8. (1 pt) set1/golden-math1220spring q8.pg Let f (x) = ln(cosx). Assume that x is restricted so that ln is defined. Find

Transcription

1 Hsiang-Ping Huang WeBWorK assignment number is due : 0/7/009 at 06:00pm MST math0spring009-3 ( pt) set/srw pg Evaluate the ( following expressions (a) log 3 43) = (b) log 7 = (c) log 4 56 = (d) 9 log 9 9 = ( pt) set/srw pg Evaluate the following expressions (a) lne 9 = (b) e ln5 = (c) e ln = (d) ln(/e ) = ( pt) set/srw4 3 46pg Solve the given equation for x 5 x 6 = ( pt) set/srw4 3 48pg If ln(4x + ) = 5, then x = ( pt) set/srw4 3 5pg If e 3x = 6, then x = ( pt) set/srw4 4 9pg is equal to ln(r 6 s 6 9 r 6 s 9 ) Alnr + Blns where A = and where B = ( pt) set/mecpg The equation e x 9e x + 8 = 0 has two solutions The smaller one is: and the larger one is: ( pt) set/golden-math0spring00--q8pg Let f (x) = ln(cosx) Assume that x is restricted so that ln is defined Find Undefined ( f π ) 9 ( pt) set/golden-math0spring00--q9pg Find the integral t + t + 4t + 3 dt 0 ( pt) set/golden-math0spring00--q0pg Suppose y = (x + 3x)(x )(x + ) Find dy by logarithmic differentiation See Example 7 in dx Section 7 of your text 5*x**4 + 4*x**3-5*x** + *x - 6 ( pt) set/golden-math0spring00--qpg The rate of transmission in a telegraph cable is observed to be proportional to x ln(/x), where x is the ratio of the radius of the core to the thickness of the insulation ( 0 < x < ) What value of x gives the maximum rate of transmission? ( pt) set/golden-math0spring00--qpg Evaluate π/3 tanx dx

2 3 ( pt) set/golden-math0spring00--q3pg A ball is thrown vertically upward with initial velocity v Find the maximum height H of the ball as a function of v Then find the initial velocity v required to achieve a height of H Height H achieved with given initial velocity v: Initial velocity v required to achieve given height H: (v**)/64 8*sqrt(H) 4 ( pt) set/golden-math0spring00--q4pg Suppose y = e /x + /e x Find D x y D x y = -*exp(/(x**))/(x**3) - *x/exp(x**) 5 ( pt) set/golden-math0spring00--q5pg Suppose e x+y = x + y Find D x y D x y = - 6 ( pt) set/golden-math0spring00--q6pg Find the integral ln(abs( exp(x)- )) e x e x dx +C 7 ( pt) set/golden-math0spring00--q7pg Find the integral e 3/x x dx 8 ( pt) set/golden-math0spring00--q8pg The region bounded by y = e x, y = 0, x = 0, and x = is revolved about the y-axis Find the volume of the resulting solid ( pt) set/golden-math0spring00--q9pg lnx Let f (x) = for x in (0,) Find + (lnx) a) lim x 0 + f (x) = b) lim x f (x) = ( pt) set/golden-math0spring00--q0pg Evaluate the following limit: lim ( + x 0 x)/x ( pt) set/golden-math0spring00--qpg Suppose y = sin x + sinx Find dy/dx *sin(x)*cos(x) + **(sin(x)) * cos(x) * ln() ( pt) set/golden-math0spring00--qpg The loudness of sound is measured in decibels in honor of Alexander Graham Bell (847-9), inventor of the telephone If the variation in pressure is P pounds per square inch, then the loudness L in decibels is L = 0log 0 (3P) Find the variation in pressure caused by a rock band at 5 decibels pounds per square inch Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

