Exam Information Math 1: Final Exam Review Sheet The final exam will be given on Wednesday, December 1th from 8-1 am. The exam is cumulative and will cover sections 5., 5., 5.4, 5.5, 5., 5.9,.1,.,.4,., 7., 7., 7.4, 7.5, 7.8, 8.1, 8., 1.1, 1., and 1., with most of the material coming from the sections in bold. The exam is closed book, closed notes, and without calculator. The exam is located in the Papadakis Integrated Sciences Building, room 1. Exam Preparation Suggestions To prepare for this exam, I recommend the following: Review your lecture notes, the suggested practice problems, and the appropriate chapter contents of a calculus textbook. Specifically, make sure you have mastered the Expected Skills which are listed at the top of each set of practice problems. Depending on your learning style, it may be a good idea to form an outline of the important topics discussed in each chapter - including one or two sample examples of the types of problems that you expect to be asked. Re-work the suggested practice problems! Try to identify those problems and topics on which you have difficulty and then review the notes or chapter related to that material. 1
Form a study group with your peers. After reviewing the material, make a mock exam and trade with a friend. Try to solve the problems without the aid of a textbook or notes. Again, use this to help you learn which material is still unclear so that you can effectively use any additional study time working on these topics. Make sure that you get a good night s sleep before test day, and eat a hearty breakfast in the morning, especially one rich in protein and fiber. DON T CRAM OR PULL AN ALL-NIGHTER! DON T TEST ON AN EMPTY STOMACH! Exam Taking Tips & Advice Arrive early to the exam location. If you arrive late, then you get less time, not more time. Bring a watch; classrooms and lecture halls in Drexel tend to lack clocks. Cell phones should be turned off and packed away before the exam begins. Use a pencil instead of a pen, and bring more than one with you. Cross out or erase any work that you do not want graded. As soon as you receive your exam, write your name and section number on the front page. Exams without names will result in a loss of 5 points. Quickly skim through all of the problems on the exam before doing any one problem. Then do the problems that you think are easiest first. Pace yourself. Know when you should stop working on a problem and move on to another one. It is not worth leaving a 1-point problem blank because you struggled for ten minutes on a 5-point problem.
There is no partial credit for true/false, always/sometimes/never, and multiple choice questions. You can show as little or as much work as you want, and the work does not have to be very organized. You should try to eliminate incorrect answers and, if necessary, you can guess from the remaining choices. Never leave a true/false, always/sometimes/never, or multiple choice question blank. For all of the other questions you must show all of your work. Be sure that your work is organized and legible and that you do not skip a lot of steps. Answers even correct ones) will not receive a lot of credit without the necessary work to back them up. Partial credit will be awarded based on how much correct work that you show. If you have the time, you should check your answers. Just be sure to manage your time - see Pace yourself above.) It doesn t pay to leave twenty minutes early without checking your answers. A lot of silly mistakes can be avoided by going through your work again. EXPECTED SKILLS: Given a differentiation rule, be able to construct the associated indefinite integration rule. Know how to integrate power functions including polynomials), exponential functions, and trigonometric functions. Know how to simplify a complicated integral to a known form by making an appropriate substitution of variables. Understand sigma notation. Be able to use the basic properties and formulas involving sigma notation. You do not need to memorize the Useful Formulas listed below; if they are needed, they will be provided to you.)
