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2007 Fall Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, SPSU Math2253 - Review Problems for Final Exam Chpater 2: Limits and Continuity 1. If f ( x) x 2 2 x 2 , find (a) f (3) (5) (b) f (3 h) ( h 2 4h 5 ) f (3 h) f (3) (c) (h 4) h 2. Find the average rate of change of the function over the given interval or intervals. f ( x) x 3 1 ; a) [2, 3] (19) b) [2, 2 h] ( 12 6h h 2 ) 3. Find the limit. x 2 a) lim x 2 2 x 4 b) lim x 2 (3 x 2 5 x 8) c) d) e) f) g) h) i) j) (-1/4) (10) (4) (1/6) (1/3) (1) (5/4) (1) (0) lim x lim x 3 x2 7 x 9 3 x x 1 lim x 1 3 x 1 x lim x 3 tan 12 sin 5 x lim x 0 4x tan x lim x 0 x 1 cos x lim x 0 sin x 2 x 1, x 2 f ( x) x 2 3, x 2 0 lim f ( x) x 2 (3) (7) (0) (2/5) lim f ( x) x 2 sin 2 x x x 2x 3 l) lim x 5x 1 k) lim Page 1 of 8 Fall 2007 Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, SPSU m) lim x 2 1 x 2 ( ) 4. Discuss the continuity of each of the following functions. x 2 3x 2 a) f ( x) (continuous everywhere but x=1) x 1 2 x 1, x 2 b) f ( x) (continuous everywhere but x=2) x 2 3, x 2 5. Determine the value of c such that the function is continuous on the entire real line. x 3, x 4 (c = ) f ( x) cx 6, x 4 6. Find the vertical asymptote (if any) x2 f ( x) x2 1 7. The function f is defined as follows tan 2 x f ( x) , x 0 x a) Find lim x 0 f ( x) (if it exits) b) Can the function f be define at x If so, how? (x = -1 and x = 1) (2) 0 such that it is continuous everywhere? 8. Find an equation of the tangent line to the curve of y x 2 at the point (1, 1). (y=2x+5) 9. Graph the following functions using horizontal, vertical, or slant asymptotes if possible. 2x 1 a) f ( x) x 1 x2 1 b) f ( x) x 2 Chapter 3: Differentiation 1. Find the derivative of the function by the limit process. a) f ( x) x 2 2 x 3 ( 2x 2 ) b) f ( x) x 1 1 2 x 2. Find an equation of the line that is tangent to the graph of f ( x) ( x 2 8)( x 2 7) at (3, 2). ( y 18x 52 ) 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. 4x3 (a) f ( x) x 4 2 x 3 x 2 3x 6 (b) f ( x) (c) f ( x) (d) f ( x) (e) f ( x) x2 x x 63 x 6x 2 2x 3 4 1 4 x2 1 2 x 3 2 x2 x 1 2x 3 ( x 3 4 ) x2 1 1 (2 x 3) 2 4 (x K 3)3 ( x2 C 1 ) 4 K 8 (x K 3)4 x ( x2 C 1 ) 5 4. Find an equation of the line that is tangent to the graph of parallel to the line 4 x 2 y 3 . (y 5. Find an equation of the line that is tangent to the graph of f ( x) 3 2 x 2x 3 ) x 2 and f ( x) 3 x 2 and 1 47 perpendicular to the line 4 x 2 y 3 . (y ) x 2 16 72 3 6. If f ( x) x 2 2 x 5 , find f ( 4 ) ( x) . x5 x 7. A ball is thrown straight down from the top of a building. Its distance from the ground is given by the position function s(t ) 220 22 t 16 t 2 . Find its velocity and acceleration after 3 seconds. (-118 and -32) 8. Find the derivative of each of the following functions. (a) f ( x) sin x cos x ( cos2 x sin 2 x ) (b) f ( x) 2 x sin x cos x ( 2 sin x 2x cos x sin x ) 2 (c) f ( x) 2 x x tan x ( 2 2 x tan x x 2 sec2 x ) (d) y cscx cot x ( csc x cot x csc2 x ) (e) f ( x) (f) f ( x) 1 x3 (2 x 5) 1 ( x 2 2 sin 4 (3x 1) (4 3x) 9 5 x) 6 ( 3x 2 1 x 3 ) 2(1 x 3 ) (g) f ( x) (h) f ( x) 2( x 2 5x) 6 ) (2 x 5) 2 ( 24 sin 3 (3x 1) cos(3x 