3 Hsiang-Ping Huang WeBWorK assignment number is due : 0/03/009 at :59pm MST math0spring009-3 ( pt) set/ur le pg A certain bacteria population is known to quadruple every 90 minutes Suppose that there are initially 50 bacteria What is the size of the population after t hours? 50 * 4**( * t) ( pt) set/srw4 5 8pg If a bacteria culture starts with 4000 bateria and doubles every 5 minutes, how many minutes will it take the population to reach 6000? ( pt) set/srw4 5 0pg The count in a bacteria culture was 300 after 5 minutes and 000 after 35 minutes What was the initial size of the culture? Find the doubling period Find the population after 95 minutes When will the population reach 0000? ( pt) set/ur le pg The rat population in a major metropolitan city is given by the formula n(t) = 49e 005t where t is measured in years since 99 and n(t) is measured in millions What was the rat population in 99? What is the rat population going to be in the year 004? ( pt) set/srw4 7pg The half-life of Radium-6 is 590 years If a sample contains 400 mg, how many mg will remain after 4000 years? ( pt) set/srw4 3pg If 7000 dollars is invested in a bank account at an interest rate of 7 per cent per year, find the amount in the bank after 5 years if interest is compounded annually Find the amount in the bank after 5 years if interest is compounded quaterly Find the amount in the bank after 5 years if interest is compounded monthly Finally, find the amount in the bank after 5 years if interest is compounded continuously ( pt) set/srw4 5 4pg The 906 San Francisco earthquake had a magnitude of 83 on the Richter scale At the same time in South America there was an eathquake with magnitude 5 that caused only minor damage How many times more intense was the San Francisco earthquake than the South American one? The magnitude M on the Richter scale of an earthquake as a function of its intensity I is given by ( ) I M = log 0, I 0 where I 0 is some fixed reference level of intensity ( pt) set/golden-math0spring00--p-8pg Human hair from a grave in Africa proved to have only 76% of the carbon 4 of living tissue When was the body buried? See Problem 3 and 4 and Example 3 of Section 65 of the course text The body was buried about years ago ( pt) set/golden-math0spring00--p-9pg Newton s Law of Cooling states that the rate at which an object cools is proportional to the difference in temperature between the object and the surrounding medium Thus, if an object is taken from an oven at 94 F and left to cool in a room at 80 F, its temperature T after t hours will satisfy the differential equation dt dt = k(t 80) If the temperature fell to 94 F in 08 hour(s), what will it be after 3 hour(s)? After 3 hour(s), the temperature will be F

4 0 ( pt) set/golden-math0spring00--p-0pg Inflation between 977 and 98 ran at about 4% per year On this basis, what would you expect a car that would have cost $5000 in 977 to cost in 98? ( pt) set/golden-math0spring00--p-5pg Find the current I as a function of time for the circuit in the following figure if the switch S is closed and I = 0 at t = 0, where E = 3 volt(s), L = henry(s) and R = 5 ohm(s) ( pt) set/golden-math0spring00--p-pg Assume that () world population continues to grow exponentially with growth constant k = 003, () it takes acre of land to supply food for one person, and (3) there are 3,500,000 square miles of arable land in in the world How long will it be before the world reaches the maximum population? Note: There were 606 billion people in the year 000 and square mile is 640 acres Maximum population will be reached some time in the year 079 ( pt) set/golden-math0spring00--p-pg Use the fact that e = lim( + h) /h h 0 to find the following limits: (a) lim x 0 ( x) /7x = (b) lim x 0 ( ( + 7x) /x ) = n + 9 n (c) lim n = n ( ) n 9 n (d) lim n = n E-08 3 ( pt) set/golden-math0spring00--p-3pg Solve the following differential equation: y + ytanx = secx Instruction: Name your integration constant C sin(x) + C*cos(x) 4 ( pt) set/golden-math0spring00--p-4pg A tank initially contains 00 gallons of brine, with 50 pounds of salt in solution Brine containing pounds of salt per gallon is entering the tank at the rate of 4 gallons per minute and is flowing out at the same rate If the mixture in the tank is kept uniform by constant stirring, find the amount of salt in the tank at the end of 40 minutes Amount of salt after 40 minutes: pound(s) (Click on image for a larger view ) 3 * (-exp(-5*t/)) / 5 6 (0 pts) set/golden-math0spring00--p-6pg Megan bailed out of her plane at an altitude of 8000 feet, fell freely for 5 seconds, and then opened her parachute Assume that the drag coefficients are a = 0 for a fee fall and a = 6 with a parachute When did she land? It takes Megan about seconds to reach the ground 9 7 ( pt) set/sc3 6 6apg Let f (x) = x 3 tan (5x) f (x) = NOTE: The WeBWorK system will accept arctan(x) but not tan (x) as the inverse of tan(x) 3*x**(3-)*arctan(5*x) + x**3*5/(+5*5*x**) 8 ( pt) set/sc3 6 3apg Let f (x) = 3sin(x)sin (x) f (x) = NOTE: The webwork system will accept arcsin(x) and not sin (x) as the inverse of sin(x) 3*(cos(x)*arcsin(x) + sin(x)/sqrt(-x*x))

