Honors Calculus II Spring 2002

Transcription

1 Honors Calculus II Spring 2002 Instructors James Graham-Eagle ( Kiwi), OS215 (x2712), jamesgrahameagle@umledu Gilbert rown, E220 (x3166), gilbertbrown@umledu Office Hours TA Text James Stewart, Calculus Concepts and Contexts, 2th edition (rooks/cole, 2001) Prerequisites Calculus I Homework Homework will be handed out each Monday and collected for grading the following week You are encouraged to work on the homework problems in groups and to seek help from tutors etc, but you must write up the solutions in your own words Attendance Attendance is not mandatory but is expected Students showing a lack of commitment to the Honors section will be asked to leave In addition you are responsible for all announcements, homework and material given in class Late homework and missed tests cannot be made up without a compelling prior excuse Course Outline Text Chapter 5 Integrals Chapter 6 Applications of Integral Chapter 7 Differential Equations (maybe) Chapter Infinite Sequences and Series Grading Course grades will be based on homework (40% total), three in-class exams on dates to be announced (10% each), and a comprehensive final (30%) Scale A:0, A:75, :70, C:65, C:60, CD:55, D:50 Philosophy We continue where Calculus I left off e aware that, as for Honors Calculus I, the assigned problems will be somewhat harder than the standard course and we will pursue topics beyond the syllabus Resources There is a tutoring center in Southwick Hall on North Campus and you should become familiar with their services Hours of operation and availability of specific tutors can be obtained there In addition tutors will be available in OS219A from 10:30 3:30 each weekday

2 Homework Honors Calculus II Due February 4 1 (a) ( ˆ $ $ % (b) ( (c) ( È (d) ( sec È 1Î% (e) ( sin (f) ( 2 (a) Find the derivative of ln and use the result to help you find ln (b) Find / by looking at the derivative of E/ F/ cos cos sin and choosing the right E and F 3 Use Maple to help you find antiderivatives of the following functions and check your answers (without Maple) by differentiation (a) / (b) È (c) sin (d) tan 4 According to Torricellis law as liquid drains through a hole in the bottom of a cylindrical container, the depth 2 at time > satisfies the differential equation 2 + œ È 12 > E Here + is the area of the hole, E is the cross-sectional area of the container and 1 is the acceleration due to gravity (a) Use separation of variables to find a general solution of the equation in (a) (b) Find the particular solution if the cylinder was initially filled to a depth L (c) Use the result in (b) to find the time taken for the liquid to completely drain from the container

3 Homework Honors Calculus II Due February 11 1 Evaluate by substitution (a) (c) $ ( / (b) ( È ( È $ (d) ( / Hint: Let œ sin? / Hint: Multiply by / 2 Evaluate by parts (a) (c) 1Î ( cos (b) ( aln b ( / (d) ( tan 3 Define O œ ( Ð Ñ / for œ ß ß á (a) Find O by evaluating the integral directly (b) Use integration by parts to prove the reduction formula (c) U se (a,b) to find OßOßO and O Leave the answer in terms of $ % / onus: Prove that O for Hint: / $ for ŸŸ O œ Ð ÑO for œ ß ß ß $ß á

4 Homework Honors Calculus II Due February 25 1 The function 0ÐÑ œ È % describes a semicircle on the interval Òß %Ó (a) Write down, using D notation, the upper Riemann sum for 0 on the interval Òß Ó corresponding to equal subintervals (b) What values of will guarantee that the sum found in (a) is within of 0ÐÑ? (c) Find the limit of the sum in (a) as Ä $ 2 (a) Write the lower Riemann sum P for / with equal subintervals (b) Use the geometric formula (see Q4 below what is < in this case?) to simplify the formula for P and hence find lim P Ä (c) Verify the result in (b) using the Fundamental Theorem of Calculus + : 3 In class we found ( œ Ð+ Ñ using Riemann sums but this clearly fails for : œ How : : does the analysis proceed in this case? < 4 (a) Prove the geometric formula < < á < œ, <Á, < Hint: Write W for the sum and simplify the quantity W <W; lots of stuff will cancel Now solve for W Suppose you borrow an amount E from the bank at an annual interest rate 3 for a term of months The bank calculates the monthly payment T like this After one month the interest accrued will be E3Î and you will make a payment T so the amount you now owe is E E3Î T œe< T if we write <œ 3Î After the second month the interest is calculated on this new balance so now you will owe ae< T b< T œe< Ð <ÑT Continuing like this its easy to see that if E5 is the amount owing after months then E5 œ E< Ð < < á < ÑT (b) Use the geometric formula to simplify the expression for E 5 Ð <Ñ< (c) The loan is paid when 5œso E œ Show that this gives the formula Tœ E < (d) Find the monthly payment on a 30 year mortgage with an annual interest rate of *% on $100,000 (e) How much do you end up paying in total to the bank by the time the loan is repaid?

