Essex County College Division of Mathematics MTH-22 Assessments Last Name: First Name: Phone or email: Honor Code The Honor Code is a statement on academic integrity, it articulates reasonable expectations of students and teachers in establishing and maintaining the highest standards in academic work:. that they will not give or receive aid in taking these assessments, including the use of notes and electronic devises; 2. that they will not use any communication device while taking these assessments, either in the class room or while on a break. If you have a device that rings or vibrates during the assessment, DO NOT ANSWER IT or look at it. Prior to starting these assessments you must turn these devices off and store them away from you for the duration of these assessments. No score may be earned if you use any such device while taking these assessments;. that they will do their share and take an active part in seeing to it that others as well as themselves uphold the spirit and letter of the Honor Code; 4. that they will only turn in their assessments if they are able to honestly state I do hereby affirm, at the close of each assessment, that I had no unlawful knowledge of the questions or answers prior to the assessment and that I have neither given nor received assistance in answering any of the questions during this assessment. Please sign your name below to record that you have reviewed this Honor Code and will abide by these expectations at all times during these assessments. Signature:
Essex County College, MTH-22 Exams may have separate timed parts, one needing a calculator to complete, and one where a calculator is not allowed. Please be prepared for exams by having your own calculator, as there will be absolutely no sharing of calculators on exams. Cell phones are not allowed on exams! Pay careful attention to the Honor Code that appears on the previous page. Each in-class assessment will have this as a cover sheet and you will need to abide by this code. I strongly suggest that you read this Honor Code before coming to an in-class assessment. Understanding this guide will allow you to easily complete exams. I strongly suggest that you review these questions as we cover the material, and seek help when necessary. Using proper notation is critical, as is following through in clearly ordered steps. My exams will mirror what is in this guide. Not that you need to memorize these problems, but you should be very clear on what types of questions you ll see, as well as what typical work looks like. Again, if you are not sure what to do, please get help before the exam. It is generally too late to get help after an exam has been given. The material is divided into four sections study the material sequentially. This guide is in draft mode and certainly contain errors I typed up this document and I know I sometimes make egregious errors. I have typeset this document using L A TEX 2ε, which is what professional mathematicians/scientist use when preparing documents for publication. Students wishing to learn L A TEX 2ε may request source files for this document which will allow students a chance to see how mathematics is typeset using L A TEX 2ε. Many schools require students to use L A TEX 2ε (pronounced lay tech or perhaps lah tech ) and it is in your interest to learn it now, especially if you plan to become a professional mathematician/scientist. Waiting to learn L A TEX 2ε is not advised, and can cause a lot of stress if you re in a competitive program where it is assumed that you already know L A TEX 2ε. There s nothing like being prepared. Errors (actual of imagined) in this document should be reported to Ron Bannon. b@nnon.us This document may be shared with MTH 22 students and instructors only. This document may change over the course of time. It was last updated on January 2, 25. My MTH-22 students are welcome to stop by my office, during office hours, to discuss this document. Other MTH-22 students are encouraged to see their MTH-22 teachers if they need help. Although I have proofed this document, errors may still be there and I encourage anyone to report errors, real or imagined, to me directly. Furthermore, some problems are missing simpler steps that may be obvious to prepared students, but appear as magic to others. b@nnon.us Page of 66 http://mth22.mathography.org
Essex County College, MTH-22 Contents Exam. 6...............................................2 6.2............................................. 6. 6.............................................. 7.4 6.4............................................. 9.5 7...............................................6 7.2............................................. 7 2 Exam 2 22 2. 7.............................................. 22 2.2 7.5............................................. 28 2. 7.6............................................. 4 2.4 7.8............................................. 4 2.5 8.............................................. 4 2.6 8.4............................................. 49 Exam 55............................................... 58.2.2............................................. 72............................................... 8.4.4............................................. 88.5.5............................................. 96.6.6............................................. 4.7.7............................................. 7 4 Exam 4 28 4............................................... 28 4.2.2............................................. 9 4............................................... 44 4.4.4............................................. 52 4.5.5............................................. 6 Page 2 of 66
Essex County College, MTH-22 Exam. 6.. Find the area of the region between y sin x and y cos x over the interval [π/4, π/2]. Solution: Drawing the graph may prove helpful, especially if you need to review trigonometry. A calculator should not be used to do this problem. π 2 π 4 π 2 sin (x) cos (x) dx cos (x) sin (x) π 4 ( ( π ) ( π )) cos sin 2 ( 2 ( ) ) 2 2 2 ( ( π ) ( π )) cos sin 4 4 2. Find the area of the shaded region (Figure, page ) bounded by y x 2x 2 + and y x 2 + 4x. 5 - -2-2 -5 - Figure : Partial graph of y x 2x 2 + and y x 2 + 4x. Solution: Although this may be obvious from the graph, you still need to find the points of intersection using algebra. Graphs are to be used as guides only! A calculator should not be Page of 66