5 9 ( pt) set/golden-math0spring00--p-9pg Find the limit lim x sec x 0 ( pt) set/golden-math0spring00--p-0pg Evaluate the integral: dx x x ( pt) set/golden-math0spring00--p-pg Evaluate the integral: π/ sinθ + cos θ dθ ( pt) set/golden-math0spring00--p-pg Suppose y = sinh(x + x) Find D x y D x y = (*x+)*cosh(x**+x) 3 ( pt) set/golden-math0spring00--p-3pg Suppose y = ln(cothx) Find D x y D x y = -sech(x)/sinh(x) 4 ( pt) set/golden-math0spring00--p-4pg Evaluate the integral cosh z dz z + C *sinh(sqrt(z)) 5 (0 pts) set/golden-math0spring00--p-5pg The curve y = sinhx,0 x, is revolved about the x-axis Find the area of the resulting surface Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

6 Hsiang-Ping Huang WeBWorK assignment number 3 is due : 0/7/009 at :59pm MST math0spring009-3 ( pt) set3/golden-math0spring00--p3-pg Perform the following integration: sin(4t ) sin (4t ) dt + C sec(4*t-)/4 ( pt) set3/golden-math0spring00--p3-pg Perform the following integration: e x sec (e x ) dx + C tan(exp(x)) 3 ( pt) set3/golden-math0spring00--p3-3pg Perform the following integration: e x e x 0 e x dx + e x ( pt) set3/golden-math0spring00--p3-4pg Perform the following integration: x 4x + 9 dx + C arctan((x-)/sqrt(5))/sqrt(5) 5 ( pt) set3/golden-math0spring00--p3-5pg Perform the following integration: x + x 3 dx x + + C sqrt(x**+*x-3)-*arccos(/(x+)) 7 ( pt) set3/golden-math0spring00--p3-7pg Perform the following integration: cos 3 x dx + C sin(x)-((sin(x))**3)/3 8 ( pt) set3/golden-math0spring00--p3-8pg Integrals of the form R cot n xdx can be evaluated by factoring out cot x = csc x Use this method to evaluate the following integral: cot 4 x dx + C -(cot(x)**3)/3 + cot(x) + x 9 ( pt) set3/golden-math0spring00--p3-9pg When n is even, integrals of the form R tan m xsec n x dx can be evaluated by factoring out sec x = + tan x and using the fact that D x tanx = sec x When m is odd, integrals of this form can be evaluated by factoring out tanxsecx and using the fact that D x secx = secxtanx Use this method to evaluate the following integral: tan 3/ xsec 4 x dx + C -/sqrt(tan(x)) + (/3)*(tan(x))**(3/) 0 ( pt) set3/golden-math0spring00--p3-0pg Evaluate the following integral: L cos mπx nπx cos L L dx, L where m n and m, n are integers 0 6 ( pt) set3/golden-math0spring00--p3-6pg Perform the following integration: sinx (cosx) 5 4cosx dx + C -ln(abs((sqrt(5-4*cos(x))-sqrt(5))/(sqrt(5-4*cos(x))+sqrt(5))))/sqrt(5) ( pt) set3/golden-math0spring00--p3-pg Evaluate the following integral: t 0 t + dt