5 Homework Honors Calculus II Due March 4 1 Find (a) (c) ( (b) ( Ð ÑÐ Ñ Ð Ñ Ð Ñ ( (d) ( Ð ÑÐ Ñ Ð ÑÐ Ñ (e) ( È a b 2 Sketch the upper Riemann sum rectangles for the following functions and partitions and calculate the value of the sum in each case c (a) 0ÐÑœ ß À Î Î (b) 1ÐÑœ ß c À Î Î (c) 2ÐÑœcosß c À 1Î% 1Î Do all computations in this question to at least decimal places (a) Compute the upper, lower and midpoint Riemann sums for Î with 10 equal subintervals (b) Find the corresponding trapezoid and Simpson approximations for the integral and partition in (a) (c) Compute the errors in the approximations in (a) and (b)

6 4 Prove the formula 3 œ Ð ÑÐ Ñ by following these hints (a) Show that 3œ $ $ $ cð3 Ñ 3 d œð Ñ This is a telescoping sum write it out and youll see that 3œ almost everything cancels $ (b) Expand the Ð3 Ñ to rewrite the sum in (a) Rearrange the result to find 3 You will need to remember what 3 and are these were done in class 3œ 3œ 3œ w 5 If?œÎ œ then integration by parts says ( œ ( or œ Find the mistake 6 (a) Write down the lower Riemann sum corresponding to equal subintervals for the integral È It is not possible to simplify the result in (a) enough to find its limit as Ä However, we can calculate the integral by Riemann sums if we choose them more carefully Let be a positive integer and let < œ Î Consider the partition of ÒßÓ given by < < á < < œ Note that these subintervals are not of equal length (b) Sketch a diagram showing the function CœÈ and the lower Riemann sum for the given partition in the case œ& (c) Find the lower Riemann sum for general and write it in D notation (d) Use the geometric formula of H2Q2 to simplify the result in (d) and hence find its limit as Ä (e) Check your result by antidifferentiating the function È 7 Show that ( œ ˆ + for :Á and + by using Riemann sums as follows : + : : $ Î (a) For a positive integer, consider the partition á 2 œ+ where 2œ+ Sketch a graph of the left Riemann sum P with + œ, : œ Î and œ & (b) Show that P œð2 Ñ Ð: Ñ Ð: Ñ Ð ÑÐ: Ñ 2 2 á 2 (c) Apply the geometric formula to the sum in the square brackets (see Q below what is < in this case?) to 2 show that P œ ˆ : + 2: (d) As Ä what happens to 2? What happens to P? (e) What changes in the above argument if :œ? (a) Sketch and find the area of the region bounded by the curves CœÈ, Cœ and the axis $ (b) Sketch C œ and find the area of the bounded region between the curve and the axis

7 9 Evaluate tan by following these hints (a) Sketch the graph of Cœtan and shade the region corresponding to tan 1Î% (b) Sketch the graph of Cœtan, shade the region corresponding to tanand evaluate this integral (c) Use (a) and (b) to evaluate tan Hint: The combined area in (a) and (b) is a rectangle D (d) Generalize the above procedure to find tan where Dis a fixed positive number

8 Homework Honors Calculus II Due March 11 1 (a) Sketch the curves C œ and C œ (b) Find the area of the region between the curves by integrating with respect to (c) Redo (b), this time integrating with respect to C 2 Find the volume obtained by rotating the region Cœcos, ŸŸ1Î about (a) the axis; (b) the C axis ( Hint: substitute C œ cos) 3 (a) Show that the length of the curve Cœ ߟŸ is È (b) Evaluate the integral in (a) using the substitution œ sinh? 4 (a) Use slicing to prove that the volume of a circular cone of radius < and height 2 is 1< 2 (b) Use slicing to prove that the surface area of a circular cone of radius < and height 2 is 1<6 where 6 is the slant height (ie the distance from the apex of the cone to a point on the circumference of its base) Hint: Generate the cone by rotating the line Cœ<Î2, ŸŸ2about the axis $

9 Homework Honors Calculus II Due April 1 1 The force required to extend a given spring 5 cm from equilibrium is 20 Newtons Find the work done in extending the spring 10 cm from equilibrium 2 A 10 foot chain weighing 40 pounds is lying on the floor Find the work done in lifting it by one end so that the other end just touches the ground Hint: What is the work required to lift a segment of length C to a height C? 3 A tank in the shape of a circular cylinder of radius 2 ft and length 6 ft is lying horizontally and is full of gasoline Find the work done in pumping the gas out through a hole at the top of the tank (Gasoline weighs 42 pounds per cubic foot) Hint: Its easiest to put the origin at the center of the tank If youre really clever youll be able to get the answer without doing any integration otherwise youll need to make the trig substitution Cœ sin? 4 A cylinder and piston with cross-sectional area 2 square inches contains 32 cubic inches of gas under a pressure of 40 pounds per square inch If the pressure T and volume Z of the gas are related by the adiabatic formula Þ% TZ œ G (a constant), how much work is done by the piston in compressing the gas to 4 cubic inches? Hint: Let be the distance of the piston from the end of the cylinder The initial conditions tell you G and then the adiabatic formula tells you the pressure which in turn gives you the force in terms of