1) ) ( 27 (4 3x) 8 ) ( 6( x 2 5 x) 5 9. Use implicit differentiation to find dy . dx Page 3 of 8 Fall 2007 Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, SPSU (a) x 3 (b) y 2 y3 (x 18 xy y )( x 2 y) ( ( 6y x2 ) y 2 6x 3x 2 x2 2 xy y ) 4y x y (c) x tan(xy) 0 cos2 ( xy) x 10. Find the equation of the tangent line and the normal line to the graph of the equation at the indicated point. x 2 2 y 2 9 , (1, 2) 1 4 (tangent line: y , normal line: y 4x 2 ) x 4 9 11. The radius r of a circle is increasing at a rate of 3 centimeters per second. Find the rate of change of the area when r = 6 centimeters. ( 36 ) 12. The radius r of a sphere is increasing at a rate of 5 inches per minute. Find the rate of change of the volume when r = 24 inches. ( 11520 ) 13. Find dy : a) y x 2 2 x 5 b) y sin 2 x c) y cos(x 2 ) 2 x 2 dx 2 cos 2 x dx 2 x sin( x 2 ) dx Chapter 4: Applications of Derivatives 1. Find the absolute maximum and minimum values of each function on the given interval. a) f ( x) x 2 1, 1 x 2 (abs. Max = 3, abs. min = -1) b) f ( x) 2 x 4 cos x , [0, ] (abs. Max = 2 , abs. min = 4) 2. Find the extreme values of the function and where they occur. a) f ( x) x 3 2 x 4 2 2 4 6 , local min 4 at .) 3 3 9 x 1 1 b) f ( x) (local Max at 1 ; local min at 1) 2 2 2 x 1 f (b) f (a) f (c) in the 3. Find the value or values of c that satisfy the equation b a conclusion of the Mean Value Theorem for the following functions and intervals. (local Max 4 4 6 at 9 Page 4 of 8 Fall 2007 Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, f SPSU a) ( x) b) f ( x) x2 3x 2, [0, 1] c c x 1, [1, 3] 1 2 3 2 4. Find all possible functions with the following derivative. x 2 C C is a constant. a. y 2 x x2 x C b. y 2x 1 x3 x2 x C c. y 3 x 2 2 x 1 1 1 x 2 cos x C d. y 2 x sin x 2 x x 5. Use the Second Derivative Test to find any local extrema of the function 1 1 a) f ( x) 2 x 2 (1 x 2 ) (Local min 0 at x=0. local Max at and 2 2 b) f ( x) 4x 2 x4 1 (Local Max 4 at 2 and 2 2 , local min 0 at x = 0 ) ) 6. Find the length and width of a rectangle that has the perimeter 100 meters and the maximum area. (25, 25) 7. What is the largest possible area for a right triangle whose hypotenuse is 5 inches 25 long? ( ) 4 8. For the given functions i) Find open intervals on which the function is increasing or decreasing; ii) Locate all local extrema; iii) Find open intervals on which the function is concave upward or downward; iv) Find the points of inflection; v) Graph this function. a) f ( x) ( x 2) 2 ( x 4) b) f ( x) x cos x, [0, 2 ] c) f ( x) x 3 3x 2 9 x 12 x 1 d) f ( x) x 3 9. Find antiderivative for each function. Check your answers by differentiation. Note: Verify your answers by differentiation: F ( x) f ( x) 1 1/ 3 1 3/ 2 3 5/ 2 x dx x dx x dx a) 2 2 2 x x 4 sec3x tan 3xdx sec tan dx b) sec x tan xdx 2 2 Page 5 of 8 Fall 2007 Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, SPSU c) d) x 2 1 7 1 2 x x5/ 4 dx dx e) f) sec2 x dx 3 2 tan 2 x dx 10. Solve the initial value problems. dy a) 9 x 2 4 x 5, y ( 1) 5 dx d 2 s 3t ds b) ; 3, s(4) 4 8 dt t 4 dt 2 y 3x 3 2x 2 5 x 10 s t3 16 Chapter 5: Integration 1. Find the average value of the function