7 ( pt) set3/golden-math0spring00--p3-pg Perform the following integration: x 6 x dx + C 8*arcsin(x/4) - x*sqrt(6-x**)/ 3 ( pt) set3/golden-math0spring00--p3-3pg Perform the following integration: x x + 4x + 5 dx + C *sqrt(x**+4*x+5) - 5*ln(abs(sqrt(x**+4x+5)+x+)) 4 ( pt) set3/golden-math0spring00--p3-4pg The region bounded by y = /(x + x + 5), y = 0, x = 0, and x =, is revolved about the y-axis Find the volume of the resulting solid Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

8 Hsiang-Ping Huang WeBWorK assignment number 4 is due : 03/03/009 at :59pm MST The (* replace with url for the course home page *) for the course contains the syllabus, grading policy and other information math0spring009-3 This file is /conf/snippets/setheaderpg you can use it as a model for creating files which introduce each problem set The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake Usually you can attempt a problem as many times as you want before the due date However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA s or your professor for help Don t spend a lot of time guessing it s not very efficient or effective Give 4 or 5 significant digits for (floating point) numerical answers For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 3 instead of 8, sin(3 pi/)instead of -, e (ln()) instead of, ( +tan(3)) (4 sin(5)) 6 7/8 instead of , etc Here s the list of the functions which WeBWorK understands You can use the Feedback button on each problem page to send to the professors ( pt) set4/golden-math0spring00--p4-pg Use integration by parts to evaluate the following integral: (t + 7)e t+3 dt + C (/)*t*exp(*t+3) + (3/4)*exp(*t+3) ( pt) set4/golden-math0spring00--p4-pg Use integration by parts to evaluate the following integral: arctan5x dx + C x * arctan(5*x) - (/0)*ln(+5*x**) 3 ( pt) set4/golden-math0spring00--p4-3pg Use integration by parts to evaluate the following integral: sec 3 x dx + C (/)*sec(x)*tan(x) + (/)*ln(abs(sec(x)+tan(x))) 4 ( pt) set4/golden-math0spring00--p4-4pg Use integration by parts twice to evaluate the following integral: cos(lnx) dx + C (x/)*( cos(ln(x))+sin(ln(x)) ) 5 ( pt) set4/golden-math0spring00--p4-5pg Find the volume of the solid obtained by revolving the region under the graph of y = sin(x/) from x = 0 to x = π about the y-axis ( pt) set4/golden-math0spring00--p4-6pg Use the method of partial fraction decomposition to perform the following integration: x 7 x x dx + C -(3/7)*ln(abs(x-4)) + (0/7)*ln(abs(x+3)) 7 ( pt) set4/golden-math0spring00--p4-7pg Use the method of partial fraction decomposition to perform the following integration: x x 0 x + x 6 dx + C *x - (/5)*ln(abs(x+3)) - (4/5)*ln(abs(x-)) 8 ( pt) set4/golden-math0spring00--p4-8pg Use the method of partial fraction decomposition to perform the following integration: 5x + 7 x + 4x + 4 dx + C 5*ln(abs(x+)) + 3/(x+)