10 Homework Honors Calculus II Due April 1 At the bottom of a 30 ft dam is a trapezoidal gate with base 6 ft, top 2 ft and height 4 ft Find the force on the $ gate due to the water pressure Use 62 lbs/ft for the weight density of the water 30 ft 2 ft 4 ft 6 ft 2 (a) Sketch the region ŸCŸ (b) Find the area of the region in (a) (c) Find the coordinates of the centroid of the region in (a) (d) Find the volume obtained by rotating the region in (a) about the axis (e) Verify the theorem of Pappus for this example È $ 3 The density 3 of a sphere of radius 2 is given by the formula 3 œ < where < is the distance from the center Find the mass of the sphere Hint: Slice the disk into concentric circles

11 Homework Honors Calculus II Due April 12 1 Use telescoping series to find the sum, if it exists, of each of the following series (a) ln 5 (b) 5 %5 5œ 5œ 2 Determine whether each of the following geometric series converge and, if so, find the sum (a) (c) 5 5 *Î (b) a b 5œ 5œ Ð *ÎÑ (d) $ Î% 5œ 5œ& 3 Consider the series 5œ 5 (a) Use the integral test to prove the series converges (b) Estimate the sum of the series by explicitly finding W and using the formula given in class to improve the result Compare your answer with the actual value of

12 Homework Honors Calculus II Due April 22 Ð Ñ 1 Consider the series 5 5œ 5 (a) Use the alternating series test to prove the series converges (b) Estimate the sum of the series by explicitly finding W and using the formula given in class to improve the result Compare your answer with the actual value of Use the ratio test to decide whether or not the following series converge (a) (c) 5 5 (b) 5 5 5œ 5œ 5 Ð5xÑ (d) Ð5Ñx 5x 5œ 5œ 3 Test for convergence 5 È Ð Ñ 5 (a) (b) (c) 5/ È5 5 5œ 5œ 5œ 5 Î $5 Ð Ñ 5 Ð Ñ ln5 (d) (e) (f) ln5 $ 5 5 5œ 5œ 5œ 5

13 Homework Honors Calculus II Due April 29 1 Substituting for in the Taylor approximation for the exponential function and integrating gives erfðñ œ È $ & ( á Ð Ñ 1 V ÐÑ $ x & x ( $x Ð Ñx ÎÈ1 $ where kv ÐÑk Ÿ kk Ð $ÑÐ Ñx (a) Estimate erf ÐÑ using the above polynomial with œ & and verify that this estimate is within kv ÐÑk Maples value Þ)%((** (b) Redo (a) with œ of 2 Find the 5th order MacLaurin polynomial for tan by (a) using MacLaurins formula; (b) dividing the 5th order MacLaurin polynomials for sin and cos; $ & w (c) substituting tanœ+ + $ + & áinto the equation tan œ tan and equating coefficients of like powers of to solve successively for the + s 3 We wish to evaluate lnð Ñ but the integral is too hard to do directly (a) Find the MacLaurin polynomial TÐÑ for lnð Ñto order (b) Show that the remainder satisfies kvðñÿ k for (c) Use the results in (a) and (b) to show that ( with error no greater than Ð Ñ Ð Ñ lnð Ñ œ á $ (d) Calculate the sum to œ in (c) terms and show that this does indeed differ from the exact answer of 1 by less than (e) y how much is the answer in (d) improved if you add half of the 101st term?

14 Homework Honors Calculus II Due May 1 Find the center and radius of convergence of the following power series (a) (c) Ð5 Ñ 5 Ð Ñ Ð $Ñ (b) 5Ð5 Ñ ) 5 5œ 5œ 5 5x (d) á % 5œ $5 % 2 (a) Write down a power series centered at the origin with radius of convergence 3 (b) Write down a power series centered at 2 with radius of convergence (c) Find a power series whose derivative is 5œ (d) Find a power series centered at the origin with leading coefficient whose derivative has each coefficient multiplied by 3 Consider the function 0 defined by the power series 0ÐÑ œ 5 Ð5 Ñ Þ (a) Find the radius of convergence of the series (b) Obtain the series for 0ÐÑ and find its sum using the geometric formula (c) Differentiate the result in (b) to find a simple formula for 0ÐÑ 5œ 4 Find the first three nonzero terms and the radius of convergence of the MacLaurin series for each of the following functions (a) 0ÐÑ œ cos Hint: cos œ use the binomial È (b) 1ÐÑ œ Hint: Start by expanding the log then divide lnð Ñ sin> (c) 2ÐÑ œ ( > > Hint: Start with the sine series

15 Name: Exam Honors Calculus II Spring The differential equation which models the decay of a radioactive material is U > œ 5U where U is the amount of the material present at time > and 5 is a positive constant (a) (4 pts) Use separation of variables to find the solution assuming there is an initial amount U (b) (2 pts) Find the time taken for half of the material to decay 2 (a) (2 pts) Find ( ˆ cos sec (b) (3 pts) Find ( sin )) (c) (3 pts) Find ( È% $ % 3 (a) (2 pts) Find ( ln (b) (3 pts) Find ( (c) (3 pts) Find ( È$ 4 (3 pts) Prove the reduction formula ( / œ / ( /