Know how to denote the approximate area under a curve and over an interval as a sum, and be able to find the exact area using a limit of the approximation. Be able to find the net signed area between the graph of a function and the x-axis on an interval using a limit. Be able to evaluate the definite integral of a function over a given interval using geometry. Be familiar with the interpretation of the definite integral of a function over an interval as the net signed area between the graph of the function and the x-axis. Know how to use linearity properties of the definite integral to evaluate scalar multiples, sums, and differences of integrable functions. Be able to use one part of the Fundamental Theorem of Calculus FTC) to evaluate definite integrals via the antiderivative. Know how to use another part of the FTC to compute derivatives of functions defined as integrals. Understand the connections between the definite integral, area, and both parts of the FTC. Be able to evaluate definite integrals using a substitution of variables. Be able to find the area between the graphs of two functions over an interval of interest. Know how to find the area enclosed by two graphs which intersect. 4
Know how to use the method of disks and washers to find the volume of a solid of revolution formed by revolving a region in the xy-plane about the x-axis, the y-axis, or any other horizontal or vertical line. Be able to find the volume of a solid that consists of known crosssectional areas. Be able to find the arc length of a smooth curve in the plane described as a function of x or a function of y. Be able to use integration to compute the work done by a variable force in moving an object along a straight path from x a to x b. Specifically, be able to solve spring problems, lifting problems, and pumping problems. Be able to use integration by parts to evaluate various integrals, including integrands involving products of functions, isolated logarithmic functions, or isolated inverse trigonometric functions. Know antiderivatives for all six elementary trigonometric functions. Be able to evaluate integrals that involve powers of sine, cosine, tangent, and secant by using appropriate trigonometric substitutions. Be able to evaluate integrals that involve particular expressions by making the appropriate trigonometric substitution. Know how to evaluate integrals that involve quadratic expressions by first completing the square and then making the appropriate substitution. Be able to recognize an improper rational function, and perform the necessary long division to turn it into a proper rational function. 5
Know how to write down the partial fraction decomposition for a proper rational function, compute the unknown coefficients in the partial fractions, and integrate each partial fraction. Given an improper integral, which either has an infinite interval of integration or an infinite discontinuity, be able to evaluate it using a limit. Know how to determine if such an integral converges and if so, what it converges to) or diverges. Be able to verify that a given function is a solution to a differential equation. Be able to solve first-order separable equations by separating and integrating. Be able to solve initial-value problems for first-order separable equations. Be able to sketch a parametric curve by eliminating the parameter, and indicate the orientation of the curve. Given a curve and an orientation, know how to find parametric equations that generate the curve. Without eliminating the parameter, be able to find dy dx and d y dx at a given point on a parametric curve. Be able to find the arc length of a smooth curve in the plane described parametrically. Be able to describe points and curves in both polar and rectangular form, and be able to convert between the two coordinate systems.
