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2004 Spring - 125FinalSpring04short math125

2004 Spring - 125FinalSpring04short math125 - Math 125...

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Math 125, Final Exam (common) Spring 2004 1. [20] Find the limits if they exist. (You may not use L’Hospital’s Rule.) (a) [5] lim x 0 tan 2 x sin (1 /x ) (b) [5] lim x →∞ e 3 x + 5 e 2 x 2 e 3 x - 1 (c) [5] lim x 1 x + 3 - 2 x + 2 x - 1 (d) [5] lim x 2 ln 1 5 x - 2 x 2 + x - 6 2. [20] Consider f ( x ) = ( x - 1)( x - 2) | x - 2 | . (a) [10] Sketch the graph of f . (b) [10] Describe the set of points on the real axis at which f is continuous, e.g., x (7 , 9] or 7 < x 9. 3. [20] Find the derivatives of the following functions: (a) [5] f ( x ) = x 5 - 2 x (b) [5] g ( x ) = 3 1 - xe x 2 (c) [10] h ( x ) = x sin x 4. [15] Use implicit differentiation to find the equation of the tangent line to the curve y cos( x 2 ) = x cos( y 2 ) at the point (0,0). 5. [15] If a bacteria population starts with 400 bacteria and doubles every 5 hours, then the number, f ( t ), of bacteria after t hours is f ( t ) = 400 · 2 t/ 5 . (a) [5] Find the derivative of f ( t ). (b) [5] Find the value of f (5). (c) [5] Find a linear approximation to f ( t ) at t = 5. 6. [15] Consider the polynomial p ( x ) = x 3 + x + 1.
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