16 5 (2 pts each) Find 0ÐÑ if $ (a) 0ÐÑ œ (b) 0ÐÑ œ sinð1îñ (c) 0ÐÑ œ È$ (d) 0ÐÑ œ / 10 y (e) 0ÐÑ œ x

17 6 (a) (3 pts) Sketch the region in the first quadrant bounded by the C axis, C œ cos and C œ sin (b) (4 pts) Find the area of the region in (a) (c) (3 pts) Write down the definite integral which gives the volume generated by rotating the region in (a) about the axis onus: Evaluate the integral in (c)

18 Name: Exam Honors Calculus II Spring Evaluate the following integrals (a) (4 pts) ( Hint: Let œ tan? a b $Î (b) (4 pts) ( 2 The graph shown below is of the function 0ÐÑ œ $ (a) (3 pts) On the graph sketch the right Riemann sum on the interval subintervals Òß Ó corresponding to 4 equal y 2 1 x (b) (3 pts) Use D notation to write out the sum in (a) be as explicit as possible (c) (3 pts) If V is the right Riemann sum corresponding to equal subintervals of ÒßÓ, find lim V Ä $Î 3 Consider the curve GÀCœ ߟŸ% Write down the integral which gives (a) (3 pts) the length of G; (b) (3 pts) the volume of the bowl generated by rotating G about the C axis

19 (c) (2 pts) Evaluate one of the above integrals

20 Name: Exam Honors Calculus II Spring (a) (2 pts) Show using partial fractions that œ (b) (3 pts) Use the integral test to show that œ converges 2 (c) (3 pts) Given œ œ Þ%*%, estimate the sum using the integral test formula (d) ( pts) Use telescoping series to find the sum of the series in (b)

21 2 Use an appropriate test to decide whether the following series converge or diverge (a) (3 pts) 5œ Ð Ñ (b) (3 pts) È 5 5œ (c) (3 pts) 5 5œ Ð Ñ Ð5 Ñx

22 3 The fourth order Taylor polynomial T% for tancentered at 1Î% is 1 1 ) 1 $ 1 % T% ÐÑœ Š Š Š Š % % $ % $ % (a) (2 pts) Use T % to find tan www Ð1Î%Ñ without differentiating tan (b) (2 pts) Find the second order Taylor polynomial for tan centered at 1Î% (c) (2 pts) Find the third order Taylor polynomial for sec centered at 1Î%

23 Name: Final Exam Honors Calculus II Spring 2002 Answer all eight questions You must show all work to receive full credit 1 (a) (4 pts) Write, in D notation, the right Riemann sum V for 0ÐÑ œ on the interval Òß %Ó using equal subintervals (b) (3 pts) Prove that» ( È ) V» for % È (c) (3 pts) Verify by calculating V% and the integral that the inequality in (b) holds in the case œ % 2 (a) (3 pts) Evaluate by parts ( sin (b) (3 (c) (4 pts) Evaluate by substitution ( sin Ð Ñ pts) Evaluate by partial fractions ( Ð ÑÐ Ñ 3 Evaluate the following definite integrals (a) (3 pts) ( 1Î sin $ (b) (3 pts) ( È * (c) (4 pts) ( / 4 Consider the region bounded by the axis, the line œ and the parabola C œ È (a) (2 pts) Find the area of the region (b) (5 pts) Find the centroid of the region (c) (3 pts) Find the volume generated by rotating the region about the axis

24 $ 5 Suppose the container shown (all dimensions are in feet) is filled with a liquid of weight density 50 lbs/ft (a) (5 pts) Find the work required to pump the liquid out over the top of the container (b) (5 pts) Find the force due to pressure on the end of the container y x y = x sin % sin 6 We wish to evaluate ( to within We cannot integrate so we proceed as follows sin (a) (4 pts) Use the MacLaurin series for sin to prove œ á $ $x & &x ( (x (b) (3 pts) Use an appropriate test to show that the series obtained in (a) converges absolutely (c) (3 pts) How many terms of the series in (a) will guarantee the desired accuracy? Justify your answer ( 7 Find the center and radius of convergence of the following power series (if you can find the sum of the series you will get extra credit) (a) (3 pts) Ð Ñ 5 5œ (b) (3 pts) 5œ Ð Ñ 5x (c) (3 pts) 5 5œ (d) (1 pt each) Find the sum of the above series Find the first three nonzero terms of the MacLaurin series for the following functions and find the radius of convergence (a) (3 pts) / (b) (3 pts) È$ % (c) (4 pts) sin

