finalps3 - Math 16A Short Calculus, section 2 Practice...

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Unformatted text preview: Math 16A Short Calculus, section 2 Practice Final Exam Answers 1. Continuity For each of the following functions, state: the range, the inter- val(s) on which the function is continuous, and for each disconti- nuity whether or not the discontinuity is removable. (a) y = x 2 + 4 Range: (4 , ) Continuous on: (- , ) No discontinuities (b) y = x +2 x 2- 4 Range: (- , 0) (0 , ) Continuous on: (- ,- 2) (- 2 , 2) (2 , ) Removable discontinuity at x =- 2 Nonremovable discontinuity at x = 2 (c) y = 2 b x c Range: all even integers Continuous on: [ x,x + 1) for all integers x Nonremovable discontinuities at each integer; no other discontinuities (d) y = 4 cos( ) + 1 Range: (-3,5) Continuous on: (- , ) No discontinuities 2. Slope and Derivative (a) Find an equation of the line tangent to f ( x ) = x 2 + 1 with slope- 2. We have f ( x ) = 2 x , so slope- 2 occurs when- 2 = f ( x ) = 2 x = x =- 1. When x =- 1, f (- 1) = 2. The tangent line is: ( y- 2) = (- 2)( x + 1) = y =- 2 x. (b) Use the limit definition to find the derivative of the following functions. i. f ( x ) = x 2- 1 f ( x ) = lim x f ( x + x )- f ( x ) x = lim x ( x + x ) 2- 1- x 2 + 1 x = lim x x 2 + 2 x x + ( x ) 2- x 2 x = lim x ( x )(2 x + x ) x = lim x (2 x + x ) = 2 x. ii. h ( t ) = 2 t h ( x ) = lim t h ( t + t )- h ( t ) t = lim t 2 t + t- 2 t t = lim t 2( t + t- t ) t t + t + t t + t + t = lim t 2(( t + t )- t ) ( t )( t + t + t ) = lim t 2 t ( t )( t + t + t ) = lim t 2 t + t + t = 2 2 t = 1 t . 3. Rates of Change (a) The profit P from selling x units of a product is given by...
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This note was uploaded on 05/22/2011 for the course MATH 16A taught by Professor Sabalka during the Fall '08 term at UC Davis.

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finalps3 - Math 16A Short Calculus, section 2 Practice...

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