Know the formulas for the basic shapes in polar coordinates: circles, lines, limaçons, cardioids, rose curves, and spirals. Know how to compute the slope of the tangent line to a polar curve at a given point. Be able to find the arc length of a polar curve. Be able to calculate the area enclosed by a polar curve or curves. OLD EXAMS WITH SOLUTIONS: http://www.math.drexel.edu/classes/math1/15/ go to the Course Material section) http://www.math.drexel.edu/classes/math1/115/final/math1_ Winter_1_Final_Version_1.pdf Solutions: http://www.math.drexel.edu/classes/math1/115/ Final/Math1_Winter_1_Final_Version_1_Solutions.pdf PRACTICE PROBLEMS: For problems 1-5, compute the rectangular coordinates of the points whose polar coordinates are given. 1.. 1, ), ). 5, ) 4., 9 ) 4 5., 11 ) 7
For problems -1, find two pairs of polar coordinates for the point whose rectangular coordinates are given. The first pair should satisfy r and θ. The second pair should satisfy r and θ.. 5, 5) 7., ) 8., ) 9., 1) 1. 4, 4) 11. Consider the point with rectangular coordinates 1, ). a) Find a pair of polar coordinates which satisfy r and θ <. b) Find a pair of polar coordinates which satisfy r and θ <. c) Find a pair of polar coordinates which satisfy r and < θ. d) Find a pair of polar coordinates which satisfy r and < θ. For problems 1-1, identify the curve by transforming the polar equation into rectangular coordinates. 1. r 1 1. r cos θ 14. r sin θ 15. r cos θ sin θ 8
1. r sec θ For problems 17-, express the given equation in polar coordinates. 17. y 18. x 19. x + y 1. x + y + 8y For problems 1-, find an equation in polar coordinates for each of the given graphs. 1. Circle 5 1 7 11 4 5. Circle 5 1 4 7 11 4 5 9
. Cardioid 5 1 4 7 11 4 5 For problems 4-, sketch the curve in polar coordinates. 4. r 5. r cos θ. r sin θ 7. r + sin θ 8. r 1 cos θ 9. r θ, θ. r sin θ 1. r 1 + cos θ). r 4 cos θ. r sin θ For problems 4-, find the slope of the tangent line to the polar curve for the given value of θ. 4. r θ; θ 1
5. + sin θ; θ. r 1 sin θ; θ 7. Consider the circle r cos θ. Find all values of θ in θ < for which the curve has either a horizontal or vertical tangent line. For problems 8-4, find the arc length of the given curves. 8. The entire circle r 4 sin θ. 9. The spiral r e θ for θ. 4. The entire cardioid r 1 + cos θ. [Hint: It may be useful to use symmetry and the identity cos θ 1 1 + cosθ)).] For problems 41-49, find the area of each of the specified regions. 41. The region in the first quadrant within the circle r cos θ. 4. The region enclosed by the cardioid r + sin θ. 4. The region inside the circle r and outside the cardioid r 1 + cos θ. 44. The region inside the circle r but outside the cardioid r + cos θ. 45. The region outside the circle r but inside the cardioid r + cos θ. 4. The region in common between the two circles r sin θ and r cos θ. 47. The region inside the circle r and to the right of the line r sec θ. 48. The region enclosed by the rose r cos θ. 49. The region enclosed by the rose r sin θ. 11
5. Find the area of the shaded region shown below) which is enclosed between the circle r cos θ and the cardioid r + cos θ. 51. Consider the limaçon r + cos θ. a) Compute the area enclosed by the inner loop of the limaçon. b) Compute the area enclosed by the outer and inner loops of the limaçon. Design a general strategy to integrate rational functions where the powers of x in the integrand are arbitrary rational numbers, and not necessarily just integers. Test your strategy on some nontrivial examples. Four bugs are placed at the four corners of a square with side length a. The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach the center of the square along spiral paths. 1
a) Find the polar equation of a bug s path assuming the pole is at the center of the square. Use the fact that the line joining one bug to the next is tangent to the bug s path.) b) Find the distance traveled by a bug by the time it meets the other bugs at the center. 1