9 9 ( pt) set4/golden-math0spring00--p4-9pg Use the method of partial fraction decomposition to perform the following integration: x 4 6 dx + C (/3)*ln(abs(x-)) - (/3)*ln(abs(x+)) - (/6)*arctan(x/) infinity 3 ( pt) set4/golden-math0spring00--p4-3pg Find the following limit using l Hopital s Rule: R x 0 t cost dt lim x 0 + x Enter the word infinity if the answer is 0 ( pt) set4/golden-math0spring00--p4-0pg In many population growth problems, there is an upper limit beyond which the population cannot grow Let us suppose that the earth will not support a population of more than 6 billion and that there were billion people in 95 and 4 billion people in 975 Then, if y is the population t years after 95, an appropriate model is the differential equation dy = ky(6 y) dt (a) Solve this differential equation Solution: y(t) = (b) Find the population in 05 Population in 05: billion (c) When will the population be 9 billion? The year the population will be 9 billion: 6 / ( +7*exp(-t*ln(7/3)/50) ) ( pt) set4/golden-math0spring00--p4-pg Find the following limit using l Hopital s Rule: lnx lim x x ( pt) set4/golden-math0spring00--p4-pg Find the following limit using l Hopital s Rule: 3sinx lim x 0 x 0 4 ( pt) set4/golden-math0spring00--p4-4pg Find the following limit using l Hopital s Rule: 3x lim x ln(00x + e x ) 3 5 ( pt) set4/golden-math0spring00--p4-5pg Find the following limit using l Hopital s Rule: lim x 0 3x csc x 3 6 ( pt) set4/golden-math0spring00--p4-6pg Find the following limit using l Hopital s Rule: lim x 0 (cosx)/x ( pt) set4/golden-math0spring00--p4-7pg Find the following limit using l Hopital s Rule: lim lnx) x 0 +(x/ 0 8 ( pt) set4/golden-math0spring00--p4-8pg Find the following limit: lim x 0 + xx

10 9 ( pt) set4/golden-math0spring00--p4-9pg Find the following limit: lim x 0 (xx ) x + 0 ( pt) set4/golden-math0spring00--p4-0pg Evaluate the following improper integral: e 4x dx If the integral diverges, enter diverge as answer ( pt) set4/golden-math0spring00--p4-pg Evaluate the following improper integral: x 0 + x dx If the integral diverges, enter diverge as answer diverge ( pt) set4/golden-math0spring00--p4-pg Evaluate the following improper integral: lnx x dx e If the integral diverges, enter diverge as answer diverge 3 ( pt) set4/golden-math0spring00--p4-3pg Evaluate the following improper integral: xe x dx If the integral diverges, enter diverge as answer ( pt) set4/golden-math0spring00--p4-4pg Evaluate the following improper integral: 4 dx (π x) /3 If the integral diverges, enter diverge as answer diverge 5 ( pt) set4/golden-math0spring00--p4-5pg Find the area of the region under the curve y = x + x to the right of x = Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

11 Hsiang-Ping Huang WeBWorK assignment number 5 is due : 03/4/009 at :59pm MDT math0spring009-3 ( pt) set5/golden-math0spring00--p5-pg Evaluate the following improper integral If the integral is divergent, enter divergent as answer 5 dx 5 x/ ( pt) set5/golden-math0spring00--p5-pg Evaluate the following improper integral If the integral is divergent, enter divergent as answer x 3 dx 0 x ( pt) set5/golden-math0spring00--p5-3pg Evaluate the following improper integral If the integral is divergent, enter divergent as answer π/ cscx dx 0 divergent 4 ( pt) set5/golden-math0spring00--p5-4pg Evaluate the following improper integral If the integral is divergent, enter divergent as answer 3 x ln( x) dx ( pt) set5/golden-math0spring00--p5-5pg Evaluate the following improper integral If the integral is divergent, enter divergent as answer 0 xln 00 x dx divergent 6 ( pt) set5/golden-math0spring00--p5-6pg Evaluate the following improper integral If the integral is divergent, enter divergent as answer x ln x dx divergent 7 ( pt) set5/golden-math0spring00--p5-7pg Consider the sequence a n = ncos(nπ) n Write the first five terms of a n, and find lim n a n If the sequence diverges, enter divergent in the answer box for its limit a) First five terms:,,,, b) lim n a n = divergent 8 ( pt) set5/golden-math0spring00--p5-8pg Consider the sequence a n = ln(/n) n Write the first five terms of a n, and find lim n a n If the sequence diverges, enter divergent in the answer box for its limit a) First five terms:,,,, b) lim n a n =