Essex County College, MTH-22 used to do this problem. x 2 + 4x x 2x 2 + x 5x 2 4x + 2 x 2 (x 5) 4 (x 5) (x 5) ( x 2 4 ) (x 5) (x + 2) (x 2) There are three solutions, clearly, from the provided graph (Figure, page ), our limits of are from x 2 to x 2. Yes, the graphs of these two functions intersects at x 5, but that is not the region indicated in the graph (Figure, page ). 2 2 ( x 2x 2 + ) ( x 2 + 4x ) dx 2 2 x 5x 2 4x + 2 dx x4 4 5x 2 2x2 + 2x 2 ( 6 4 ) ( ) 4 44 8 8 6. Find the area between the graphs x sin (y) and x cos (y) over the interval y π/2. Solution: My suggestion, especially if you are confused with the variables in the problem is to rewrite an equivalent problem by interchanging x and y. The problem now reads, find the area between the graphs y sin (x) and y cos (x) over the interval x π/2. You should now be able to visualize this problem. A calculator should not be used to do this problem. π/2 sin (x) ( cos (x)) dx π/2 sin (x) + cos (x) dx cos (x) + sin (x) x ( π ) ( ) 2 π/2 2 π 2 4. Find the area of the region lying to the right of x y 2 5 and to the left of x y 2. Page 4 of 66
Essex County College, MTH-22 Solution: You need to find the points of intersection using algebra. A calculator should not be used to do this problem. y 2 5 y 2 2y 2 8 y 2 4 y ±2 Now the integration. 2 2 ( y 2 ) ( y 2 5 ) dy 2 2 8 2y 2 dy 8y 2y ( 6 6 2 2 ) ( ) 6 6 64 5. Find the area of the region enclosed by the graphs of x y 8y and y + 2x. Solution: You need to find the points of intersection using algebra. A calculator should not be used to do this problem. x y 8y 8x 5x x ( 8x 2 5 ) x ( 2x) 8 ( 2x) x 8x + 6x There are three points of intersection, (, ), ( 7/4, 7/2 ) and ( 7/4, 7/2 ). Graphing may be helpful in setting up the following integration. The two regions are symmetric and that is why I am doubling one. 2 7/2 ( y ) ( y 8y ) 7/2 5y dy 2 2 2 y dy ( ) 5y 2 2 y4 7/2 4 4 52 8 225 8 Page 5 of 66
Essex County College, MTH-22.2 6.2. Find the volume V of the solid whose base is the region enclosed by y x 2 and y, and whose cross sections perpendicular to the y-axis are squares. Solution: Your ability to visualize the region is essential. Once you see the region you should be able to easily set-up the following integral that represents the indicated volume. A calculator should not be used to do this problem. (2 y) 2 dy 2y 2 8 4y dy 2. Calculate the average over the given interval. f (x) x, [, 4] Solution: A calculator should not be used to do this problem. 4 4 x dx x4 6 6 4. Calculate the average over the given interval. f (x) x 2, [, ] + Solution: A calculator should not be used to do this problem. 2 x 2 + arctan (x) dx 2 ( π 2 4 + π ) 4 π 4 Page 6 of 66
Essex County College, MTH-22. 6.. Calculate the volume of the solid obtained by revolving the region under the graph of f (x) x about the x-axis over the interval [, 4]. Solution: A calculator should not be used to do this problem. 4 π ( x ) 2 π dx x π 4 4 2. Find the volume of the solid obtained by rotating the region under the graph of f (x) sin (x) cos (x) about the x-axis over the interval [, π/2]. Solution: A calculator should not be used to do this problem. π/2 ( ) 2 π/2 π sin (x) cos (x) dx π sin (x) cos (x) dx π 2 π 2 π/2 π/2 π cos (2x) 4 2 sin (x) cos (x) dx sin (2x) dx π ( ) 4 π/2 π 2. Find the volume of the solid obtained by rotating the region enclosed by the curves f (x) x 2 and g (x) 2x + about the x-axis. Solution: Find the points of intersection first. A calculator should not be used to do this problem. x 2 2x + x 2 2x (x ) (x + ) Page 7 of 66
Essex County College, MTH-22 So our limits along the x-axis are from to. Here s the integration. π (2x + ) 2 π ( x 2) 2 π (2x + ) dx πx5 6 5 ( 24π 24π ) ( π 2 5 6 + π ) 5 88π 5 Page 8 of 66
Essex County College, MTH-22.4 6.4. Use the Shell Method to compute the volume obtained by rotating the region enclosed by the graphs as indicated, about the y-axis. f (x) x 2, g (x) 6 x, x Solution: Yep, you ll need to be able to visualize this before setting up and evaluating the integral. A calculator should not be used to do this problem. 2 2πx [(6 x) (x 2)] dx 8π 2 2x x 2 dx ) 8π (x 2 x 2 2π 2. Use the Shell Method to compute the volume, V, of the solid obtained by rotating the region enclosed by the graphs as indicated, about the y-axis. y x 2, y x /2 Solution: Yep, you ll need to be able to visualize this before setting up and evaluating the integral. A calculator should not be used to do this problem. ( 2πx x /2 x 2) dx 2π 2π π x /2 x dx ) x4 5 4 ( 2x 5/2. Use the Shell Method to compute the volume of the solid obtained by rotating the region underneath the graph of y x 2 + over the interval [, 2], about the line x. Page 9 of 66
Essex County College, MTH-22 Solution: Yep, you ll need to be able to visualize this before setting up and evaluating the integral. A calculator should not be used to do this problem. 2 2πx x 2 + dx You ll need to use u-substitution (learned in MTH 2). 2 u x 2 + du 2x dx 5 2πx x 2 + dx π u /2 du 5 2πu /2 ( ) 2π 5 Page of 66
Essex County College, MTH-22.5 7.. Evaluate using Integration by Parts. (2x + ) e x dx Solution: A calculator should not be used to do this problem. Let s proceed: u 2x + and dv e x dx, this easily leads to (forgetting about C), du 2 dx and v e x. Let s rewrite our original example. (2x + ) e x dx (2x + ) e x + 2 e x dx (2x + ) e x 2e x + C e x (2x + ) + C 2. Evaluate using Integration by Parts. x 2 e x dx Solution: A calculator should not be used to do this problem. Let s proceed: u x 2 and dv e x dx, this easily leads to (forgetting about C), du 2x dx and v e x. Let s rewrite our original example. x 2 e x dx x 2 e x 2 xe x dx Now, do it again. u x and dv e x dx, Page of 66
Essex County College, MTH-22 this easily leads to (forgetting about C), du dx and v e x, Let s rewrite again. x 2 e x dx x 2 e x 2 xe x dx [ x 2 e x 2 xe x x 2 e x 2xe x + 2e x + C e x ( x 2 2x + 2 ) + C ] e x dx. Evaluate using Integration by Parts. x sin ( x) dx Solution: A calculator should not be used to do this problem. Let s proceed: u x and dv sin ( x) dx, this easily leads to (forgetting about C), du dx and v cos ( x). Let s rewrite our original example. x sin ( x) dx x cos ( x) cos ( x) dx x cos ( x) + sin ( x) + C 4. Evaluate using Integration by Parts. x 2 cos (x) dx Solution: A calculator should not be used to do this problem. Let s proceed: u x 2 and dv cos (x) dx, Page 2 of 66
Essex County College, MTH-22 this easily leads to (forgetting about C), du 2x dx and v sin (x). Let s rewrite our original example. x 2 cos (x) dx x2 sin (x) 2 x sin (x) dx Now, do it again. u x and dv sin (x) dx, this easily leads to (forgetting about C), du dx and v Let s rewrite again. cos (x), x 2 cos (x) dx x2 sin (x) x2 sin (x) x2 sin (x) 2 x sin (x) dx 2 [ x cos (x) + + 2x cos (x) 9 2 sin (x) 27 ] cos (x) dx + C 5. Evaluate using Integration by Parts. 7x ln (8x) dx Solution: A calculator should not be used to do this problem. Let s proceed: u ln (8x) and dv 7x dx, this easily leads to (forgetting about C), du x dx and v 7x2 2. Let s rewrite our original example. 7x ln (8x) dx 7x2 ln (8x) 7 2 2 7x2 ln (8x) 2 x dx 7x2 4 + C Page of 66
