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Fall 2007 Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, SPSU Math2253 - Review Problems for Final Exam Chpater 2: Limits and Continuity 1. If 2 2 ) ( 2 +- = x x x f , find (a) ) 3 ( f (5) (b) ) 3 ( h f + ( 5 4 2 + + h h ) (c) h f h f ) 3 ( ) 3 (- + ( 4 + h ) 2. Find the average rate of change of the function over the given interval or intervals. 1 ) ( 3 + = x x f ; a) ] 3 , 2 [ (19) b) ] 2 , 2 [ h + ( 2 6 12 h h + + ) 3. Find the limit. a) 4 2 lim 2 2- +- x x x (-1/4) b) ) 8 5 3 ( lim 2 2 +- x x x (10) c) 7 lim 2 3 + x x (4) d) x x x 3 9 lim- + (1/6) e) 1 1 lim 3 1-- x x x (1/3) f) 12 tan lim 3 x x (1) g) x x x 4 5 sin lim (5/4) h) x x x tan lim (1) i) x x x sin cos 1 lim- (0) j) + - = 2 , 3 2 , 1 2 ) ( 2 x x x x x f ) ( lim 2 x f x- (3) ) ( lim 2 x f x- (7) k) x x x 2 sin lim (0) l) 1 5 3 2 lim +- x x x (2/5) Page 1 of 8 Fall 2007 Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, SPSU m) 2 1 lim 2- + x x ( ) 4. Discuss the continuity of each of the following functions. a) 1 2 3 ) ( 2- +- = x x x x f (continuous everywhere but x=1) b) + - = 2 , 3 2 , 1 2 ) ( 2 x x x x x f (continuous everywhere but x=2) 5. Determine the value of c such that the function is continuous on the entire real line. + + = 4 , 6 4 , 3 ) ( x cx x x x f (c = ) 6. Find the vertical asymptote (if any) 1 ) ( 2 2- = x x x f (x = -1 and x = 1) 7. The function f is defined as follows , 2 tan ) ( = x x x x f a) Find ) ( lim x f x (if it exits) (2) b) Can the function f be define at = x such that it is continuous everywhere? If so, how? 8. Find an equation of the tangent line to the curve of 2 x y = at the point (1, 1). (y=2x+5) 9. Graph the following functions using horizontal, vertical, or slant asymptotes if possible. a) 1 1 2 ) (- + = x x x f b) 2 1 ) ( 2-- = x x x f Chapter 3: Differentiation 1. Find the derivative of the function by the limit process. a) 3 2 ) ( 2 +- = x x x f ( 2 2- x ) b) 1 ) ( + = x x f x 2 1 2. Find an equation of the line that is tangent to the graph of ) 7 )( 8 ( ) ( 2 2-- = x x x f at (3, 2). ( 52 18 + = x y ) Page 2 of 8 Fall 2007 Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, SPSU 3. Find the derivative of each of the following functions. (a) 6 3 2 ) ( 2 3 4- +- + = x x x x x f ( 29 3 2 6 4 2 3 +- + x x x (b) x x x f 4 ) ( 2- = + 2 4 1 x (c) 3 6 ) ( x x x f- = - 3 2 2 2 1 x x (d) 3 2 1 ) (-- = x x x f -- 2 ) 3 2 ( 1 x (e) 4 2 ) 1 3 ( ) ( +-... View Full Document