12 9 ( pt) set5/golden-math0spring00--p5-9pg Suppose a =,a = 3,a 3 = 3 3 4,a 4 = ,a 5 = a) Find an explicit formula for a n : b) Determine whether the sequence is convergent or divergent: (Enter convergent or divergent as appropriate) c) If it converges, find lim n a n = (n**+n)/(n**+*n) convergent 0 ( pt) set5/golden-math0spring00--p5-0pg Suppose a =,a n+ = ) (a n + an Find lim n a n = Hint: Let a = lim n Then, since a n+ = have a = ( a + ) Now solve for a a (a n + an ), we ( pt) set5/golden-math0spring00--p5-pg Consider the series: [ ( ) k ( ) ] k+ 5 3 k= 7 a) Determine whether the series is convergent or divergent: (Enter convergent or divergent as appropriate) b) If it converges, find its sum: If the series diverges, enter here divergent again convergent ( pt) set5/golden-math0spring00--p5-pg Consider the series: 3 k= k a) Determine whether the series is convergent or divergent: (Enter convergent or divergent as appropriate) b) If it converges, find its sum: If the series diverges, enter here divergent again divergent divergent 3 ( pt) set5/golden-math0spring00--p5-3pg Consider the series: k= k(k + ) a) Determine whether the series is convergent or divergent: (Enter convergent or divergent as appropriate) b) If it converges, find its sum: If the series diverges, enter here divergent again convergent 5 4 ( pt) set5/golden-math0spring00--p5-4pg Consider the series: k=9 ( 3 (k ) 3 ) k a) Determine whether the series is convergent or divergent: (Enter convergent or divergent as appropriate) b) If it converges, find its sum: If the series diverges, enter here divergent again convergent ( pt) set5/golden-math0spring00--p5-5pg A ball is dropped from a height of 9 feet Each time it hits the floor, it rebounds to 4 its previous height Find the total 5 distance it travels before coming to rest feet 89 6 ( pt) set5/golden-math0spring00--p5-6pg How large must N be in order for S N = N k= to exceed 4? Note: Computer calculations show that for S N to exceed 0, N = 7,400,600 and for S N to exceed 00, N N = 3 k

13 7 ( pt) set5/golden-math0spring00--p5-8pg Use the Integral Test to decide the convergence or divergence of the following series: converge k k= e k (Enter converge or diverge ) 8 ( pt) set5/golden-math0spring00--p5-9pg Use the Integral Test to decide the convergence or divergence of the following series: divergent 000k k= + k 3 (Enter convergent or divergent ) 9 ( pt) set5/golden-math0spring00--p5-0pg Use the Integral Test to decide the convergence or divergence of the following series: k=5 000 k(lnk) (Enter convergent or divergent ) convergent 0 ( pt) set5/golden-math0spring00--p5-pg Decide the convergence or divergence of the following series: convergent k= ( ) 3 k π (Enter convergent or divergent ) ( pt) set5/golden-math0spring00--p5-pg Decide the convergence or divergence of the following series: ( k ) k + convergent k= (Enter convergent or divergent ) ( pt) set5/golden-math0spring00--p5-3pg Decide the convergence or divergence of the following series: divergent n=3 n lnn ln(lnn) (Enter convergent or divergent ) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

14 Hsiang-Ping Huang WeBWorK assignment number 6 is due : 03/3/009 at :59pm MDT math0spring009-3 ( pt) set6/golden-math0spring00--p6-pg Determine the convergence or divergence of the following series A convergent B divergent A n + n= n ( pt) set6/golden-math0spring00--p6-pg Determine the convergence or divergence of the following series A convergent B divergent A 3 k + k n= k! 3 ( pt) set6/golden-math0spring00--p6-3pg Determine the convergence or divergence of the following series A convergent B divergent A 5 n n= n! 4 ( pt) set6/golden-math0spring00--p6-4pg Determine the convergence or divergence of the following series A convergent B divergent A [ ( )] n cos n= n 5 ( pt) set6/golden-math0spring00--p6-5pg Determine whether the following series is n= ( ) n+ A absolutely convergent B conditionally convergent C divergent A 5n 6 ( pt) set6/golden-math0spring00--p6-6pg Determine whether the following series is n= ( ) n n A conditionally convergent B absolutely convergent C divergent A 7 ( pt) set6/golden-math0spring00--p6-7pg Determine whether the following series is n= ( ) n sinn n n A absolutely convergent B conditionally convergent C divergent A 8 ( pt) set6/golden-math0spring00--p6-8pg Find the convergence set of the given power series: n x n n= n! The above series converges for < x < Enter infinity for and -infinity for -infinity infinity

