2 + 1 . ±ind the intervals where w ( x ) is increasing, decreasing, concave up and concave down. Solution: We have w ( x ) = ( x 2 + 1)(2 x ) x 2 (2 x ) ( x 2 + 1) 2 = 2 x ( x 2 + 1) 2 . Note that w ( x ) exists for every real number x . Also w ( x ) = 0 if and only if 2 x ( x 2 + 1) 2 = 0 and this occurs if and only if x = 0. We see that f ( x ) > 0 for x in the interval (0 , ) and f ( x ) < 0 for x on the interval ( −∞ , 0). Therefore w is increasing on the 1

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Solutions Homework 8 - MATH 106 CALCULUS I FOR BIO SOC SCI...

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