25 9 (a) (4 pts) Prove that lim for all Apply lhopitals rule to lim ln ln and use the Ä a b œ Hint: Ä Î result to answer the question given (b) (5 pts) Use integration by parts to prove the reduction formula aln b œ aln b, (c) (3 pts) Use induction to prove a b ln œ Ð Ñ x for œ ß ß 10 (a) (3 pts) Find the radius of convergence of the power series 0ÐÑ œ % œ (b) (3 pts) Find the MacLaurin series for c0ðñd and give its radius of convergence (c) (3 pts) The series in (c) is geometric find its sum (d) (3 pts) Integrate the result in (b) to find 0ÐÑ in closed form (e) (3 pts) Show that for œ Þ the first terms of the series in (a) approximates 0ÐÞÑ within &Þ 11 It is known that if a rope of uniform density 3 is suspended from its ends then it satisfies the equation ww C œ E É Cw where E œ 31ÎX (a) Show that CÐÑ œ E cosh ÐEÑ satisfies the above equation This curve is called a catenary (b) If a rope of length 20m is suspended by posts of equal height 16m apart, how far down does the bottom of the rope hang? % 12 (a) Show that, for, ( Ð D Ñ Dœ Hint: Substitute Dœ cos? and then $ & ( use the reduction Formula?? on p??? of the text as we did in class when deriving Walliss product (b) Now show, by making the substitution œ ÈD in the integral, that ( Œ œè % $ & ( È % % 1 (c) Recall Walliss product Ä as Ä Take the $ $ & & ( square root of both sides and compare the result with (b) as Ä to obtain the final result (see 2(f) also) ( / œ È1 13 The volume obtained by rotating the graph of Cœ,, about the -axis is, according to the disk method, given by

26 1( Evaluate this improper integral ( Hint: Make a suitable trig substitution immediately) 14 Let Wbe the surface obtained by rotating the graph of the function Cœ ß ŸŸ about the line Cœ (a) Find the volume enclosed by W (b) Write down an integral which gives the surface area of W (but dont evaluate it) 15 A man jumps out of a stationary hot air balloon If his downward velocity then Newtons Second shows that > œ1 5@ where 1 is the acceleration due to gravity and 5 is the wind resistance coefficient (both constant) (a) Use separation of variables to solve (b) Find the limiting velocity of the man; ie lim 16 (a) ( (b) ( tan (c) ( % È 17 Find the following indefinite integrals (a) ( sin (b) ( sin È (c) ( (d) ( Ð Ñ 1 Find the area bounded by the two parabolas Cœ and Cœ 19 (a) Solve for À ln ln Ð Ñ œ (b) Simplify À lnðñ ln lnðþ&ñ (c) sin / (d) ( / 20 ( 21 ( % $ sin cos

27 / 22 ( 23 ( È / 24 ( ln 25 ( È % 1 26 ( sin 27 ( Ð% Ñ $Î 2 (a) ( ln (b) ( (c) ( È $ 29 Prove the absolute convergence, conditional convergence or divergence of the following series 5 5 cos5 Ð Ñ Ð Ñ ln5 (a) (b) (c) 5 È $Î œ 5œ 5œ GRAPHDAT 30 The manufacturer of a boat needs to find the area of a cross-section of the hull A coordinate system is superimposed on the prototype (as shown on the right) Use an appropriate method to approximate the area Hint: Write down the integral you need to evaluate and use one of the numerical methods covered in class You will have to read the function values off the graph as best you can Compare the integrals ( œ sin and ( œ lnš È Its curious that such È È similar integrals have such different results Maybe lnš È is the inverse of the simple function (a) Sketch the graph of lnš È and give its domain and range (b) Write =ÐÑ for the inverse Sketch the graph of =ÐÑ and give its domain and range

28 w (c) Show (using the derivative of the inverse) that =ÐÑœ È= ÐÑ w w (d) Define -ÐÑ œ = ÐÑ Sketch the graph of -ÐÑ Show that - ÐÑ = ÐÑ œ and - ÐÑ œ =ÐÑ (e) Prove the identities =ÐÑ œ =ÐÑ-ÐÑ and -ÐÑ œ - ÐÑ œ = ÐÑ C 32 (a) Use integration to prove that the area of the ellipse equals +, œ 1 +, (b) Show that the perimeter of the ellipse is given by Tœ% ( È+ cos?, sin??þ elliptic integral and cannot be done in terms of the functions we have studied so far (c) Calculate T in the cases + œ, and, œ 1Î This is called an +, (d) Comment on the approximate formula given by Larson T 1Ê How did he think of it? Is it always pretty accurate or is it better for some values of + and, than for others? 33 Consider the quarter disk of radius 2, ie W œeðßcñà ßC ß C Ÿ% f (a) Find the coordinates of the centroid (b) Find the volume obtained by rotating W about the line Cœ (c) Find the surface area obtained by rotating W about the -axis 34 Suppose 0 is strictly increasing (so in particular 0 is one-to-one) and 0ÐÑ œ +, (a) Prove Youngs inequality +, Ÿ ( 0ÐÑ ( 0 ÐCÑ Cß +ß, Hint: Draw the graph of 0 and interpret each term in the inequality as an area (b) What condition on + and, gives equality in (a)? (c) Write out Youngs inequality in the case 0ÐÑ œ (d) Write out Youngs inequality for another function of your choice R 35 (a) The trapezoid rule says 0ÐÑ c0ðñ 0Ð Ñd Using œ RßR ßR in this formula and adding gives the approximation R formula from class 0ÐÑ 0ÐÑ 0ÐRÑ 0ÐÑ œ œ +, R 0ÐÑ 0ÐRÑ 0ÐÑ Use this to derive the œr (b) An improved trapezoid rule says 0ÐÑ c0ð+ñ 0Ð,Ñd c0 Ð+Ñ 0 Ð,Ñd Compare this and the original trapezoid rule with the exact answer for Î w w