15 9 ( pt) set6/golden-math0spring00--p6-9pg Find the convergence set of the given power series: (x ) n= n The above series converges for x Enter infinity for and -infinity for 3 0 ( pt) set6/golden-math0spring00--p6-0pg A famous sequence f n, called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around AD 00, is defined by the recursion formula f = f =, f n+ = f n+ + f n Find the radius of convergence of f n x n n= Radius of convergence: ( pt) set6/golden-math0spring00--p6-pg Find the power series representation for f (x) = ( + x) and specify the radius of convergence f (x) = ( ) e n a n x p n, n= where e n =, where a n =, and p n = Radius of convergence: n- n n- ( pt) set6/golden-math0spring00--p6-pg Find the power series representation for f (x) = xe x f (x) = n=0 a n! xp n, where a n = and p n = n *n+ 3 ( pt) set6/golden-math0spring00--p6-3pg Find the power series representation for x tan t f (x) = dt 0 t f (x) = ( ) e n a n x p n, n= where e n =, and a n =, and p n = n- /((*n-)**) *n- 4 ( pt) set6/golden-math0spring00--p6-4pg Find the sum of n= n(n + )xn = for < x < *x/((-x)**3) - 5 ( pt) set6/golden-math0spring00--p6-5pg Find the terms through x 5 in the Maclaurin series for f (x) = +O(x 6 ) f (x) = e x cosx -x+/3*x**3-/6*x**4+/30*x**5 6 ( pt) set6/golden-math0spring00--p6-6pg Find the terms through x 5 in the Maclaurin series for f (x) = +O(x 6 ) f (x) = sinx +*x+*x**+5/6*x**3+/3*x**4+6/0*x**5 7 ( pt) set6/golden-math0spring00--p6-7pg Find the Taylor series in (x a) through (x a) 3 for ( f (x) = + (x ) ) O π f (x) = tanx, a = π 4 ( ) ( ) x π 4 + x π ( ) 4 + x π 3 4 +

16 8 ( pt) set6/golden-math0spring00--p6-8pg Find the Taylor series in (x a) through (x a) 3 for f (x) = x + 3x x 3, a = f (x) = + (x + )+ (x + ) + (x + ) ( pt) set6/golden-math0spring00--p6-9pg Calculate the following integral, accurate to five decimal places: sin x dx Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

17 Hsiang-Ping Huang WeBWorK assignment number 7 is due : 04//009 at :59pm MDT math0spring009-3 ( pt) set7/golden-math0spring00--p7-pg Solve the following differential equation: y 3y 0y = 0; y =,y = 0 at x = 0 y(x) = (/7)*exp(5*x) - (5/7)*exp(-*x) ( pt) set7/golden-math0spring00--p7-pg Solve the following differential equation: y + 0y + 5y = 0 y(x) = C +C exp(-5*x) x*exp(-5*x) 3 ( pt) set7/golden-math0spring00--p7-3pg Solve the following differential equation: y + 9y = 0; y = 3, y = 3 at x = π/3 y(x) = -sin(3*x)-3*cos(3*x) 4 ( pt) set7/golden-math0spring00--p7-4pg Solve the following differential equation: y + y + y = 0 y(x) = C +C exp(-x/)*cos(sqrt(3)*x/) exp(-x/)*sin(sqrt(3)*x/) 5 ( pt) set7/golden-math0spring00--p7-5pg Solve the following differential equation: y y + y = 0 and express your answer in the form ce αx sin(βx + γ) α =, β = 6 ( pt) set7/golden-math0spring00--p7-6pg Use the method of undetermined coefficients to solve the following differential equation: y + y = 4x y(x) = +C +C *x** - 4*x exp(-x) 7 ( pt) set7/golden-math0spring00--p7-7pg Use the method of undetermined coefficients to solve the following differential equation: y + 6y + 9y = e x y(x) = +C +C exp(-x)/ exp(-3*x) x*exp(-3*x) 8 ( pt) set7/golden-math0spring00--p7-8pg Use the method of undetermined coefficients to solve the following differential equation: y + 4y = cosx y(x) = +C +C (-/7)*cos(x) + (4/7)*sin(x) exp(-4*x) 9 ( pt) set7/golden-math0spring00--p7-9pg Solve the following differential equation: y + 4y = sin 3 x y(x) = +C +C sin(x)/4 + sin(3*x)/0 cos(*x) sin(*x)