over the given interval. a) f ( x) 3x 2 3 on [0, 1] ( 2 ) 3 b) f ( x) x 2 x on [ 2, 1] ( ) 2 2. Evaluate each of the following definite integrals using the fundamental theorem of Calculus. a) b) c) d) e) f) 4 0 1 1 (2 x) dx (2 (2 x 3 (16) 14 3 x 2 ) dx 2 x) dx 1 1 (0) 1 8 2 1 ( 1 x2 1 ) dx x3 0 4 4 sin t dt sec2 t dt (2) (2) 3. Find the total area between the region and the x-axis. x 2 a) y cos x, (6) 2 b) y 3x 3, 3 x 2 ( 28 ) 2 c) y 2 x , 0 x 2 (2) 4. Use the integration by substitution to find the indefinite integrals. 1 (3 x 3) 8 C (2 x 3) 7 dx a) 16 Page 6 of 8 Fall 2007 Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, SPSU b) c) d) e) f) g) x 2 x 3 10 dx 2x 3 dx (2 x 6 x 1) 2 2 2 3 ( x 10 ) 3 / 2 9 C 1 2(2 x sin 4 x 4 2 6 x 1) C C sin 3 x cos xdx tan 7 x sec2 xdx x x 2 dx 3x 5 x 3 1 dx tan 8 x 8 C 5 2 2 4 ( x 2) 3 C ) ( ( x 2) 2 5 3 2 5 2 3 2 3 ( x 1) 2 ( x 1) 3 C ) ( 5 3 5. Use the integration by substitution to find the definite integrals. a) b) c) d) e) 3 1 x( x 2 4) 3 dx ( 68 ) (2) 7 2 3 0 1 x 1 dx 1 0 x 2 ( x 3 1) 5 dx x cos dx 2 /2 0 0 (2) t t sec2 dt (3) 2 2 3 /2 x x cot 2 sec2 dx f) ( 12 ) 6 6 6. Find the areas of the region enclosed by the lines and curves. 16 x , y 0 and x 4 a) y 3 9 b) y x 2 2 x and y x 2 32 c) y x 2 2, y 2 3 32 d) y x 2 2 x 1, y 5 2 x 3 2 tan Chapter 6: Applications of Definite Integrals Page 7 of 8 Fall 2007 Math2253 Fianl Exam Review Dr. Taixi Xu Department of Mathematics, SPSU 1. Find the volume of the solid generated by revolving the region between y 8 0 x 4 , and the x-axis about the x-axis. x, x, 2. Find the volume of the solid generated by revolving the region between y 128 0 x 4 , and the x-axis about the y-axis. 5 3. Find the volume of solid formed by revolving the region bounded by the graphs 278 of f ( x) x 2 2, g ( x) 1 from x 1 to x 2 about the x-axis. 15 4. Find the volume of solid formed by revolving the region bounded by the graphs 40 of y x 2 , y 0 from x 1 to x 3 about the y-axis. 5. Find the volume of solid formed by revolving the region bounded by the graph 256 of f ( x) x 2 and the x-axis from x 1 to x 4 about the line x 5 . 3 6. Find the volume of the solid generated by revolving region bounded on the left by the parabola x y 2 1 and on the right by the line x 5 about (a) the x-axis; (b) the 1088 512 8 , , y-axis; (c) the line x 5 . 15 15 x 7. Given density function ( x) 2 of thin rod lying along 0 x 4 of the x-axis. 4 16 Find the rod's moment about the origin, mass, and center of mass. 9 1 8. Given density function ( x) 1 of thin rod lying along 1 x 4 of the x-axis. x 73 Find the rod's moment about the origin, mass, and center of mass. 30 9. Lifting equipment A rock climber if about to haul up 100 N (about 22.5 lb ) of equipment that has been hanging beneath her on 40 m of rope that weights 0.8 newton per meter. How much work will it take? ( 4640 J ) 10. A spring has a natural length of 10 in. An 800-lb force stretches the spring to 14 in. 200 a) Find the force constant. b) How much work is done in stretching the spring from 10 in. to 12 in. c) How far beyond natural length will a 1600-lb force stretch the spring? 400 8 Page 8 of 8 ... View Full Document