29 (c) Follow the steps in (a) using the improved trapezoid rule to obtain a better approximation for the sum of the series (d) Apply this result to the series $ with R œ and compare the result with H7Q3 œ 36 Evaluate the following limits / (a) (2 pts) lim (b) (3 pts) lim clnð Ñ lnð Ñd Ä / Ä 37 Evaluate the following improper integrals (a) (5 pts) ( (b) (5 pts) $Î ( Ð% Ñ Ð Ñ Ð Ñ 3 Þ (a) (5 pts) Find the centroid of the region ŸCŸ (b) (5 pts) Find the volume obtained by rotating the above region about the -axis 39 Consider the series + and suppose + and that + Î+ and decreasing œ (a) Use the ratio test to show that this series converges You will have to deduce that given assumptions + (b) Show that the remainder after adding terms of the series is less than + Î+ Hint: Look at how we estimated the remainder for the ratio test in class and show that lim Ä + + 5œ + + from the works (c) Show that, for the series, the remainder after truncating the series is less than the last included x œ term Hint: Use (b) after verifying the appropriate assumptions hold in this case 5 (d) How many terms of the series in (c) will guarantee a remainder less than 10? 40 (a) (5 pts) Find the centroid of the sector of the unit disk shown Hint: ecause of symmetry you need only do one integral (b) (2 pts) Where does the centroid go as Ä 1Î? (c) (3 pts) Use the theorem of Pappus to find the volume obtained when the shown region is rotated about the C axis (d) (2 pts) Show that œ 1Î gives the correct answer in (c)

30 C 41 Use any means to find antiderivatives of the following functions of (Do not forget the G) È $ $ (a) Î, Á (b) Î, Á (c) sin (d) / (e) / (f) sinh 42 (a) Find the upper and lower Riemann sums corresponding to equal subintervals for the integral $ Þ (b) Show that both sums converge to the same limit as Ä and find that limit ( You will need the formulas on page 250 of the text) (c) Evaluate the integral by antidifferentiating to check your answer in (b) 43 For a sequence of numbers? œ c? ß? ß? $ ß? % ßád we define new sequences W? and H? by W? œ cß? ß?? ß??? ßád and H? œ c?? ß?? ß?? ßád $ $ % $ For example, W cß %ß *ß ß á d œ cß ß &ß %ß $ß á d and H cß %ß *ß ß á d œ c$ß &ß (ß á d (a) If H? œ cßßßßád, what can you say about the sequence?? (b) Find all sequences? which satisfy H? œ cß $ß &ß (ß á d (c) If H? what relation exists between the sequences? (d) Find all sequences which satisfy H? œ? (e) Show that if each number in the sequence H? is positive then the sequence? is increasing (f) Let? œ c?ß?ß?ß?ßá $ % d Find WÐH? Ñ and HÐW? Ñ 44 A 15 ft 30 ft swimming pool is 3 ft deep at one end and 5 ft deep at the other When full of water (weight density 624 lbs per cubic ft) (a) (6 pts) What is the force due to pressure on each of the two ends of the pool? (b) (4 pts) What work is required to empty the top 3 ft of water through a drain level with the top of the pool?

31 (c) Pressure? 45 Let 0 be a continuous function on Decide whether the following statements are true or false These are not obvious think carefully e sure to justify your claim in each case (a) If 0ÐÑ Ä as Ä then 0ÐÑ converges (b) If 0ÐÑ converges then 0ÐÑ Ä as Ä (c) If 0 is decreasing and 0ÐÑ converges then 0ÐÑ Ä as Ä (d) If k0ðñ k converges, then 0ÐÑ converges (e) If 0ÐÑ converges, then k0ðñ k converges 46 A ball is thrown straight up and attains an initial height of 2 % inches After each bounce, it rises to 75% of the height of the previous bounce so the sequence of bounces (in inches) is %)ß $ß (ß )Î%ß á (a) Find the total distance traveled by the ball after it has bounced 10 times (b) How many bounces does it take for the ball to travel a total of 190 inches? (c) Find the total distance traveled by the ball as the number of bounces tends to infinity 47 Decide which of the following sequences are geometric, harmonic, increasing, decreasing, bounded above, bounded below, alternating and convergent (a) ß Þß Þß Þß Þß á (b) š (c) œ œ e+ f defined inductively by + œß+ œ (d) ß $ ß ß * ß á + 4 For each of the following sequences, find R such that k+ k Þ for all R Prove your choice works (a) + œ (b) + œ (c) + œ Consider the sequence + œ ß œ ß ß $á È (a) Show that the sequence Ö+ is decreasing for $ ( Hint: Look at the derivative)