18 0 ( pt) set7/golden-math0spring00--p7-0pg A spring with a spring constant k of 00 pounds per foot is loaded with -pound weight and brought to equilibrium It is then stretched an additional inch and released Find the equation of motion, the amplitude, and the period Neglect friction y(t) =, where t is time and y(t) is displacement in time Amplitude: inch(es) Period: second(s) (/)*cos(40*sqrt()*t) ( pt) set7/golden-math0spring00--p7-pg A spring with a spring constant k of 0 pounds per foot is loaded with a 0-pound weight and allowed to reach equilibrium It is then displaced foot downward and released If the weight experiences a retarding force in pounds equal to four times the velocity at every point, find the equation of motion y(t) =, where t is time and y(t) is displacement in time exp(-64*t)*( cos(48*t) + (4/3)*sin(48*t) ) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

19 Hsiang-Ping Huang math0spring009-3 WeBWorK assignment number Demo is due : 0/30/009 at :58pm MST The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making some kind of mistake Usually you can attempt a problem as many times as you want before the due date However, if you are having trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA s or your professor for help Don t spend a lot of time guessing it s not very efficient or effective Give 4 or 5 significant digits for (floating point) numerical answers For most problems when entering numerical answers, you can if you wish enter elementary expressions such as 3 instead of 8, sin(3 pi/)instead of -, e (ln()) instead of, ( +tan(3)) (4 sin(5)) 6 7/8 instead of , etc Here s the list of the functions which WeBWorK understands You can use the Feedback button on each problem page to send to the professors ( pt) setdemo/demo prpg Evaluate the expression (7 9) = -4 ( pt) setdemo/demo prpg Evaluate the expression 5/(4 + 3) = Enter you answer as a decimal number listing at least 4 decimal digits (WeBWorK will reject your answer if it differs by more than one tenth of percent from what it thinks the answer is) ( pt) setdemo/demo pr3pg Let r = 5 Evaluate 4/π r = Next, enter the expression 4/(π r) = WorK compute the result ( pt) setdemo/demo pr4pg Enter here the expression a + b Enter here the expression a+b /a+/b /(a+b) 5 ( pt) setdemo/demo pr5pg Enter here Enter here a + + b the expression the expression and let WeB- a + b c + d If WeBWorK rejects your answer use the preview button to see what it thinks you are trying to tell it (a+)/(+b) (a+b)/(c+d) 6 ( pt) setdemo/demo pr6pg Enter here the expression a + b Enter here Enter here sqrt(a+b) a/sqrt(a+b) (a+b)/sqrt(a+b) 7 ( pt) setdemo/demo pr7pg Enter here Enter here Enter here a a + b a + b a + b x + y x x + y x + y x + y sqrt(x**+y**) x*sqrt(x**+y**) (x+y)/sqrt(x**+y**) the expression the expression the expression the expression the expression

20 8 ( pt) setdemo/demo pr8pg Enter here b + b 4ac a the expression Note: this is an expression that gives the solution of a quadratic equation by the quadratic formula (-b+sqrt(b**-4*a*c))/(a) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester