MATH 140 EXAM 1 Oct-Dec 2016 NAME Professor:B. Stec INSTRUCTIONS The exam is worth 100 points. There are 10 problems and one extra credit worth 5pts. This exam is open book and open notes with unlimited time. This means you may refer to your text book, notes, and online classroom materials. You may take as much time as you wish provided you submit your exam no later than the due date posted in our course syllabus. To be fair to others, late exams will not be accepted. You must show your work to receive full credit. If you do not show your work, you may earn only partial or no credit at the discretion of the instructor. If you have any questions, please feel free to send me a PAGER message in LEO or email me Boguslaw.Stec@faculty.umuc.edu. Best wishes! (1) (2) (3) (4) (5) (6) (7) (8) ___ (9) ___ (10) (11) (12) (13) (14) (15) (16) (17) (18) ___ ___ ___ 1 ___ ___ ___ ___ (10) ___ (19) (11)Bonus question ___ (20) (21) (22) (23) (1) (10 pts)___ Find the slope of midpoint of (24) (25) (26) (27) (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) OM in the figure below if M is the XY : Y ( 0, y ) M O X ( x , 0) of t he a bove y x y x ( A ) ( B ) ( C ) ( D ) ( E ) x y x y (38) (39) (40) (41) (42) (43) (44) (45) (46) (47) (2) (10 pts)____ If f ( x )=ax +b and g ( x ) =cx +d , then when is ( fog ) ( x ) =( gof )( x ) ? You must show your work in order to receive credit. (48) (49) ( A ) f ( a ) =g ( c ) of the above (E ) (50) (51) (52) (53) (54) (55) (56) (57) ( B ) f ( d )=g ( b ) ( C ) f ( b )=g ( d ) ( D ) f ( c )=g ( d ) (58) (59) (60) (61) (62) X (3) (10 pts)___ Two scientists positioned at and Y are 3 miles apart simultaneously measure the angle of elevation of a hot air balloon to be 55 and 65 , respectively. Suppose the balloon is directly above a point on the line segment connecting X Y . Find the elevation of the balloon. and (63) ( A ) h 1.3miles (64) ( D ) h 5.4 miles ( B ) h 2.0 miles of the above (E ) (65) (66) (67) (68) (69) (70) (71) (72) (73) (74) (75) (4) (10 pts)___ True/False. If then f og ( C ) h 2.6 miles is continuous at f and g are both continuous at x=0 , x=0 . If true, then explain why; if false, give a counterexample. (76) (77) ( A ) True ( B ) False ( C ) not enoughinformation of t h e above ( D) (78) (79) (80) (81) (82) (83) (84) (85) (86) (87) (88) (5) (10)____ True/False. Suppose a function to 0 over an interval . I , then the sign of f f is continuous and never equal never changes on that interval I (89) ( A ) True (90) ( B ) False ( C ) not enoughinformation of t h e above ( D) (91) (92) (93) (94) (95) (96) (97) (98) (99) (100) (6) (10 pts)____ Find the trigonometric limit: lim (101) x 0 tan5 x . 4x (A)0 (102) ( B ) 4/5 ( C ) 5/4 ( D)1 of the above (E ) (103) (104) (105) (106) (107) (108) (109) (110) (7) (10 pts)____ If lim f ( x ) g ( x ) x c lim ( f ( x )+ g ( x ) )=2 x c and lim ( f ( x )g ( x ) ) =2 x c . (111) (112) (A)4 (B )2 ( D ) not enough information (113) (114) (115) (116) (117) (118) (119) (120) (121) (122) (C) 0 of the above (E ) , then find (123) (124) (8) (10pts)___ Compute the limit lim (125) x 1 (126) (127) sin ( 1 x ) x1 of the above ( A ) 0 ( B ) 1 ( C ) 1/2 ( D ) 1/2 ( E ) (128) (129) (130) (131) (132) (133) (134) (9) (10pts)____ Suppose the point (0, -1). (135) (136) ( A ) y= x 24 x 2+ 4 . Find the equation of the tangent line at of the above 16 x ( x 2+ 4 ) y= 2 ( B ) y=x ( C ) y=1 ( D ) y =x1 ( E ) (137) (138) (139) (140) (141) (142) (143) (144) (145) (146) (147) (148) (10) (10 pts)___ Find the point(s) on the curve y=x 1 2x where the slope is 11/2 . (149) (150) (151) (152) (153) (154) (A) ( 12 , 12 )( 12 , 12 )( B ) ( 76 , 13 )( 76 , 13 )( C ) ( 13 , 76 )(13 , 76 ) ( D ) ( 13 , 13 ) of the above (E ) (155) (156) (157) (158) (159) (160) (11) (Extra credit 5 pts)____ In Figure 3, a 7 a crank of radius 3 cm connection rod is fastened to cm . The crank shaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find an approximation to the velocity of the piston when = /3 . Use usual conventions with x positive direction to the right. (161) (162) ( A ) 1257 cm/min (163) ( B ) 2467 cm/min (164) ( C ) 3918 cm/min (165) ( D )4018 cm /min (166) of the above (E ) (167) (168) Figure 3: The velocity of the crankshaft is related to the angle of the crankshaft (169) (170) (171) (172) (173) (174) (175) 1. lim ( f ( x )+ g ( x ) )=2 If x c (A)4 and lim ( f ( x )g ( x ) ) =2 x c (B )2 (C) 0 ( D ) not enough information of the above (E ) , then find lim f ( x ) g ( x ) x c . 2. Suppose y= x 24 x 2+ 4 . Find the equation of the tangent line at the point (0, -1). of the above ( A ) y= 16 x ( x 2+ 4 ) 2 ( B ) y=x ( C ) y=1 ( D ) y =x1 ( E ) 3. In Figure 3, a 7 cm connection rod is fastened to a crank of radius 3 cm . The crank shaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find an approximation to the velocity of the piston when = /3 . Use usual conventions with x positive direction to the right. ( A ) 1257 cm/min ( B ) 2467 cm/min ( C ) 3918 cm/min ( D )4018 cm /min of the above (E ) Figure 3: The velocity of the crankshaft is related to the angle of the crankshaft