WW Prob Lib1 Math course-section, semester year

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1 Peter Alfeld WeBWorK problems WW Prob Lib Math course-section, semester year WeBWorK assignment due /5/06 at :59 PM ( pt) Evaluate the following expressions (a) log ( 6 ) = (b) log = (c) log 5 65 = (d) 9 log 9 = ( pt) Evaluate the following expressions (a) lne 5 = (b) e ln5 = (c) e ln 4 = (d) ln(/e 4 ) = 3( pt) Solve the given equation for x 5 4x 5 = 34 4( pt) If ln(4x + 4) = 3, then x = 5( pt) If e 4x = 5, then x = 6( pt) ln(r 7 s 7 8 r s 0 ) is equal to Alnr + Blns where A = and where B = 7( pt) The equation e x 8e x + 5 = 0 has two solutions The smaller one is: and the larger one is: 8( pt) Let ln(cosx) Assume that x is restricted so that ln is defined Find f ( π 9( pt) Find the integral ) t + 0 t + 4t + 3 dt 0( pt) Suppose y = (x + 3x)(x )(x + ) Find dy by logarithmic differentiation See Example 7 in Section 7 of your text dx ( pt) 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) Evaluate π/3 tanx dx 0 3( pt) 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: 4( pt) Suppose y = e /x + /e x Find D x y D x y = 5( pt) Suppose e x+y = x + y Find D x y D x y = 6( pt) Find the integral e x e x dx 7( pt) Find the integral e 3/x x dx +C 8( pt) 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 9( pt) Let lnx for x in (0,) + (lnx) Find a) x 0 + b) x 0( pt) Evaluate the following it:

2 ( + x 0 x)/x ( pt) Suppose y = sin x + sinx Find dy/dx ( pt) 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 Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR

3 Peter Alfeld WeBWorK problems WW Prob Lib Math course-section, semester year WeBWorK assignment due /7/06 at :59 PM ( pt) A certain bacteria population is known to triple every 90 minutes Suppose that there are initially 70 bacteria What is the size of the population after t hours? ( pt) If a bacteria culture starts with 4000 bateria and doubles every 5 minutes, how many minutes will it take the population to reach 3000? 3( pt) The count in a bacteria culture was 00 after 5 minutes and 400 after 35 minutes What was the initial size of the culture? Find the doubling period Find the population after 00 minutes When will the population reach 4000? 4( pt) The rat population in a major metropolitan city is given by the formula n(t) = 59e 004t 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 00? 5( pt) The half-life of Radium-6 is 590 years If a sample contains 00 mg, how many mg will remain after 3000 years? 6( pt) If 4000 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 7( pt) 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, where I 0 is some fixed reference level of intensity 8( pt) Human hair from a grave in Africa proved to have only 7% of the carbon 4 of living tissue When was the body buried? See Problem 3 of Section 75 of the course text The body was buried about years ago 9( pt) 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 300 F and left to cool in a room at 80 F, its temperature T after t hours will satisfy the differential equation dt = k(t 80) dt If the temperature fell to 03 F in 08 hour(s), what will it be after 5 hour(s)? After 5 hour(s), the temperature will be F 0( pt) Inflation between 977 and 98 ran at about 35% per year On this basis, what would you expect a car that would have cost $500 in 977 to cost in 98? ( pt) 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 ( pt) Use the fact that I 0 e = h 0 ( + h) /h

4 to find the following its: (a) x 0 ( x) /7x = (b) x 0 ( ( + 8x) ) /x = n + 5 n (c) n = n ( ) n 7 n (d) n = n 3( pt) Solve the following differential equation: y + ytanx = secx Instruction: Name your integration constant C 4( pt) 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) 5( pt) 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 = 4 volt(s), L = henry(s) and R = 4 ohm(s) (Click on image for a larger view ) 6( pt) 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 reach the ground 7( pt) Let x 3 tan (6x) seconds to f (x) = NOTE: The WeBWorK system will accept arctan(x) but not tan (x) as the inverse of tan(x) 8( pt) Let cos(x)sin (x) f (x) = NOTE: The webwork system will accept arcsin(x) and not sin (x) as the inverse of sin(x) 9( pt) Find the it x sec x 0( pt) Evaluate the integral: dx x x ( pt) Evaluate the integral: π/ sinθ 0 + cos θ dθ ( pt) Suppose y = sinh(x + x) Find D x y D x y = 3( pt) Suppose y = ln(cothx) Find D x y D x y = 4( pt) Evaluate the integral cosh z z 5( pt) The curve y = sinhx,0 x, is revolved about the x-axis Find the area of the resulting surface dz Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR

