MTH 201 Final, Fall 2011

Lebanese American University Calculus III Fall 2005 Byblos Final Exam 2 Hours ----------------------------------------------------------------------------------------------------------------- 1. Evaluate the integral I � � 1 x�x�1���x�2� � x ( 7 pts) ----------------------------------------------------------------------------------------------------------------- 2. Let D be the region bounded by the x-axis, the y-axis and the curve y � 2� 1 � x2 , 0 � x � 1. a) Find the area of D. ( 8 pts) b) Find the volume generated by revolving D about the: (i) x-axis (ii) y-axis (12 pts) 1 2 ----------------------------------------------------------------------------------------------------------------- 3. Determine if the series is convergent or divergent: a) � n�1 �� �an � � n�1 �� � 1�sin2�n n� n b) � n�1 �� �an � � n�1 �� � ��5�n 3n�1 c) � n�1 �� �an � � n�1 �� � 3n n� (15 pts) ----------------------------------------------------------------------------------------------------------------- 4. Using the binomial series, find the first 4 nonzero terms of the Maclaurin series of f�x� � 1 1�x2 and g�x� � Ln x � 1 � x2 (10 pts) ----------------------------------------------------------------------------------------------------------------- 5. a) Find the MacLaurin series of f�x� � tan�1�x x b) Estimate the integral � 0 1 � tan�1�x x � x with an error � Rn � � 1 50 (10 pts) ----------------------------------------------------------------------------------------------------------------- SOCIAL b) Estimate the integral � 0 1 � tan�1�x x � x with an error � Rn � � 1 50 (10 pts) ----------------------------------------------------------------------------------------------------------------- Turn Over ----------------------------------------------------------------------------------------------------------------- 6. Let f�x, y� � x2�y2�y x2�2�y . a) Using polar coordinates, is it possible to find the limit of f as �x, y� � �0, 0� b) Determine if this limit exists or not. ( 8 pts) --------------------------------------------------------------------------------------------------------------- 7. Let f�x, y� � sin�x2�y�. Verifythat fxy � fyx ( 7 pts) ----------------------------------------------------------------------------------------------------------------- 8. Find the cartesian equation and sketch the ellipse r � 2 1�3�sin2�Θ ( 8 pts) ----------------------------------------------------------------------------------------------------------------- 9. Evaluate the double integral I � � �D xy�� x��y where D is the domain bounded by the lines y � x , y � 2�x and x � 2. ( 7 pts) 2 2 4 ----------------------------------------------------------------------------------------------------------------- 10. Evaluate the double integral I � �0 1�x 1ey2��y�� x. Hint: the function ey2 can't be integrated. ( 8 pts) ----------------------------------------------------------------------------------------------------------------- Calc3 Fa 2005Final(Library).nb 1 SOCIAL Lebanese American University Calculus III Spring 2002 Byblos Final Exam 2 hours ------------------------------------------------------------------------------------------- 1. Let D be the region in the 1st quadrant bounded by the coordinate axes and the curve y � e�x , 0 � x � �. 1 Find the volume generated by revolving D about: a) the x-axis. b) the y-axis. (15 pts) --------------------------------------------------------------------------------------------- 2. Let D be the region in the 1st quadrant bounded by the y-axis, the line y � Π2 and the parabola y � x2 , 0 � x � Π. p p2 Find the mass of the plate D, if the density is: a) ∆ � y b) ∆ � sin x (Tabular integration) (15 pts) --------------------------------------------------------------------------------------------- 3. Evaluate the improper integral I � � 1 �� � 1 x2�x�1� �x (10 pts) -------------------------------------------------------------------------------------------- 4. a) Find the sum of the geometric series � n�1 �� xn�1 b) Integrate both sides then deduce the sum of the series � n�1 �� 1 n 2n (10 pts) -------------------------------------------------------------------------------------------- 5. Find the interval of convergence of the series � n�1 �� �2�x�1�n n (10 pts) -------------------------------------------------------------------------------------------- SOCIAL -------------------------------------------------------------------------------------------- 6. Estimate the integral I � � 0 1 � sin x x �x with an error � Rn � � 1 5000 (10 pts) -------------------------------------------------------------------------------------------- 7. Find the Taylor polynomial of order 3 of f�x� � Ln�sec x� at a � Π 4 ( 6 pts) -------------------------------------------------------------------------------------------- 8. Use series to evaluate the limit L � l i m x�0 x�Ln�1�x� 1�cos�5�x� ( 7 pts) -------------------------------------------------------------------------------------------- 9. Find the cartesian equation and sketch the curve r � 3 2�sin Θ ( 7 pts) -------------------------------------------------------------------------------------------- 10. Discuss and sketch the curve r � 2 � 2�sin Θ. �1�unit � 1�cm� (10 pts) -------------------------------------------------------------------------------------------- x y 15° 30° 45° 60° 75° 105° 120° 135° 150° 165° 1 unit = 1 cm -------------------------------------------------------------------------------------------- Calc3 Sp 2002Final(Library).nb 1 SOCIAL