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124quiz5solutions

# 124quiz5solutions - Math 124 Quiz 5 Oct 20 2006 SOLUTIONS 1...

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Math 124 Quiz 5 Oct. 20, 2006 SOLUTIONS 1) Let g ( x ) = a sin( x + 1) + 3, where a is a non-zero constant. (a) Find the local linearization L ( x ) for values of x near - 1. (Note your answer will be in terms of a ). (b) Use the local linearization to approximate g ( - . 95). Solutions: (a) First, g 0 ( x ) = a cos( x + 1). Then L ( x ) = g ( - 1) + g 0 ( - 1)( x - ( - 1)) = 3 + a ( x + 1) (b) g ( - 0 . 95) L ( - 0 . 95) = 3 + a ( - 0 . 95 + 1) = 3 + 0 . 05 a 2) The graph of a function f ( x ) is given below. The line y = x + 1 is the tangent line to f ( x ) at the point (3 , 4). (a) What is a in the tangent line approximation formula in this situation? (b) Find f ( a ) and f 0 ( a ). (c) Estimate f (2 . 9). (d) Is your estimate for (c) an underestimate or overestimate? Give an explanation. Solutions: (a) 3 (b) f ( a ) = 4 and f 0 ( a ) = 1 , i.e. the slope of the tangent line. (c) Note that here L ( x ) = x + 1, the y -value of the tangent line. Hence, f (2 . 9) L (2 . 9) = 2 . 9 + 1 = 3 . 9 (d) The above estimate is an

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124quiz5solutions - Math 124 Quiz 5 Oct 20 2006 SOLUTIONS 1...

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