SOLUTIONS 1. x r cos θ cos 1 and y r sin θ sin rectangular coordinates are 1 ),. so that the. x r cos θ cos and y r sin θ sin ) the rectangular coordinates are,. so that. x r cos θ 5 cos ) 5 and y r sin θ 5 sin ) so that the rectangular coordinates are 5, ). 4. x r cos θ cos 9 4 and y r sin θ sin 9 4 so that the rectangular coordinates are, ). 5. x r cos θ cos 11 and y r sin θ sin 11 so that the rectangular coordinates are, ).. r x + y 5) + 5) 5 and tan θ y x 5 1. Note 5 that 5, 5) lies in the third quadrant so that we have θ 5 so that 4 the polar coordinates with θ are 5, 5 ). In the case of 4 θ, simply subtract to get 5, ). 4 7. r x + y ) + and tan θ y x 1. Note that, ) lies in the second quadrant so that we have θ so that 4 the polar coordinates with θ are, ). In the case of 4 θ, simply subtract to get, 5 ). 4 14
8. r x + y + and tan θ y x, which is undefined. Note that, ) lies on the positive y-axis so that we have θ so that the polar coordinates with θ are,, ). θ, simply subtract to get 9. r ). In the case of ) + 1) + 1 and tan θ y x 1 1. Note that, 1) lies in the fourth quadrant so that we have θ 11 so that the polar coordinates with θ are, 11 ). In the case of θ, simply subtract to get, ). 1. r 4 ) + 4) 4 + 1 8 and tan θ y x 4 4 1. Note that 4, 4) lies in the third quadrant so that we have θ 7 so that the polar coordinates with θ are 8, 7 ). In the case of θ, simply subtract to get 8, 5 ). 11. Here x 1 and y so that tan θ y x 1. Note that 1, ) lies in the first quadrant. a) r x + y coordinates are, 1 + ). Here θ. Thus the polar ). b) r x + y. Since r is negative, we are interested in the reflection of the point through the origin, which lies in the third quadrant. This can be accomplished by adding to the value of θ from part a). Thus the desired polar coordinates are, 4 ). 15
c) r. For this one, subtract from the answer in part a) to get the polar coordinates, 5 ). d) r. For this one, subtract from the answer in part b) to get the polar coordinates, ). 1. r 1 r 1 x + y 1, which is the unit circle. 1. r cos θ r rθ x + y x x x + 1 + y 1 x 1) + y 1, which is the circle of radius 1 centered at 1, ). 14. r sin θ y, which is the horizontal line with y-intercept. 15. r cos θ sin θ r r cos θ r sin θ x + y x y x x + 9 ) + y + y + 1) 9 4 4 + 1 x ) + y + 1) 1 4, ) 1 which is the circle of radius centered at, 1. 1. r sec θ r cos θ x, which is the vertical line with x-intercept. 17. y r sin θ r csc θ 18. x r cos θ r sec θ 19. x + y 1 r 1 r 1. x + y + 8y x + y 8y r 8r sin θ r 8 sin θ 1. This is a circle of radius centered at the pole, so we have r.. This is a circle of radius 4 centered at, ), so we have r 4 cos θ.. This is a cardioid intersecting the y-axis at, ) and, ) with its farthest point from the pole and unique x-axis at 4, ), so we have r cos θ. 1
4. This is a circle of radius centered at the pole. 5.5 1 1.5 7 11 4 5 5. This is a circle of radius 1 centered at 1, ). 5.5 1 7 11 4 5. This is a circle of radius centered at, ). 5 1 7 11 4 5 17
7. This is a cardioid intersecting the x-axis at, ) and, ) with its farthest point from the pole and unique x-intercept at, ). 5 4 7 11 4 5 8. This is a limaçon whose axis of symmetry is the x-axis with its farthest point from the pole at, ) when θ and the peak of its inner loop at 1, ). 5 1 7 11 4 5 9. This is a spiral 5 5 1 15 7 11 4 5 18
. This is a limaçon whose axis of symmetry is the y-axis with its farthest point from the pole at, 5) when θ and the peak of its inner loop at 1, ). 5 4 7 11 4 5 1. This is a cardioid intersecting the y-axis at, ) and, ) with its farthest point from the pole and unique y-intercept at, 4). 5 1 4 7 11 4 5. This is a rose curve with 4 petals with the coordinates axes going through the petals tips. 5 1 4 7 11 4 5 19
. This is a rose curve with petals with the petal tips at θ 7, θ, and θ 11. 5 1 7 11 4 5 4. 5. dy dr sin θ + r cos θ dθ dx dr θ/ cos θ r sin θ dθ sin θ + θ cos θ cos θ θ sin θ sin + cos cos sin θ/ θ/ 1 + 1 1 dy dr sin θ + r cos θ dθ dx dr θ/ cos θ r sin θ dθ θ/ cos θ sin θ + + sin θ) cos θ cos θ cos θ + sin θ) sin θ 4 sin θ cos θ + cos θ cos θ sin θ sin θ θ/ 4 sin ) cos ) + cos ) cos ) ) sin sin ) 4 1 + 1 4 4 θ/
