szTtrX-a151tfau

szTtrX-a151tfau - M ATH 151 Answers to Final Examination y...

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MATH 151 Answers to Final Examination Autumn 2009 x y 5 - 5 5 - 5 1. The graph of a certain function f is below, on the right. Use this graph to Fnd the following limits. Use and -∞ as appropriate. lim x →-∞ f ( x ): 0 lim x →- 1 - f ( x ): 2 lim x →- 1 + f ( x ): 2 lim x →- 1 f ( x ): 2 lim x 1 - f ( x ): 4 lim x 1 + f ( x ): - 1 lim x 1 f ( x ): DNE lim x 3 - f ( x ): lim x 3 + f ( x ): -∞ lim x 3 f ( x ): DNE lim x 5 - f ( x ): 0 (over) Typeset using A M S -T E X.
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MATH 151 Answers to Final Examination , p .2 Autumn 2009 2. Find the following limits, where f ( x )= 1 /x if x< 0 2 x if 0 2 x 2 if 2 <x . (i) Find lim x 0 - f ( x ) lim x 0 - f ( x ) = lim x 0 - 1 x = -∞ . Answer : -∞ (ii) Find lim x 0 + f ( x ) lim x 0 + f ( x ) = lim x 0 + 2 x = 2(0). Answer : 0 (iii) Find lim x 0 f ( x ) Answer : Does not exist (iv) Find lim x 2 - f ( x ) lim x 2 - f ( x ) = lim x 2 - 2 x = 2(2). Answer : 4 (v) Find lim x 2 + f ( x ) lim x 2 + f ( x ) = lim x 2 + x 2 =2 2 . Answer : 4 (vi) Find lim x 2 f ( x ) Answer : 4 (vii) Find lim x →-∞ f ( x ) lim x →-∞ f ( x ) = lim x →-∞ 1 x . Answer : 0 (viii) Find lim x →∞ f ( x ) lim x →∞ f ( x ) = lim x →∞ x 2 . Answer : (go to the next page)
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MATH 151 Answers to Final Examination , p .3 Autumn 2009 3. Find the equation of the tangent line to the given curve at the given point. (i) The tangent line to y = 2 x +1 x 2 at (0 , 1). y ± = 2( x 2 + 1) 1 / 2 - (2 x + 1) 1 2 ( x 2 + 1) - 1 / 2 (2 x ) { ( x 2 + 1) 1 / 2 } 2 . y ± (0) = 2(0 2 + 1) 1 / 2 - (0 + 1)(0 2 + 1) - 1 / 2 (1) 1 2 = 1. Answer : y - 1 = 1( x - 0) or y = x (ii) The tangent line to ( x 2 + y 2 ) 2 =4 xy at (1 , 1).
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szTtrX-a151tfau - M ATH 151 Answers to Final Examination y...

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