Essex County College, MTH-22 6. Evaluate using Integration by Parts. ln (x) x 4 dx Solution: A calculator should not be used to do this problem. Let s proceed: u ln (x) and dv x 4 dx, this easily leads to (forgetting about C), du x dx and v x. Let s rewrite our original example. ln (x) ln (x) x 4 dx x + x 4 dx ln (x) x 9x + C 7. Evaluate the integral, using Integration by Parts. (Remember to use ln u where appropriate.) x sec 2 (7x) dx Solution: A calculator should not be used to do this problem. Let s proceed: u x and dv sec 2 (7x) dx, this easily leads to (forgetting about C), du dx and v tan (7x) 7 sin (7x) 7 cos (7x). Let s rewrite our original example. x sec 2 (7x) dx x tan (7x) 7 7 sin (7x) cos (7x) dx Page 4 of 66
Essex County College, MTH-22 Now using u-substitution on the remaining integral. u cos (7x) du 7 sin (7x) dx sin (7x) cos (7x) dx 7 u du ln u + C 7 ln cos (7x) + C 7 x sec 2 x tan (7x) ln cos (7x) (7x) dx + 7 49 + C 8. Evaluate using Integration by Parts. arccos (5x) dx Solution: A calculator should not be used to do this problem. Let s proceed: u arccos (5x) and dv dx, this easily leads to (forgetting about C), 5 du dx and v x. 25x 2 Let s rewrite our original example. arccos (5x) dx x arccos (5x) + 5 Now using u-substitution on the remaining integral. x 25x 2 dx u 25x 2 du 5x dx x dx u /2 du 25x 2 5 u 25 + C 25x 2 + C 25 arccos (5x) dx x arccos (5x) + 5 x 25x 2 dx x arccos (5x) 25x 2 5 + C Page 5 of 66
Essex County College, MTH-22 9. Evaluate using Integration by Parts. x7 x dx Solution: A calculator should not be used to do this problem. Let s proceed: u x and dv 7 x dx, this easily leads to (forgetting about C), du dx and v 7x ln 7. Let s rewrite our original example. x7 x dx x7x ln 7 ln 7 7 x dx x7x ln 7 7x (ln 7) 2 + C. Compute the definite integral. 9 xe x dx Solution: A calculator should not be used to do this problem. Let s proceed: u x and dv e x dx, this easily leads to (forgetting about C), du dx and v ex. Let s rewrite our original example. 9 xe x dx xex xex 9 ex 9 + 26e27 9 9 9 e x dx Page 6 of 66
Essex County College, MTH-22.6 7.2. Use the method for odd powers to evaluate the integral. 7 cos (x) dx Solution: A calculator should not be used to do this problem. u sin (x) du cos (x) dx 7 cos (x) dx 7 cos x cos 2 (x) dx 7 cos x ( sin 2 (x) ) dx ( 7 u 2 ) du ) 7 (u u + C ( ) 7 sin (x) sin (x) + C 2. Use the method for odd powers to evaluate the integral. sin (x) cos 4 (x) dx Solution: A calculator should not be used to do this problem. u cos (x) du sin (x) dx sin (x) cos 4 (x) dx sin (x) sin 2 (x) cos 4 (x) dx sin (x) ( cos 2 (x) ) cos 4 (x) dx sin (x) ( cos 4 (x) cos 6 (x) ) dx ( u 6 u 4) du u7 7 u5 5 + C cos7 (x) 7 cos5 (x) 5 + C Page 7 of 66
Essex County College, MTH-22. Evaluate the integral. π/2 cos (x) dx Solution: A calculator should not be used to do this problem. π/2 cos (x) dx u sin (x) du cos (x) dx π/2 π/2 2 9 cos (x) cos 2 (x) dx cos (x) ( sin 2 (x) ) dx (u u u 2 du ) 4. Evaluate the integral using double angle formulas to reduce the powers of the cosine function. 8 cos 4 (y) dy Solution: A calculator should not be used to do this problem. 8 cos 4 (y) dy 8 cos 2 (y) cos 2 (y) dy ( ) ( ) cos (2y) + cos (2y) + 8 2 2 dy 2 (cos (2y) + ) (cos (2y) + ) dy 2 cos 2 (2y) + 2 cos (2y) + dy ( ) cos (4y) + 2 + 2 cos (2y) + dy 2 cos (4y) + 4 cos (2y) + dy sin (4y) 4 + 2 sin (2y) + y + C Page 8 of 66
Essex County College, MTH-22 5. Use the equation sin m (x) cos n (x) dx sinm+ (x) cos n (x) m + n + n m + n sin m (x) cos n 2 (x) dx to evaluate. sin (x) cos 2 (x) dx Solution: A calculator should not be used to do this problem. Finishing. sin (x) cos 2 (x) dx sin+ (x) cos 2 (x) + 2 sin (x) cos 2 2 (x) dx + 2 + 2 sin4 (x) cos (x) + sin (x) dx 5 5 u cos (x) du sin (x) dx sin (x) dx sin 2 (x) sin (x) dx ( cos 2 (x) ) sin (x) dx (u 2 ) du u u + C cos (x) + C cos (x) sin (x) cos (x) dx sin4 (x) cos (x) 5 + cos (x) 5 cos (x) 5 + C 6. Evaluate the integral. cos 6 (x) sin (x) dx Page 9 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. u cos (x) du sin (x) dx Using u-substitution. cos 6 (x) sin (x) dx u 6 du u7 7 + C cos7 (x) 7 + C 7. Evaluate the integral. π/2 sin 2 (2x) dx Solution: A calculator should not be used to do this problem. example. Let s rewrite our original 2π 2π sin 2 cos (4x) (2x) dx 2 x sin (4x) 2π 2 8 π dx 8. Evaluate the integral. π/2 cos (9x) dx Solution: A calculator should not be used to do this problem. u sin (9x) du 9 cos (9x) dx Page 2 of 66
Essex County College, MTH-22 Let s rewrite our original example. π/2 cos (9x) dx 9 9 2 27 π/2 π/2 cos 2 (9x) cos (9x) dx ( sin 2 (9x) ) cos (9x) dx ( u 2 ) du (u u ) 9. Evaluate the integral. π/ sin 6 (x) cos (x) dx Solution: A calculator should not be used to do this problem. u sin (x) du cos (x) dx Let s rewrite our original example. π/ sin 6 (x) cos (x) dx π/ π/ /2 /2 u7 7 u9 9 45 584 sin 6 (x) cos 2 (x) cos (x) dx sin 6 (x) ( sin 2 (x) ) cos (x) dx u 6 ( u 2) du u 6 u 8 du /2 Okay, the arithmetic might be tough here, but you should nonetheless be able to do this without using a calculator. Page 2 of 66
Essex County College, MTH-22 2 Exam 2 2. 7.. Use the indicated substitution to evaluate the integral. 6 x 2 dx, x 4 sin (θ) Solution: A calculator should not be used to do this problem. x dx 6 x 2 dx 4 sin (θ) 4 cos (θ) dθ 6 6 sin 2 4 cos (θ) (θ) dθ 6 sin 2 (θ) cos (θ) dθ 6 cos 2 (θ) dθ 8 cos (2θ) + dθ 8 ( ) sin (2θ) + θ + C 2 8 (sin (θ) cos (θ) + θ) + C ( 8 ( x 6 x 2 x + arcsin 4 4 4 ( ) x x 6 x 2 2 + 8 arcsin 4 + C )) + C 2. Use the indicated substitution to evaluate the integral. /2 x 2 9 x 2 dx, x sin (θ) Page 22 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. /2 x sin (θ) dx cos (θ) dθ x 2 π/6 dx 9 sin 2 (θ) cos (θ) dθ 9 x 2 9 9 sin 2 (θ) 9 2 9 2 8 π/6 π/6 ( θ 9 sin 2 (θ) dθ cos (2θ) dθ ) sin (2θ) π/6 2 ) ( 2π. Evaluate using the substitution u x 2 4. 9x x 2 4 dx Solution: A calculator should not be used to do this problem. u x 2 4 du 2x dx 9x x 2 4 dx 9 u /2 du 2 9 u + C 9 x 2 4 + C 4. Evaluate using trigonometric substitution. 9x x 2 4 dx Page 2 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. x 2 sec (θ) dx 2 sec (θ) tan (θ) dθ 9x x 2 4 dx 6 sec (θ) sec (θ) tan (θ) dθ 4 sec 2 (θ) 4 8 sec 2 (θ) dθ 8 tan (θ) + C 9 x 2 4 + C 5. Evaluate using trigonometric substitution. x 2 7 x 2 dx Solution: A calculator should not be used to do this problem. x 2 7 x 2 dx x 7 sin (θ) dx 7 cos (θ) dθ 7 cos (θ) 7 sin 2 (θ) 7 7 sin 2 (θ) dθ 7 csc 2 (θ) dθ cot (θ) + C 7 7 x 2 + C 7x 6. Evaluate using trigonometric substitution. x 4 x 2 dx Page 24 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. x 2 sin (θ) dx 2 cos (θ) dθ x 4 x 2 dx 8 sin (θ) 4 4 sin 2 (θ) 2 cos (θ) dθ 2 sin (θ) cos 2 (θ) dθ 2 sin (θ) ( cos 2 (θ) ) cos 2 (θ) dθ u cos (θ) du sin (θ) dθ 2 sin (θ) ( cos 2 (θ) ) cos 2 (θ) dθ 2 u 4 u 2 du ( ) u 5 2 5 u + C ( cos 5 ) (θ) 2 cos (θ) + C 5 ( ) 5 ( ) 2 4 x 2 4 x 2 + C 5 2 2 ( 4 x 2 ) 5/2 4 ( 4 x 2) /2 + C ( 5 x 2 + 8 ) ( 4 x 2) 4 x 2 + C 5 7. Evaluate using trigonometric substitution. You ll need to complete the square first. 