+ 1 5 1 5. 7. dy dr sin θ + r cos θ dθ dx dr θ cos θ r sin θ dθ θ cos θ sin θ + 1 sin θ) cos θ cos θ cos θ 1 sin θ) sin θ θ cos) sin) + 1 sin)) cos) cos) cos) 1 sin)) sin) 1)) + 1 )1) 1) 1) 1 )) 1 dr dy dx sin θ + r cos θ dθ cos θ r sin θ dr dθ sin θ sin θ + cos θ cos θ sin θ cos θ cos θ sin θ cos θ sin θ sin θ cos θ 1 cos θ sin θ 1 cot θ We get horizontal tangent lines when dy dx, which occurs when θ 4 or θ dy, and we get vertical tangent lines when is undefined, 4 dx which occurs when θ or. 8. One full rotation of the circle goes from θ to θ. So L r + 1 ) dr dθ dθ
4 sin θ) + 4 cos θ) dθ 1 sin θ + 1 cos θ dθ 1 dθ 4 dθ 4 9. L t lim t r + ) dr dθ dθ e θ ) + e θ ) dθ t lim e θ + e θ dθ t t lim e θ dθ t lim [ ] t e θ t [ lim e t )] e t 4. One full rotation of the cardioid goes from θ to θ. So L r + ) dr dθ dθ 1 + cos θ) + sin θ) dθ 1 + cos θ + cos θ + sin θ dθ + cos θ dθ
1 + cos θ dθ ) θ cos dθ ) θ cos dθ θ cos dθ ) 4 θ cos dθ ) ) θ 4 cos dθ [ )] θ 4 sin ) 8 sin 8 sin) 8 41. This is simply 1 / r dθ 1 / / 9 cos θ dθ 9 1 + cosθ) dθ 9 [ 1 θ + 1 ] / 4 sinθ) 9 1 ) 9 8 4. 1 r dθ 1 1 + sin θ) dθ 9 + 18 sin θ + 9 sin θ) dθ
1 9 + 18 sin θ + 9 ) 1 cosθ)) dθ 1 [9θ 18 cos θ + 9 θ 94 ] sinθ) 1 9 18 + 9 ) 1 7 18) 4. Note that 1 cos θ 1 1 + cos θ so that the circle and the cardioid do not intersect. Therefore, the area is, by symmetry, simply 1 dθ 1 1 + cos θ) dθ 9 1 cos θ cos θ) dθ 8 cos θ 1 + cosθ) [ 8θ sin θ 1 θ 1 4 sinθ) ] 8 1 15 44. The circle and the cardioid intersect when + cos θ cos θ 1 cos θ 1 θ or θ 5 [. Note that for θ on, 5 ], cos θ 1 cos θ 1 + cos θ. For this, reason, the area is, by symmetry, ) dθ 1 5/ / dθ 1 5/ / + cos θ) dθ / / 9 4 8 cos θ 4 cos θ) dθ 5 8 cos θ 1 + cosθ))) dθ [5θ 8 sin θ θ sinθ)] / [θ 8 sin θ sinθ)] / 4
8 sin sin ) + 8 + + 9 45. Based on our work on the previous problem, we are interested in [ integrating over the intervals, ] [ ] 5 and,. It is easier, however, [ ] 5 to think of, as [ ], and use symmetry to get the area as 1 / + cos θ) / dθ 1 / / dθ / + cos θ) ) dθ [sinθ) + 8 sin θ θ] / sin + 8 sin + 8 9 4. Here is the graph of the two circles. 5 1 7 11 4 5 They intersection in the first quadrant when sin θ cos θ, or at θ /4. Note that the lower half of the region in common is the area under r sin θ from θ to θ /4 and the upper half of the region in common is the area under r cos θ from θ /4 to θ /. So the total area is 5
1 /4 9 sin θ dθ + 1 /4 9 cos θ dθ 1 9 /4 1 cosθ)) dθ + 1 9 1 + cosθ)) dθ /4 [ 9 4 θ 9 ] /4 [ 9 8 sinθ) + 4 θ + 9 ] 8 sinθ) /4 9 1 9 ) + 9 9 8 4 1 + 9 ) 9 8 4 9 4 47. The graph and the circle intersect when sec θ cos θ 1, which occurs when θ or θ 5. Since we wish to stay to the right of r sec θ, it is easier to look at θ 5 as θ area is, by symmetry, instead, and so the 1 / / dθ 1 / / sec θ dθ / 4 sec θ) dθ [4θ tan θ] / 4 tan 4. 48. The rose has ) 4 equally sized petals, so it suffices to find the area enclosed by one petal and then multiply by 4. 5 1 7 11 4 5
The leftmost petal goes from θ 4 to θ. So, by symmetry, the 4 total area is 4 1 /4 /4 9 cos θ) dθ 4 /4 /4 /4 9 cos θ) dθ 1 + cos4θ) dθ 18 + 18 cos4θ)) dθ [18θ + 9 ] /4 sin4θ) 18 4 9 49. The rose has equally sized petals, so it suffices to find the area enclosed by one petal and then multiply by. 5.5 1 1.5 7 11 4 5 The leftmost petal goes from θ to θ. So the total area is 1 / / 4 sin θ) dθ 7 / 1 cosθ) dθ cosθ)) dθ
[θ 1 ] / sinθ) 5. Just take the area of the semicardioid and subtract the area of the semicircle to get 1 + cos θ) dθ 1 / 4 cos θ dθ which is 1 4 + 8 cos θ + 4 cos θ) dθ 1 + 4 cos θ + cosθ)) dθ / / [ θ + 4 sin θ + 1 sinθ) ] 1 + cosθ)) dθ 1 + cosθ)) dθ [θ + 1 sinθ) ] / 5 Beware! Note that the cardioid and the circle intersect at different values of θ; θ and θ /, respectively.) 51. a) We first need the values of θ for which r + cos θ intersect the pole. This occurs when + cos θ cos θ 1 θ or θ 4. So the desired area is, by symmetry, 1 4/ / ) 1 + cos θ dθ 4/ / / / + 1 cos θ + 1 cos θ ) dθ + 1 cos θ + 1 + cosθ))) dθ 9 + 1 cos θ + cosθ)) dθ [9θ + 1 sin θ + sinθ)] / 8
9 + 1 sin + sin 4 ) 1 + 9 b) This area can be found by integrating from θ to θ, subtracting half of the answer from part a), and finally doubling. So 1 / + cos θ ) dθ [ 9 θ + sin θ + sinθ) ] / 9 + sin + sin 4 + + 15 + 4 Hence, the final answer is + 15 4 9 4 ) + ) +. 9