32 (b) Find two different upper bounds and two different lower bounds for the sequence Ö+ (c) Show that lim + œ Ä (d) Given &, find a positive integer R such that k+ k & for R Hint: Rearrange as Ð & Ñ and use the binomial theorem to show that Ð & Ñ Ð Ñ& 50 (a) Let 0ÐÑ be decreasing for and define, œ 0ÐÑ 0ÐÑ â 0ÐÑ ( 0ÐÑ y y = f(x) What area on the graph does, represent? (b) Use the picture to explain why e, f is increasing (c) Use the picture to explain why e above So the sequence e f converges,, f is bounded n n + 1 x (d) What is, if 0ÐÑ œ Î and what is lim, in this case? Ä Ð Ñ 51 (a) For which real values of does the series converge? œ Hint: egin with the ratio test this will answer the question for all but two values of Do these last two values separately by some other means (b) Denote the sum of the series (where it converges) by 0ÐÑ y considering 0Ð Ñ and using the geometric formula, find 0ÐÑ in closed form 52 Test whether the following series converge absolutely, conditionally or not at all (a) (c) Ð Ñ Ð Ñ (b) x œ œ sin Ð Ñ (d) œ œ 53 (3 pts each) Write down sequences satisfying each of the following conditions and state in each case whether the given conditions always guarantee convergence or divergence of the sequence (a) ounded; (b) Not bounded; (c) Strictly increasing and bounded;

33 (d) Geometric and alternating with ratio greater than ; (e) Harmonic $ 54 (5 pts) Consider the geometric series Find the sum of the series and the sum of the first 100 Ð %Ñ œ st terms and verify that they differ by less than the 101 term 55 The graph of CœÎß ŸŸP is rotated about the axis to produce a funnel (a) Sketch a diagram of the funnel and find its volume Z (b) Show that the surface area Wœ 1( È % $ (c) Evaluate the integral in (b) using the substitution œ tan? (d) Evaluate the integral in (b) using the substitution œ sinh? (e) What happens to Z and W as P Ä? P 56 (a) Pick a positive function 0 defined on the interval [0ß2] and sketch its graph there (b) Sketch the graph of C œ 0ÐÑ on the interval Òß Ó (c) What is the relation between the areas under these graphs on their domains? 57 Heres a problem from Vol 32 No 1 (January 2001) of The College Mathematics Journal For a positive integer greater than 2, define L by Prove that the sequence L L œ â Þ $ $Ð Ñ is convergent and determine its limit Find the limit of L by showing that this is the upper Riemann sum for 0ÐÑ for the right 0 $ 5 (a) Write â in D notation $ (b) The expression in (a) is a Riemann sum find the integral from which it is derived (this means finding the function and the interval in question) (c) What value of guarantees that the sum in (a) is within $ of the value of the integral?

34 59 Use Maple throughout this question (a) Sketch the upper, lower and midpoint Riemann sums for ( $ using 10 equal subintervals (b) Evaluate the sums in (a) and the trapezoid approximation also with 10 equal subintervals (c) Calculate the value of the integral and so find the errors in the estimates in (b) (d) Find Simpsons approximation to the integral tell Maple to use 10 subintervals (e) Redo (d) with 20 subintervals (f) Call the answer in (d) W and the answer in (e) W Calculate V œ & a W W b Find the errors in W, W and V 60 The area E of a circle of unit radius is given by E œ % È (a) Use Maple to find an antiderivative of È (b) Check the answer in (b) by differentiation (c) Show that Eœ1 61 The worlds sneakiest substitution is due to Weierstrass and deals with integrals of rational combinations of the trig functions Consider the problem ( $ sin % cos We make the substitution?œtanaî b? (a) Show (just use a triangle) that cosaî b œ and sinaî b œ È? È??? (b) Use the double angle formulas for sin and cos to show that cosœ and sinœ?? (c) Show that œ?? (d) Verify that the above integral is transformed to (e) Return to the variable to obtain the final result (?? $? and use partial fractions to evaluate it $ 62 Use induction to prove that is divisible by $ for every 63 (a) Use the trapezoid rule with subintervals of unit length to show ln ln5 ln Hint: Do not use the formula in the book just draw a picture (b) Evaluate the integral in (a) and deduce ln5 ln ln 5œ 5œ