2x x 2 dx Page 25 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. 2x x 2 ( x 2 4x ) 2x x 2 ( x 2 4x + 4 ) + 2 2x x 2 2 (x 2) 2 dx 2x x 2 2 (x 2) 2 dx 4 (x 2) 2 dx (x 2) 2 sin (θ) dx 2 cos (θ) dθ dx 4 (x 2) 2 4 4 sin 2 (θ) dθ 2 cos (θ) dθ θ + C arcsin ( x 2 2 ) + C 8. Evaluate the integral by completing the square and using trigonometric substitution. (Remember to use ln u where appropriate.) 84x + 7x 2 dx Page 26 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. 7 84x + 7x 2 7 ( x 2 + 2x ) 84x + 7x 2 7 ( x 2 + 2x + 6 ) 6 7 ( ) 84x + 7x 2 7 (x + 6) 2 6 dx 84x + 7x 2 7 (x + 6) 6 sec (θ) ( ) dx 7 (x + 6) 2 6 dx (x + 6) 2 6 dx 6 tan (θ) sec (θ) dθ dx 6 tan (θ) sec (θ) dθ (x + 6) 2 6 7 6 sec 2 (θ) 6 sec (θ) dθ 7 7 ln sec (θ) + tan (θ) + C ln x + 6 + x 2 + 2x + C 7 The anti-derivative of sec (θ) is, in my opinion, not worth remembering. This was derived in class, but I will give this anti-derivative if needed on an exam. Page 27 of 66
Essex County College, MTH-22 2.2 7.5. Find the partial fraction decomposition for the rational function. 8x 2 4x 68 (x 2 + 7) (x 5) Solution: A calculator should not be used to do this problem. 8x 2 4x 68 (x 2 Ax + B + 7) (x 5) x 2 + 7 + C x 5 8x 2 4x 68 (Ax + B) (x 5) + C ( x 2 + 7 ) 8x 2 4x 68 (x 2 + 7) (x 5) x 5 288 2C C 9 x 68 5B 6 B x 8 (A + ) ( 4) 72 8 4A 4 A x + x 2 + 7 9 x 5 2. Determine the constants A and B. 5x 8 (x 4) (x ) A x 4 + B x Solution: A calculator should not be used to do this problem. 5x 8 A (x 4) (x ) x 4 + B x 5x 8 A (x ) + B (x 4) x B B x 4 2 A A 4 Page 28 of 66
Essex County College, MTH-22. Clear denominators in the following partial fraction decomposition and determine the constant B (substitute a value of x or use the method of undetermined coefficients). 6x 2 x 5 (x + ) (x + 4) 2 x + B x + 4 (x + 4) 2 Solution: A calculator should not be used to do this problem. 6x 2 x 5 (x + ) (x + 4) 2 x + B x + 4 (x + 4) 2 6x 2 x 5 (x + 4) 2 B (x + ) (x + 4) (x + ) x 2 2 4 + 2B + B 7 Using the method of undetermined coefficients. 6x 2 x 5 (x + 4) 2 B (x + ) (x + 4) (x + ) x 2 + 8x + 6 B ( x 2 + 5x + 4 ) x x 2 Bx 2 x + 8x 5Bx + 6 4B x 2 Bx 2 + 5x 5Bx + 4B ( B) x 2 + 5 ( B) x + ( 4B) 6 B 5 ( B) 5 4B B 7 4. Find the constants in the given partial fraction decomposition. 5x + (x ) (x 2 + 4) A x + Bx + C x 2 + 4 Page 29 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. 5x + (x ) (x 2 + 4) A x + Bx + C x 2 + 4 5x + A ( x 2 + 4 ) + (Bx + C) (x ) x 6 5A A 6 5 x 24 5 C C 9 5 x 4 6 + 5 9 5 B B 6 5 ( ) 9 5 B ( 2) 5. Evaluate using long division first to write f (x) as the sum of a polynomial and a proper rational function. (Remember to use ln u where appropriate.) x x 2 dx Solution: A calculator should not be used to do this problem. x x 2 + 2/ x 2 x x 2 dx + 2/ x 2 dx dx + 2 9x 6 dx x x 2 ln 9x 6 + 9 2 ln x 2 + 9 + C + C 6. Evaluate the integral. (Remember to use ln u where appropriate.) x 2 9x + 8 dx Page of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. x 2 9x + 8 (x 6) (x ) A x 6 + B x A (x ) + B (x 6) x B B x 6 A A x 2 9x + 8 dx / x 6 / x dx ln x 6 x + C 7. Evaluate the integral. (Remember to use ln u where appropriate.) 6x x 2 5x + 6 dx Solution: A calculator should not be used to do this problem. 6x 6x x 2 5x + 6 (x ) (x 2) A x 2 + B x 6x A (x ) + B (x 2) x 7 B x 2 A A 6x x 2 5x + 6 dx 7 x x 2 dx 7 ln x ln x 2 + C Page of 66
Essex County College, MTH-22 8. Evaluate the integral. (Remember to use ln u where appropriate.) x 2 + 7x + 2 (x ) (x + ) 2 dx Solution: A calculator should not be used to do this problem. x 2 + 7x + 2 (x ) (x + ) 2 A x + B x + + C (x + ) 2 x 2 + 7x + 2 A (x + ) 2 + B (x + ) (x ) + C (x ) x 2 4A A 8 x 2 2C C x 2 8 B B 5 x 2 + 7x + 2 (x ) (x + ) 2 dx 8 x + 5 x + + (x + ) 2 dx 8 ln x + 5 ln x + x + + C 9. Evaluate the integral. x 2 4x 2 + 49 dx Page 2 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. x 2 4x 2 + 49 4 49/4 4x 2 + 49 x 2 4x 2 + 49 dx 4 dx 49/4 4x 2 + 49 dx 4 dx 49 4 4x 2 + 49 dx 4 dx 4 4x 2 /49 + dx x 4 4 (2x/7) 2 + dx x 4 4 7 ( ) 2x 2 arctan + C 7 x 4 7 ( ) 2x 8 arctan + C 7. Evaluate the integral using the appropriate method or combination of methods covered thus far in the text. 8 x x + dx Solution: A calculator should not be used to do this problem. u x /2 u 2 x du 2 x dx 8 x x + dx 6 6 u 2 + du arctan (u) + C 6 arctan (x /2) + C Page of 66
Essex County College, MTH-22 2. 7.6. Let f (x) x 8/5. (a) Evaluate. R f (x) dx Solution: A calculator should not be used to do this problem. R (b) Evaluate f (x) dx f (x) dx by computing the limit R lim R f (x) dx R x 8/5 dx 5 R x /5 5 ( ) R /5 Solution: A calculator should not be used to do this problem. R ( 5 lim f (x) dx lim ) R R R /5 5 2. Determine if ( x) /2 dx converges by computing R lim R ( x) /2 dx Page 4 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. dx lim /2 ( x) R u x du dx R lim R ( x) R 2 /2 dx u /2 du 2 R lim u R 2 ( 2 lim 2 ) R R 2 The limit does not exists, therefore the integral diverges.. Determine whether the improper integral converges and, if so, evaluate it. x 7/6 dx Solution: A calculator should not be used to do this problem. x 7/6 dx lim R R 6 lim 6 lim R 6 x 7/6 dx R ( R /6 R x /6 ) 4. Determine whether the improper integral converges and, if so, evaluate it. 4 4 x dx Page 5 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. 4 R dx lim dx 4 x R 4 4 x R 2 lim 4 x R 4 ( lim 4 2 ) 4 R 4 R 4 5. Determine whether the improper integral converges and, if so, evaluate it. e 2x dx Solution: A calculator should not be used to do this problem. e 2x dx lim R R 2 lim e 2x dx R e 2x R ( lim R 2e 6 2e 2R ) 2e 6 6. Use the Comparison Test to determine whether the integral converges or not. 4 5 x 8 + dx Solution: A calculator should not be used to do this problem. 5 x 8 + > 5 x 8 5 x 8 + The integral 4 < 5 x 8 5 dx dx x 8 x8/5 4 converges because 8/5 >. Therefore, by the Comparison Test, 4 also converges. 5 x 8 + dx Page 6 of 66
Essex County College, MTH-22 7. Determine whether the improper integral converges and, if so, evaluate it. 9 x + 9 dx Solution: A calculator should not be used to do this problem. 9 dx lim x + 9 R 9 + R 2 2 5 dx x + 9 lim (x + 9)2/ R 9 + ( lim 2 2/ (R + 9) 2/) R 9 + R 8. Evaluate the improper integral. e 4x cos (x) dx Solution: A calculator should not be used to do this problem. First do the Integration by Parts. u e 4x du 4e 4x dx dv cos (x) dx v sin (x) e 4x cos (x) dx e 4x sin (x) + 4 e 4x sin (x) dx Page 7 of 66
Essex County College, MTH-22 Yikes, we need to do Integration by Parts again! 4 25 9 u e 4x du 4e 4x dx dv sin (x) dx cos (x) v e 4x sin (x) dx 4 [ e 4x cos (x) 4 ] e 4x cos (x) dx 4e 4x cos (x) 6 e 4x cos (x) dx 9 9 e 4x cos (x) dx e 4x sin (x) 4e 4x cos (x) 6 e 4x cos (x) dx 9 9 u e 4x e 4x cos (x) dx e 4x sin (x) 4e 4x cos (x) + C 9 e 4x cos (x) dx e 4x sin (x) 25 4e 4x cos (x) 25 + C 2 Okay, I know some have trouble with taking more than one-step, but you ll need to try! Moving forward. R And now the limit. e 4x cos (x) dx e 4x sin (x) 25 4 25 e 4x cos (x) dx lim R 4 25 + sin (R) 25e 4R 4e 4x cos (x) 25 4 sin (R) 25e 4R ( 4 sin (R) + 25 25e 4R R ) 4 cos (R) 25e 4R 9. Determine whether the improper integral converges and, if so, evaluate it. 4 e x x dx Page 8 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. u x du 2 x dx e x dx 2 e u du x Now with limits. 4 2e u + C 2e x + C e x R dx lim x R 2 lim R The integral does not converge. 4 ( e e x x dx R e 2). Determine whether the improper integral converges and, if so, evaluate it. ln (2x) x 2 dx Solution: A calculator should not be used to do this problem. u ln (2x) du x dx dv x 2 dx v x ln (2x) ln (2x) x 2 dx + x 2 dx x ln (2x) x x + C Now with limits. ln (2x) x 2 dx lim R lim R R + ln (2) ln (2x) x 2 dx ( + ln (2) ln (2R) R R ) Page 9 of 66
Essex County College, MTH-22 2.4 7.8 Although I do not allow cheat sheets on exams, if I ask questions from this section I will give you the following two rules. Trapezoid Rule Given a definite integral b a f (x) dx, the n th trapezoidal approximation is: where b a 2n [f (a) + 2f (c ) + + 2f (c n ) + f (b)], c i a + (b a) i. n Simpson s Rule Given a definite integral b a f (x) dx, the n th (n needs to be even) Simpson s Rule approximation is: where b a n [f (a) + 4f (c ) + 2f (c 2 ) + 4f (c ) + + 2f (c n 2 ) + 4f (c n ) + f (b)], c i a + (b a) i. n A calculator may be necessary to complete these problems as written, however I may not require a decimal approximation on exams, and under those circumstances you will only be expected to write out the necessary computations.. Calculate T N and M N for the value of N indicated. (Round your answers to two decimal places.) 4 5 x dx, N 4 Page 4 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem unless an approximate answer is requested. That is, a calculator is not needed until the final step! Divide [, 4] into 4 equal subintervals. The width of each interval is ; the endpoints are,, 2,, 4, and the midpoints are.5,.5, 2.5,.5. [ 5 M 4 5.5 +.5 + 5 2.5 + 5 ].5 4.44 T 4 [ 2 5 5 + 2 + 2 5 2 + 2 5 + 5 ] 4 4.5 2. Calculate T 8 and M 8 for the following. (Round your answers to three decimal places.) 7 e x x + 2 dx Solution: A calculator should not be used to do this problem unless an approximate answer is requested. That is, a calculator is not needed until the final step! Here I will assign f (x) ex x + 2 and use summation notation. M 8 4 T 8 8 7 f i [ ( 8 + i ) 26.627 4 f () + f (7) + 2 7 f i ( + i ) ] 44.27 4. Calculate S N given by Simpson s Rule for the value of N indicated. (Round your answer to one decimal place.) 4 2 9 x 2 dx, N 4 Solution: A calculator should not be used to do this problem unless an approximate answer is requested. That is, a calculator is not needed until the final step! Here I will assign f (x) 9 x 2 S 4 [f (2) + 4f (2.5) + 2f () + 4f (.5) + f (4)].7 6 Page 4 of 66
Essex County College, MTH-22 4. Use S 4 to estimate π/6 sin x x dx, taking the value of sin x x at x to be. (Round your answer to three decimal places.) Solution: A calculator should not be used to do this problem unless an approximate answer is requested. That is, a calculator is not needed until the final step! Here I will assign f (x) sin x x, x f () S 4 π 72 [ ( π ( π ( π ) ( π )] f () + 4f + 2f + 4f + f.56 24) 2) 8 6 Page 42 of 66
Essex County College, MTH-22 2.5 8.. Express the arc length of the curve y 2 tan (x) for x π/ as an integral (but do not evaluate). Solution: A calculator should not be used to do this problem. y 2 tan (x) y 2 sec 2 (x) π/ s + 4 sec 4 (x) dx 2. Calculate the arc length of y x /2 over the interval [, ]. Solution: A calculator should not be used to do this problem. y x /2 y x 2 s 4 9 + 9x 4 dx /4 u /2 /4 8u u 27 /4 /4 du 27. Calculate the arc length of y 6 ln (cos (6x)) over the interval [, π/48]. Page 4 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. y 6 ln (cos (6x)) y tan (6x) π/48 s + tan 2 (6x) dx π/48 π/48 6 π/ sec 2 (6x) dx sec (6x) dx sec (u) du 6 ln sec (u) + tan (u) π/ ln ( 2 + ) 6 4. Calculate the length of the astroid (Figure 2, page 44) of x 2/ + y 2/ 5 2/. 6 5 4 2-9 -8-7 -6-5 -4 - -2-2 4 5 6 7 8 9 - -2 - -4-5 -6 Figure 2: Graph of x 2/ + y 2/ 5 2/. Solution: A calculator should not be used to do this problem. The graph indicates four equal Page 44 of 66
Essex County College, MTH-22 regions, I will only do the one arc in the first quadrant, and then multiply by four. x 2/ + y 2/ 5 2/ 2 x / + 2 y / y ( y ) 2 + ( y ) 2 y y/ x / y2/ x 2/ + y2/ x 2/ x2/ + y 2/ x 2/ 52/ x 2/ + (y ) 2 5 / x / s 4 5 6 5x 2 5 5 / x / dx 5. Find the arc length of the curve 9y 2 x (x ) 2 shown below (Figure, page 45) over the interval [, 2]. 2 4 - Figure : Graph of 9y 2 x (x ) 2. Page 45 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. Some of the easy steps are being skipped. You should be able to do this work. 9y 2 x (x ) 2 y x/2 x/2 y x/2 2 x /2 2 + ( y ) 2 (x + )2 4x + (y ) 2 x + 2 x x/2 2 + x /2 2 2 x /2 s 2 + x /2 2 ( x ) x + 2 5 2 dx 6. Compute the surface area of revolution about the x-axis over the interval. y x, [, 6] Solution: A calculator should not be used to do this problem. Some of the easy steps are being skipped. You should be able to do this work. The arithmetic may prove too difficult for some, but I really do hope that you can do 9 6 4 without the aid of a calculator. However, if you can not do this you should just write down the indicted arithmetic. y x y x 2 s 6 u + 9x 4 du 6x dx 2πx + 9x 4 dx 665 s π u /2 du 8 πu u 665 27 π ( 665 ) 665 27 Page 46 of 66
Essex County College, MTH-22 7. Compute the surface area of revolution of y ( 9 x 2/) /2 about the x-axis over the interval [8, 27]. Solution: A calculator should not be used to do this problem. Some of the easy steps are being skipped. You should be able to do this work. + ( y ) 2 y y s (9 x 2/) /2 9x 2/ 27 6π ( 9 x 2/) /2 x / 8 27 2π (9 x 2/) /2 9x 2/ dx 8 u 9 x 2/ du 2 x / dx s 9π 5 8πu2 u 5 9π 5 ( 9 x 2/) /2 x / dx u /2 du 5 8. Compute the surface area of revolution about the x-axis over the interval. y x2 4 ln (x), [, e] 2 Solution: A calculator should not be used to do this problem. Some of the easy steps are Page 47 of 66
Essex County College, MTH-22 being skipped. You should be able to do this work. y x2 4 ln (x) 2 y x 2 2x + (y ) 2 x2 + 2x s e 2π ( x 2 2π 4 e ( x 4 ln (x) 2 ) ( x 2 ) + dx 2x x 8 + x x ln (x) ln (x) 8 4 4x 2π 2 + x2 6 x2 ln (x) 8 ( x 4 π 6 + x2 4 x2 ln (x) 4 π ( e 4 9 ) 6 dx + x2 (ln (x))2 6 8 ) (ln (x))2 4 e ) e 9. Use a computer algebra system (CAS) to find the approximate surface area of the solid generated by rotating the curve about the x-axis. (Round your answer to three decimal places.) y e x2 /2, [, ] Solution: I am using a TI-89, but you may use any computer algebra system [CAS] you like. Stop by my office if you want to see it done on a variety of systems. Mathematica is great, but costly. Sage is also very good, and it is free. y e x2 /2 y xe x2 /2 s 2π (e ) x2 /2 + ( xe ) x2 /2 2 dx 8.565 Page 48 of 66