5 Peter Alfeld WeBWorK problems WW Prob Lib Math course-section, semester year WeBWorK assignment 3 due /4/06 at :59 PM ( pt) Perform the following integration: sin(4t ) sin (4t ) dt ( pt) Perform the following integration: e x sec (e x ) dx 3( pt) Perform the following integration: e x e x 0 e x dx + e x 0( pt) Evaluate the following integral: 4( pt) Perform the following integration: L x 4x + 9 dx cos mπx nπx cos L L L dx, 5( pt) Perform the following integration: x + x 3 dx x + 6( pt) Perform the following integration: sinx (cosx) 5 4cosx dx 7( pt) Perform the following integration: cos 3 x dx 8( pt) 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 R 9( pt) When n is even, integrals of the form 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 where m n and m, n are integers ( pt) Evaluate the following integral: t 0 t + dt ( pt) Perform the following integration: x 6 x dx 3( pt) Perform the following integration: x x + 4x + 5 dx 4( pt) 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

6 Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR

7 Peter Alfeld WeBWorK problems WW Prob Lib Math course-section, semester year WeBWorK assignment 4 due 3/6/06 at :59 PM ( pt) Use integration by parts to evaluate the following integral: (t + 7)e t+3 dt ( pt) Use integration by parts to evaluate the following integral: arctan5x dx 3( pt) Use integration by parts to evaluate the following integral: sec 3 x dx 4( pt) Use integration by parts twice to evaluate the following integral: cos(lnx) dx 5( pt) 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 6( pt) Use the method of partial fraction decomposition to perform the following integration: x 7 x x dx 7( pt) Use the method of partial fraction decomposition to perform the following integration: x x 0 x + x 6 dx 8( pt) Use the method of partial fraction decomposition to perform the following integration: 5x + 7 x + 4x + 4 dx 9( pt) Use the method of partial fraction decomposition to perform the following integration: x 4 6 dx 0( pt) In many population growth problems, there is an upper it 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: ( pt) Find the following it using l Hopital s Rule: x lnx x ( pt) Find the following it using l Hopital s Rule:

8 3sinx x 0 x 3( pt) Find the following it using l Hopital s Rule: R x 0 t cost dt x 0 + x Enter the word infinity if the answer is 4( pt) Find the following it using l Hopital s Rule: x 3x ln(00x + e x ) 5( pt) Find the following it using l Hopital s Rule: x 0 3x csc x 6( pt) Find the following it using l Hopital s Rule: x 0 (cosx)/x 7( pt) Find the following it using l Hopital s Rule: lnx) x 0 +(x/ 8( pt) Find the following it: x 0 + xx 9( pt) Find the following it: x 0 (xx ) x + 0( pt) Evaluate the following improper integral: e 4x dx If the integral diverges, enter diverge as answer ( pt) Evaluate the following improper integral: x 0 + x dx If the integral diverges, enter diverge as answer ( pt) Evaluate the following improper integral: lnx e x dx If the integral diverges, enter diverge as answer 3( pt) Evaluate the following improper integral: xe x dx If the integral diverges, enter diverge as answer 4( pt) Evaluate the following improper integral: dx 4 (π x) /3 If the integral diverges, enter diverge as answer 5( pt) Find the area of the region under the curve y = x + x to the right of x = Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR

9 Peter Alfeld WeBWorK problems WW Prob Lib Math course-section, semester year WeBWorK assignment 5 due 3/7/06 at :59 PM ( pt) Evaluate the following improper integral If the integral is divergent, enter divergent as answer 5 dx 5 x/3 ( pt) Evaluate the following improper integral If the integral is divergent, enter divergent as answer 0 x 3 x dx 3( pt) Evaluate the following improper integral If the integral is divergent, enter divergent as answer π/ 0 cscx dx 4( pt) Evaluate the following improper integral If the integral is divergent, enter divergent as answer 3 x ln( x) dx 5( pt) Evaluate the following improper integral If the integral is divergent, enter divergent as answer a n = ncos(nπ) n Write the first five terms of a n, and find n a n If the sequence diverges, enter divergent in the answer box for its it a) First five terms:,,,, b) n a n = 8( pt) Consider the sequence a n = ln(/n) n Write the first five terms of a n, and find n a n If the sequence diverges, enter divergent in the answer box for its it a) First five terms:,,,, b) n a n = 9( pt) Suppose a =,a = 3 3,a 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 n a n = 0( pt) Suppose a =,a n+ = ) (a n + an 0 xln 00 x dx 6( pt) Evaluate the following improper integral If the integral is divergent, enter divergent as answer x ln x dx 7( pt) Consider the sequence Find n a n = Hint: Let ) a = n Then, since a n+ = (a n + an, we have a = ( a + ) Now solve a for a ( pt) Consider the series: [ ( ) k ( ) ] k k= a) Determine whether the series is convergent or divergent:

10 (Enter convergent or divergent as appropriate) N = b) If it converges, find its sum: 7( pt) If the series diverges, enter here divergent again ( pt) Consider the series: Use the Integral Test to decide the convergence or divergence of the following 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 3( pt) 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 4( pt) Consider the series: k=6 ( 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 5( pt) A ball is dropped from a height of 95 feet Each time it hits the floor, it rebounds to 4 its previous 5 height Find the total distance it travels before coming to rest feet 6( pt) How large must N be in order for N S N = k= 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 ) k k= e k (Enter converge or diverge ) 8( pt) Use the Integral Test to decide the convergence or divergence of the following series: 000k k= + k 3 (Enter convergent or divergent ) 9( pt) Use the Integral Test to decide the convergence or divergence of the following series: k=5 000 k(lnk) (Enter convergent or divergent ) 0( pt) Decide the convergence or divergence of the following series: k= ( ) 3 k π (Enter convergent or divergent ) ( pt) Decide the convergence or divergence of the following series: ( k= k ) k + (Enter convergent or divergent ) ( pt) Decide the convergence or divergence of the following series:

11 n=3 n lnn ln(lnn) (Enter convergent or di- vergent ) Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR 3

12 Peter Alfeld WeBWorK problems WW Prob Lib Math course-section, semester year WeBWorK assignment 6 due 4/7/06 at :59 PM ( pt) Determine the convergence or divergence of the following series n + A convergent B divergent n= ( pt) Determine the convergence or divergence of the following series n 3 k + k n= k! A convergent B divergent 3( pt) Determine the convergence or divergence of the following series A convergent B divergent 5 n n= n! 4( pt) Determine the convergence or divergence of the following series [ ( )] n cos n= n A convergent B divergent 5( pt) Determine whether the following series is n= ( ) n+ 5n A conditionally convergent B absolutely convergent C divergent 6( pt) Determine whether the following series is n= ( ) n n A conditionally convergent B absolutely convergent C divergent 7( pt) Determine whether the following series is n= ( ) n sinn n n A conditionally convergent B absolutely convergent C divergent 8( pt) Find the convergence set of the given power series: x n n= n! The above series converges for < x < Enter infinity for and -infinity for 9( pt) Find the convergence set of the given power series: (x ) n n= n The above series converges for x Enter infinity for and -infinity for 0( pt) 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) Find the power series representation for ( + x) and specify the radius of convergence n= ( ) e n a n x p n,

13 where e n =, where a n =, and p n = Radius of convergence: ( pt) Find the power series representation for xe x n=0 a n! xp n, where a n = and p n = 3( pt) Find the power series representation for x 0 n= tan t t dt ( ) e n a n x p n, where e n =, and a n =, and p n = 4( pt) Find the sum of n= n(n+)xn = for < x < 5( pt) Find the terms through x 5 in the Maclaurin series for e x cosx +O(x 6 ) 6( pt) Find the terms through x 5 in the Maclaurin series for sinx +O(x 6 ) 7( pt) Find the Taylor series in (x a) through (x a) 3 for tanx, a = π 4 ( ) ( ) + x π 4 + x π 4 + ( ) ( x π 3 (x ) ) 4 + O π 4 4 8( pt) Find the Taylor series in (x a) through (x a) 3 for x + 3x x 3, a = + (x + )+ (x + ) + (x + ) 3 9( pt) Calculate the following integral, accurate to five decimal places: 05 0 sin x dx Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR

14 Peter Alfeld WeBWorK problems WW Prob Lib Math course-section, semester year WeBWorK assignment 7 due 4/8/06 at :59 PM ( pt) Solve the following differential equation: 7( pt) Use the method of undetermined coefficients to solve the following differential equation: y 3y 0y = 0; y =,y = 0 at x = 0 y + 6y + 9y = e x y(x) = y(x) = +C +C ( pt) Solve the following differential equation: 8( pt) Use the method of undetermined coefficients to solve the following differential equation: y + 0y + 5y = 0 y(x) = C +C 3( pt) Solve the following differential equation: y + 9y = 0; y = 3, y = 3 at x = π/3 y(x) = 4( pt) Solve the following differential equation: y + y + y = 0 y(x) = C +C 5( pt) 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) Use the method of undetermined coefficients to solve the following differential equation: y + y = 4x y(x) = +C +C y + 4y = cosx y(x) = +C +C 9( pt) Solve the following differential equation: y + 4y = sin 3 x y(x) = +C +C 0( pt) 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) ( pt) 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 Prepared by the WeBWorK group, Dept of Mathematics, University of Rochester, c UR