35 (c) Take exponentials of the result in (b) to obtain x / È / (d) Stirlings formula gives the better result x È / Calculate x È 1 1 / for œ,, $ á (e) How about getting the È1 by using Wallis? 64 The centroid of the region e œ e+ÿÿ,, 0 ŸCŸ0ÐÑf is the point ÐßCÑwhere Here E is the area of e,, œ 0ÐÑ E (, Cœ ( 0ÐÑ + E + (a) Find the centroid of the half disk C Ÿ, C (b) Show that the volume obtained by rotating the region V about the axis is equal to the product of the area of e and the distance moved by the centroid during the rotation w 0Ð 2Ñ 0ÐÑ 65 (a) Use Taylors formula to derive the approximation to the derivative 0ÐÑœ O Ð2Ñ 2 w 0Ð 2Ñ 0Ð 2Ñ (b) Derive the central difference formula (see 141H4Q2) 0ÐÑœ O Ð2Ñ 2 w %0Ð 2Ñ 0Ð 2Ñ $0ÐÑ (c) Derive the higher order one-sided approximation 0ÐÑœ O Ð2Ñ 2 (d) Compare the results of (a,b,c) with the exact value in the case 0ÐÑ œ È, œ and 2 œ Þ 66 Without using Maple, find the first three nonzero terms of the MacLaurin series for each of the following functions (a) 0ÐÑ œ È ) $ Hint: inomial theorem (b) 1ÐÑ œ csc Hint: Divide by sin (c) 2ÐÑ œ ln Œ Hint: œ Ð Ñ Ð Ñ ln Œ ln ln 5 Þ 5œ 67 (a) Use the integral test to show that the series converges and that the sum lies between 100 and 101 (b) Show that the sum of the first one million terms of the series in (a) is less than 14 So the remainder after adding one million terms is greater than 6, even though the terms in this sum are all less than 10 and getting smaller

36 6 Consider the series á $ % & (a) Use Maple to find the sum of the series (b) According to the alternating series test, how many terms of the series are required before the remainder is less than 00001? (c) How many terms are actually required to get a remainder less than 00001? What is the sum at this point? (d) How much is the estimate in (c) improved by adding 1/2 of the first neglected term? 69 (a) (4 pts) Prove that lim / œ for all Ä pts) (b) (3 Use integration by parts to prove the reduction formula / œ /, (c) (3 pts) Use induction to prove / œx for œßß$á 70 Use Newtons Universal Law of Gravity to find the work required for a rocket of mass 20,000 kg to escape earths gravity (ie the work required to move it from the earths surface to infinity) 71 Evaluate the following improper integrals (a) ( Ð ÑÐ Ñ (b) ( Ð ÑÐ Ñ (c) ( (d) ( (e) ( È ˆ È / sech 72 A horizontal cylindrical tank with radius 2 feet is filled with gasoline Find the fluid force due to the pressure on the end of the tank (Gasoline weighs 42 pounds per cubic foot) 73 (a) Write Þ$$$á as the ratio of two integers in lowest terms (b) Write Þ$%$%$%á as the ratio of two integers in lowest terms

37 74 Consider the polynomials W ÐÑœ 5 Ð Ñ 5œ 5 +1, Ð5 Ñx œßßß$á (a) Write out the expression for WÐÑ, œ ß ß ß $ß % without D notation (b) Calculate sinðîñ and WÐÎÑ for œ ß ß ß $ß % (c) Calculate sinðñ and WÐÑfor œßßß$ß% (d) Calculate sinðñ and WÐÑfor œßßß$ß% (e) Plot (all on one graph) Cœsin and CœWÐÑfor œ$ßß, ŸŸ, ŸCŸÞ 75 Consider the polynomials X ÐÑœ 5 Ð Ñ 5œ 5 5, œßßß$á (a) Write out the expression for XÐÑ, œ ß ß ß $ß % without D notation (b) Calculate tan ÐÎÑ and XÐÎÑ for œ ß ß ß $ß % (c) Calculate tan ÐÑand XÐÑfor œßßß$ß% (d) Calculate tan ÐÑand XÐÑfor œßßß$ß% (e) Plot (all on one graph) Cœtan ÐÑand CœXÐÑfor œ$ßß, ŸŸ, ŸCŸÞ % Ð Ñ > 76 Consider the geometric formula > > > á Ð Ñ > œ > > (a) Integrate the formula in (a) to obtain $ & ( Ð Ñ á œ tan VÐÑ $ & ( > where VÐÑ œ Ð Ñ ( > > (b) Show that kvðñÿ k kk ( Hint: Use > in the integral for V ) 77 (3 pts each) Find the radius of convergence of the following power series 5 Ð Ñ (a) È 5 5œ 5 (b) Ð5Ñx 5œ (c) Ð Ñ Ð5 Ñ % 5 5œ (a) (3 pts) Find the volume generated by rotating the region bounded by both axes and the graph of Cœ È % about the axis (b) (3 pts) Find the length of the curve Cœlnacos b for ŸŸ1Î% Hint: secœlnasec tanb

38 (c) (4 pts) Calculate t he work done in emptying a conical hole of radius ft and depth & ft filled with water through a hole at the top (use Þ% pounds per cubic foot for the weight density of water) Ð5 Ñ 79 Consider the function 0 defined by the power series 5 Þ 5 (a) (4 pts) Find the radius of convergence of the series (b) (4 pts) Integrate the series and find the resulting sum using the geometric formula (c) (2 pts) Differentiate the result in (b) to find a formula for 0ÐÑ 5œ