Essex County College, MTH-22 2.6 8.4. Find the Taylor polynomial T (x) centered at x for the function f (x) 2 + x. Solution: A calculator should not be used to do this problem. f (x) 2 + x 2 + x f (x) 2 ( + x) 2 ( + x) f (x) 6 ( + x) 2 2 ( + x) 2 f (x) 6 ( + x) 4 ( + x) f (x) 24 ( + x) 4 4 ( + x) 4 f () 2 f () 2 f () 4 f () 4 T (x) f ()! (x ) + f ()! 2 2x + 2x2 2x (x ) + f () 2! (x ) 2 + f ()! (x ) 2. Find the Taylor polynomial T 2 (x) and T (x) centered at x a for the given function and value of a. f (x) + x 2, a 4 Page 49 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. f (x) + x 2 f (x) ( + x 2) f (x) 2x ( + x 2) 2 f (x) 2 ( x 2 ) ( + x 2) f (x) 24x ( x 2) ( + x 2) 4 f ( 4) 7 f ( 4) 8 289 f ( 4) 94 49 f ( 4) 44 852 T 2 (x) f ( 4) (x + 4) + f ( 4) (x + 4) + f ( 4)!! 2! 8 (x + 4) + 7 829 T (x) f ( 4)! 8 (x + 4) + 7 829 + 47 (x + 4)2 49 (x + 4) + f ( 4) (x + 4) + f ( 4)! 2! 47 (x + 4)2 24 (x + 4) + + 49 852 (x + 4) 2 (x + 4) 2 + f ( 4)! (x + 4). Find the Taylor polynomial T 2 (x) and T (x) centered at x a for the given function and value of a. f (x) x 4 2x, a Page 5 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. f (x) x 4 2x f (x) 4x 2 f (x) 2x 2 f (x) 24x f () 75 f () 6 f () 8 f () 72 T 2 (x) f () (x ) + f () (x ) + f () (x ) 2!! 2! 75 + 6 (x ) + 54 (x ) 2 T (x) f () (x ) + f () (x ) + f () (x ) 2 + f () (x )!! 2!! 75 + 6 (x ) + 54 (x ) 2 + 2 (x ) 4. Find the Taylor polynomial T 2 (x) and T (x) centered at x 5π/4 for f (x) tan (x). Solution: A calculator should not be used to do this problem. Although the higher order Page 5 of 66
Essex County College, MTH-22 derivatives may be difficult to get for some students, they are nonetheless required. f (x) tan (x) f (x) sec 2 (x) f (x) 2 tan (x) sec 2 (x) f (x) 2 sec 4 (x) + 4 tan 2 (x) sec 2 (x) ( ) 5π f 4 ( ) 5π f 2 4 ( ) 5π f 4 4 ( ) 5π f 6 4 T 2 (x) f ( ) 5π ( 4 x 5π ) + f ( ) 5π 4! 4! + 2 ( x 5π 4 ) T (x) T 2 (x) + f ( 5π 4! ( + 2 x 5π 4 ) ( + 2 x 5π 4 ( x 5π 4 ) ) ( + 2 x 5π 4 ( x 5π 4 ) 2 ) 2 + 8 ( x 5π 4 ) + f ( ) 5π ( 4 x 5π 2! 4 ) ) 2 5. Find the Taylor polynomial T 2 (x) and T (x) centered at x ln (2) for f (x) e 2x. Page 52 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. f (x) e 2x f (x) 2e 2x f (x) 4e 2x f (x) 8e 2x f (ln (2)) 4 f (ln (2)) 8 f (ln (2)) 6 f (ln (2)) 2 T 2 (x) f (ln (2)) (x ln (2)) + f (ln (2))!! 4 + 8 (x ln (2)) + 8 (x ln (2)) 2 T (x) T 2 (x) + f (ln (2)) (x ln (2))! 4 + 8 (x ln (2)) + 8 (x ln (2)) 2 + 6 (x ln (2)) + f (ln (2)) 2! (x ln (2)) (x ln (2)) 2 6. Find the Taylor polynomial T 2 (x) and T (x) centered at x for f (x) ln (x) /x. Solution: A calculator should not be used to do this problem. f (x) f (x) f (x) ln (x) x ln (x) x 2 2 ln (x) x f (x) 6 ln (x) x 4 f () f () f () f () T 2 (x) f ()! (x ) + f ()! (x ) (x )2 2 T (x) T 2 (x) + f () (x )! (x ) 2 (x )2 + 6 (x ) + f () 2! (x ) (x ) 2 Page 5 of 66
Essex County College, MTH-22 7. Find T 2 (x) and use a calculator to compute the error f (x) T 2 (x) for the given values of a (a is the center) and x. (Round your answers to five decimal places.) f (x) e x, a, x. Solution: A calculator should not be used to do this problem. f (x) e x f (x) e x f (x) e x f () f () f () T 2 (x) + x + x2 2 f (.) T 2 (.).6 8. Find T 2 (x) and use a calculator to compute the error f (x) T 2 (x) for the given values of a and x. (Round your answers to six decimal places.) f (x) cos (x), a, x π 6 Solution: A calculator should not be used to do this problem. f (x) cos (x) f (x) sin (x) f (x) cos (x) f () f () f () T 2 (x) x2 ( 2 π ) ( π ) f T 2. 6 6 Page 54 of 66
Essex County College, MTH-22 Exam Before you come to the exam, I strongly suggest that you review the following listed items.. Convergence or Divergence of a Series: An infinite series nk a n converges to the sum S if its partial sums converge to S, that is lim S N S. N If this limit does not exist, we say the series diverges. 2. Convergence Properties of Series: If then (a) (b) (a n ± b n ) converges to n n ka n converges to k a n. n n a n and n a n ± b n.. Geometric Series: The geometric series ar n a + ar + ar 2 + n is convergent if r <, and its sum is ar n a + ar + ar 2 + n n a r. If r, the geometric series is divergent. 4. p-series Series: The p-series n n p is convergent if p >, and is divergent if p. 5. Divergence Test: If the series n a n b n converge, and if k is a constant, n Page 55 of 66
Essex County College, MTH-22 is convergent, then lim a n. n This may also be stated as, if the nth term a n does not converge to zero, then the series n diverges. a n 6. Integral Test: Suppose f is a continuous, positive, decreasing function on [, ) and a n f (n). (a) If (b) If f (x) dx is convergent, then f (x) dx is divergent, then 7. Comparison Test: Suppose that (a) If (b) If a n is convergent. n a n is divergent. n a n and n b n are series with positive terms. n b n is convergent, and a n b n for all n M, then n b n is divergent, and a n b n for all n M, then n 8. Limit Comparison Test: Suppose that a n and n a n is also convergent. n a n is also divergent. n b n are series with positive terms. If a n lim c where c > is a finite positive number, then either both series converge of both n b n series diverge. You may also see these two cases: a n (a) If lim and n b n (b) a n lim and n b n a n converges, then n b n converges, then n n b n also converges. n a n also converges. 9. Alternating Series Test (Leibniz Test): If the alternating series n ( ) n a n a a 2 + a a 4 + (a n > ) n satisfies (a) a n+ a n for all n; Page 56 of 66
Essex County College, MTH-22 (b) and lim n a n, then the series converges.. The Ratio Test: (a) If lim a n+ n a n L <, then the series a n is absolutely convergent. n (b) If lim a n+ n a n L >, or lim a n+ n a n, then the series a n is divergent. n (c) If lim a n+ n a n, the ratio test fails, and we can draw no conclusion.. The Root Test: n (a) If lim an L <, then the series n (b) If lim n n an L >, or lim n a n is absolutely convergent. n n an, then the series a n is divergent. n (c) If lim an, the root test fails, and we can draw no conclusion. n n Page 57 of 66
Essex County College, MTH-22... Calculate the first four terms of the sequence, starting with n. c n n n! Solution: A calculator should not be used to do this problem. c! c 2 2 2! 9 2 c! 9 2 c 4 4 4! 27 8 2. Calculate the first four terms of the sequence, starting with n. a 2, a n+ 2a 2 n 2 Solution: A calculator should not be used to do this problem. a 2 a 2 2a 2 2 8 2 6 a 2a 2 2 2 7 a 4 2a 2 2 9798. Calculate the first four terms of the sequence, starting with n. c n 4 + 7 + + + n + Page 58 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. c 4 c 2 4 + 7 28 c 4 + 7 + 69 4 c 4 4 + 7 + + 7 82 4. Calculate the first four terms of the sequence, starting with n. c n n-place decimal approximation to 2e Solution: A calculator should not be used to do this problem. I will provide you with an approximate value of 2e (or another constant) if this question appears on an exam. 2e 5.4656657... c 5.4 c 2 5.44 c 5.47 c 4 5.466 5. Find a formula for the n th term of the following sequences (with a starting index of n.) (a) 8, 9 8, 27, Solution: A calculator should not be used to do this problem. The denominators are the third powers of the positive integers starting with n. Also, the sign of the terms is alternating with the sign of the first term being positive. Thus, a n ( )n+ (n + 7) n. (b) 5, 7 5, 9 7, Page 59 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. We see that each numerator is 2 more than the previous and the first numerator is 5. We also see that each denominator is 2 more than the previous and the first denominator is. Thus, a n + 2n + 2n. 6. Suppose that and lim a n 4, n lim b n 8. n Determine the following. lim n a2 n 2a n b n Solution: A calculator should not be used to do this problem. Using the Limit Laws. lim n a2 n 2a n b n lim n a2 n lim 2a nb n n ( ) 2 lim a n 2 lim a n lim b n n n n 4 2 2 4 8 6 64 48 7. Determine the limit as n of the sequence and state if the sequence converges or diverges. a n 24 6 n 2 Solution: A calculator should not be used to do this problem. We have a n f (n), where f (x) 24 6 x 2 lim a n lim f (x) n x lim (24 6x ) x 2 24 24 The sequence converges. Page 6 of 66
Essex County College, MTH-22 8. Determine the limit as n of the sequence and state if the sequence converges or diverges. b n 8n 5n + 8 Solution: A calculator should not be used to do this problem. We have b n f (n), where f (x) 8x 5x + 8 lim b n lim f (x) n x lim x lim x 8 5 The sequence converges. 8x 5x + 8 8 /x 5 + 8/x 9. Determine the limit as n of the sequence and state if the sequence converges or diverges. c n 4 + n 7n2 9n 2 + Solution: A calculator should not be used to do this problem. We have c n f (n), where f (x) 4 + x 7x2 9x 2 + lim n n 4 + x 7x 2 lim x 9x 2 + 4/x 2 + /x 7 lim x 9 + /x 2 7 9 The sequence converges.. Determine the limit as n of the sequence and state if the sequence converges or diverges. d n ( ) 8 n 9 Page 6 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. We have d n f (n), where ) x ( 8 f (x) 9 [ ( ) ] 8 x lim d n lim n x 9 ( ) 9 x lim x 8 The sequence diverges.. Determine the limit as n of the sequence and state if the sequence converges or diverges. z n ( ) 5 n 9 Solution: A calculator should not be used to do this problem. We have z n f (n), where f (x) ( ) 5 x 9 lim z n lim n x The sequence converges. ( ) 5 x 9 2. Determine the limit as n of the sequence and state if the sequence converges or diverges. a n /n Solution: A calculator should not be used to do this problem. We have a n f (n), where f (x) /x lim a n lim n x /x The sequence converges. Page 62 of 66
Essex County College, MTH-22. Determine the limit as n of the sequence and state if the sequence converges or diverges. t n n n + Solution: A calculator should not be used to do this problem. We have t n f (n), where f (x) x x + lim t n lim n x x x + x/x /2 lim x x /x + /x / x lim x + /x The sequence converges. 4. Determine the limit as n of the sequence and state if the sequence converges or diverges. ( ) 7n 6 y n ln 4n Solution: A calculator should not be used to do this problem. We have y n f (n), where ( ) 7x 6 f (x) ln 4x ( ) 7x 6 lim y n lim ln n x 4x ( ) 7x 6 ln lim x 4x ( ) 7 6/x ln lim x 4 /x ( ) 7 ln 4 The sequence converges. 5. Determine the limit as n of the sequence and state if the sequence converges or diverges. s n ln (n) ln ( n 2 + 5 ) Page 6 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. We have s n f (n), where f (x) ln (x) ln ( x 2 + 5 ) lim s [ ( n lim ln (x) ln x 2 + 5 )] n x ( ) x lim ln x x 2 + 5 ( ) x ln lim x x 2 + 5 The limit of the rational function goes towards zero from the right. The sequence diverges. 6. Determine the limit as n of the sequence and state if the sequence converges or diverges. r n arctan ( e 2n) Solution: A calculator should not be used to do this problem. We have r n f (n), where f (x) arctan ( e 2x) lim r n lim arctan ( e 2x) n x ( arctan arctan () The sequence converges. lim x e 2x) 7. Let a n n n + 9. Find the minimal number M such that a n. for n M. Solution: A calculator should not be used to do this problem. Yes, I know there s arithmetic here that may prove too difficult for some students to compete, but it is nonetheless very Page 64 of 66
Essex County College, MTH-22 simple grade-school arithmetic. n > a n. n n + 9 9 n + 9 9 n + 9 9 n + 9 So M 8999. 8999 n 8. Determine the limit as n of the sequence and state if the sequence converges or diverges. d n n + n Solution: A calculator should not be used to do this problem. We have d n f (n), where f (x) x + x lim d ( ) n lim x + x n x ( ) x + x x + + x lim x x + + x lim x x + + x The sequence converges. 9. Determine the limit as n of the sequence and state if the sequence converges or diverges. a n /n Solution: A calculator should not be used to do this problem. We have a n f (n), where f (x) /x lim a n lim n x /x Page 65 of 66
Essex County College, MTH-22 The sequence converges. 2. Determine the limit as n of the sequence and state if the sequence converges or diverges. v n n n! Solution: A calculator should not be used to do this problem. For n >, write v n n n! 2 2 n n Denote by C the following expression C 2 Clearly for n > (you may need to think about this) v n C n. Using the Squeeze Theorem (MTH 2) tells us that lim n Finally, we know lim n v n lim n v n lim v n. n The sequence converges. lim n C n 2. Determine the limit as n of the sequence and state if the sequence converges or diverges. b n 4 n 7 n + 4 Page 66 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. We have b n f (n), where f (x) 4 x 7 x + 4 lim b n lim n x lim x 4 7 The sequence converges. 4 x 7 x + 4 4 7 + 4/ x 22. Determine the limit as n of the sequence and state if the sequence converges or diverges. h n 5 cos (n) n Solution: A calculator should not be used to do this problem. We have h n f (n), where f (x) 5 cos (x) x I m using the Squeeze Theorem (MTH 2). The absolute value function is really not necessary here. 5 x 5 cos (x) 5 x x lim 5 x x lim 5 cos (x) lim 5 x x x x 5 cos (x) lim x x Finally we know lim n n 5 cos (x) lim x x. The sequence converges. 2. Determine the limit as n of the sequence and state if the sequence converges or diverges. f n ( )n n Page 67 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. I m using the Squeeze Theorem (MTH 2). n ( ) n n n lim n n Finally we know ( ) n lim. n n The sequence converges. ( ) n lim n n ( ) n lim n n lim n n 24. Determine the limit as n of the sequence and state if the sequence converges or diverges. e n ln ( n 2 + 6 ) ln ( n 2 ) Solution: A calculator should not be used to do this problem. We have e n f (n), where f (x) ln ( x 2 + 6 ) ln ( x 2 ) lim e [ ( n lim ln x 2 + 6 ) ln ( x 2 )] n x ( x 2 ) lim ln + 6 x x 2 ( ln ln () The sequence converges. lim x + 6/x 2 /x 2 ) 25. Determine the limit as n of the sequence and state if the sequence converges or diverges. z n arctan ( 6n) Page 68 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. We have z n f (n), where f (x) arctan ( 6x) lim z n lim arctan ( 6x) n x ( ) arctan lim [ 6x] x π 2 The sequence converges. 26. Determine the limit as n of the sequence and state if the sequence converges or diverges. b n n 2 n Solution: A calculator should not be used to do this problem. We have b n f (n), where f (x) x 2 x lim b n lim n x H lim x The sequence converges. x 2 x 2 x ln (2) 27. Determine the limit as n of the sequence and state if the sequence converges or diverges. u n n n! Solution: A calculator should not be used to do this problem. For n >, write u n n n! 2 n n Denote by C the following expression C 2 Clearly for n > (you may need to think about this) u n C n. Page 69 of 66
Essex County College, MTH-22 Using the Squeeze Theorem (MTH 2) tells us that lim n Finally, we know lim n u n lim n u n lim u n. n The sequence converges. lim n C n 28. Determine the limit as n of the sequence and state if the sequence converges or diverges. z n 8 4n 2 + 9 4 n Solution: A calculator should not be used to do this problem. We have z n f (n), where f (x) 8 4x 2 + 9 4 x lim n n 8 4 x lim x 2 + 9 4 x /4 x 8 lim x 2/4 x + 9 8 9 The sequence converges. 29. Determine the limit as n of the sequence and state if the sequence converges or diverges. t n ( + ) n n Page 7 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. We have t n f (n), where ) x ( f (x) + x ( y + ) x x ( ln (y) x ln + ) x ln ( + /x) /x y e ln(+/x) /x lim t n lim n x ( + x lim e ln(+/x) /x x H lim e e +/x x The sequence converges. ) x. Determine the limit as n of the sequence and state if the sequence converges or diverges. ( ) g n n n 2 + n Solution: A calculator should not be used to do this problem. We have g n f (n), where ( ) f (x) x x 2 + x ( lim g n lim x x 2 + x n x ( x x 2 + x lim x x lim ( x x 2 + + x) lim ( x + /x 2 + ) ) ) ( ) x 2 + + x x 2 + + x 2 The sequence converges. Page 7 of 66
Essex County College, MTH-22.2.2. Write the given series in summation notation. + 4 + 9 + 6 + Solution: A calculator should not be used to do this problem. The general term a n /n 2, hence, the series is n n 2. 2. Compute the following partial sums for the series (a) S 2 (b) S (c) S 4 ( ) n n n Solution: A calculator should not be used to do this problem. 2 S 2 ( ) n n + 2 2 n Solution: A calculator should not be used to do this problem. S ( ) n n + 2 5 6 n Solution: A calculator should not be used to do this problem. 4 S 4 ( ) n n + 2 + 4 7 2 n. Calculate S, S 4 and S 5 and then find the sum of the following telescoping series. n ( 4 n + 5 4 ) n + 6 Page 72 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. S n ( 4 n + 5 4 ) ( 4 n + 6 6 4 ) ( 4 + 7 7 4 ) ( 4 + 8 8 4 ) 9 4 6 4 9 n ( 4 n + 5 4 ) n + 6 2 9 4 ( 4 S 4 n + 5 4 ) 4 n + 6 6 4 n 4 5 5 ( 4 S 5 n + 5 4 ) 4 n + 6 6 4 n N ( 4 S N n + 5 4 ) 4 n + 6 6 4 N + 6 n 2 4 N + 6 lim N S N 2 4. Write the following sum as a telescoping series and find its sum. n 8 n 2 + n + Page 7 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. 8 8 n 2 + n + (n + 5) (n + 6) A n + 5 + B n + 6 8 A (n + 6) + B (n + 5) n 5 A 8 n 6 B 8 8 8 n 2 + n + n + 5 8 n + 6 8 ( 8 n 2 + n + n + 5 8 ) n + 6 n n N ( 8 S N n + 5 8 ) n + 6 n 8 n 2 + n + n 8 6 8 N + 6 lim N S N 4 5. Evaluate the following sum. (Hint: rewrite the fraction as sum of partial fractions.) n 4n 2 25 Page 74 of 66
Essex County College, MTH-22 Solution: A calculator should not be used to do this problem. n 4n 2 25 (2n + 5) (2n 5) A 2n + 5 + B 2n 5 A (2n 5) + B (2n + 5) n 5/2 A / n 5/2 B / 4n 2 / 25 2n 5 / 2n + 5 4n 2 ( 25 2n 5 ) 2n + 5 n 4n 2 25 n S N N n lim N S N 5 ( 2n 5 ) 2n + 5 ( 5 2N 2N 2N + 2N + 2N + 5 The S N expression is not difficult to get, but I realize that some may need to stop by my office to see how to do it. I do encourage you to get help if you need it. ) 6. Find the sum of the infinite series below. 4 9 + 9 4 + 4 9 + 9 24 +... Solution: A calculator should not be used to do this problem. I m not showing the partial fractions here, but you would be expected to show this decomposition if this question were on Page 75 of 66
Essex County College, MTH-22 an exam! n n (5n ) (5n + 4) (5n ) (5n + 4) 5 n 5n 5n + 4 S N N 5 5n 5n + 4 n ( ) 5 4 5N + 4 lim N S N 2 7. Use the formula for the sum of a geometric series to find the sum or state that the series diverges. 4 7 + 44 7 4 + 45 7 5 +... Solution: A calculator should not be used to do this problem. This is a infinite geometric series with r 4/7 and a 4 /7. n ( ) 4 n+ 4 /7 7 4/7 64 47 8. Use the formula for the sum of a geometric series to find the sum or state that the series diverges. n ( ) 6 n Solution: n ( ) 6 n n ( ) n 6 A calculator should not be used to do this problem. This is a infinite geometric series with r /6. This series diverges because r >. Page 76 of 66
Essex County College, MTH-22 9. Use the formula for the sum of a geometric series to find the sum or state that the series diverges. n2 ( 7 ) n 7 Solution: A calculator should not be used to do this problem. This is a infinite geometric series with r /7 and a 9/7. n2 ( 7 ) n 9/7 7 + /7 9. Use the formula for the sum of a geometric series to find the sum or state that the series diverges. 8 e 9n n Solution: A calculator should not be used to do this problem. This is a infinite geometric series with r e 9 and a 8e 9. n 8 e 9n 8e 9 e 9 8 e 9. Use the formula for the sum of a geometric series to find the sum or state that the series diverges. 7 ( 2) n 5 n n 8 n Solution: A calculator should not be used to do this problem. This is a difference of two infinite geometric series. The first series with r /4 and a 7; the second series with Page 77 of 66
Essex County College, MTH-22 r 5/8 and a n 7 ( 2) n 5 n 8 n n ( 7 n 4) n 7 + /4 5/8 28 5 8 44 5 ( ) 5 n 8 2. Compute the total area of the (infinitely many where x ) triangles in the figure (Figure 4, page 78). Figure 4: Partial graph an infinite sum of areas. Solution: A calculator should not be used to do this problem. A 2bh. This clearly produces a convergent geometric series. S ( 2 2 2 ) + ( 2 2 ) + ( 4 2 4 ) 8 2 ( 2 + 2 4 + 4 8 + 8 6 + 6 +... ) I am using the formula This makes sense too you should be able to visualize that it is one half the area ( ) of the rectangle defined by the limits. However, you still need to do the math by setting up an infinite sum. Page 78 of 66