5. (1 pt) set1/p1-5.pg Evaluate the integrals. e 2x dx. = (b) log 3 1 = (c) log 5. e x dx = (d) 7 log 7 10 = Correct Answers: -3

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1 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment due 5//8 at :59pm MDT Logs and Exponentials. This assignment will cover the material from Sections ( pt) set/p-.pg Evaluate the following expressions. (a) log ( 8 ) = (b) log 3 = (c) log 5 35 = (d) 7 log 7 = ( pt) set/p-.pg is equal to ln(r 8 s r 7 s ) Alnr + Blns where A = and where B = R ( pt) set/p-3.pg ln(5x) x dx = ( pt) set/p-4.pg If f (x) = xln(x), find f (x). Find f (5)..5**xˆ(-.5)*ln(x)+ *xˆ(-.5) ( pt) set/p-5.pg Evaluate the integrals e 3x+8 f (x) = e x dx f (x) = +C e x+8 g(x) = e x dx g(x) = +C exp(3*x+8-*x)/(3-) exp(8)*x 6. ( pt) set/p-7.pg Find the integral Answer: t + t + 4t + 3 dt. 7. ( pt) set/p-8.pg Suppose y = e /x + /e x. Find D x y. D x y = -*exp(/(x**))/(x**3) - *x/exp(x**) 8. ( pt) set/p-9.pg A curve is given by the equation: y = (e x + ). Find the slope of the tangent line at the point (, 5) ( pt) set/p-.pg The region bounded by y = e x, y =, x =, and x = is revolved about the y-axis. Find the volume of the resulting solid. Answer: Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

2 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment due 7/6/8 at :59pm MDT Polar Coordinates and Calculus. This assignment will cover the material from Sections ( pt) set/p-.pg Find the area of the region inside: r = 5sinθ but outside: r = ( pt) set/p-.pg Find the area of the region bounded by the given curve: r = 5e θ on the interval 6 5 π θ π ( pt) set/p-3.pg Find the area of the region bounded by: r = 6 sinθ ( pt) set/p-4.pg A circle C has center at the origin and radius 6. Another circle K has a diameter with one end at the origin and the other end at the point (,4). The circles C and K intersect in two points. Let P be the point of intersection of C and K which lies in the first quadrant. Let (r,θ) be the polar coordinates of P, chosen so that r is positive and θ π/. Find r and θ. r = θ = ( pt) set/p-5.pg Find the area inside the inner loop of the following limacon: r = 7 4sinθ ( pt) set/p-6.pg Find the exact length of the polar curve described by: on the interval 5 π θ 9π. r = 5e θ Note, you can use the standard arclength formula found in problem 6 on page ( pt) set/p-8.pg A curve with polar equation r = 37 3sinθ + 6cosθ represents a line. This line has a Cartesian equation of the form y = mx + b,where m and b are constants. Give the formula for y in terms of x. For example, if the line had equation y = x+3 then the answer would be *x + 3. (37/3) - ( 6/3)*x 8. ( pt) set/p-9.pg Find the length of the curve r = θ from θ = to θ =. Note, you can use the standard arclength formula found in problem 6 on page ( pt) set/p-.pg Find the slope of the tangent to the curve r = 5cosθ at the value θ = π/.5

3 . ( pt) set/p-9.pg Match each polar equation below to the best description. Possible answers are C,E,H,L,P,R,S,V,and. DESCRIPTIONS C. Circle centered at origin, E. Ellipse, H. Hyperbola, L. Line neither vertical nor horizontal, P. Parabola, R. Circle not centered at origin, S. Spiral, V. Vertical Line,. Horizontal Line POLAR EQUATIONS. r = 4. r = 6cosθ 3. r = sinθ 4 4. r = 6sinθ 4 5. r = 4sinθ+6cosθ V H R L Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

4 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment due 7/3/8 at :59pm MDT Linear DEs and Applications. This assignment will cover the material from Sections ( pt) set/p-.pg Solve the following differential equation: y 3y y = ; y =,y = at x = Answer: y(x) =. (/7)*exp(5*x) - (5/7)*exp(-*x). ( pt) set/p-.pg Solve the following differential equation: y + y + 5y = Answer: y(x) = C +C. NOTE: The order of your answers is important in this problem. For example, webwork may expect the answer A+B but the answer you give is B+A. Both answers are correct but webwork will only accept the former. exp(-5*x) x*exp(-5*x) 3. ( pt) set/p-3.pg Solve the following differential equation: y + 9y = ; y = 3, y = 3 at x = π/3 Answer: y(x) =. -sin(3*x)-3*cos(3*x) 4. ( pt) set/p-4.pg Solve the following differential equation: y + y + y = Answer: y(x) = C +C. NOTE: The order of your answers is important in this problem. For example, webwork may expect the answer A+B but the answer you give is B+A. Both answers are correct but webwork will only accept the former. exp(-x/)*cos(sqrt(3)*x/) exp(-x/)*sin(sqrt(3)*x/) 5. ( pt) set/p-5.pg Solve the following differential equation: y y + y = and express your answer in the form ce αx sin(βx + γ) Answer: α =, β =. 6. ( pt) set/p-6.pg Use the method of undetermined coefficients to solve the following differential equation: y + y = 4x Answer: y(x) = +C +C. NOTE: The order of your answers is important in this problem. For example, webwork may expect the answer A+B but the answer you give is B+A. Both answers are correct but webwork will only accept the former. *x** - 4*x exp(-x) 7. ( pt) set/p-7.pg Use the method of undetermined coefficients to solve the following differential equation: y + 6y + 9y = e x Answer: y(x) = +C +C. NOTE: The order of your answers is important in this problem. For example, webwork may expect the answer A+B but the answer you give is B+A. Both answers are correct but webwork will only accept the former. exp(-x)/ exp(-3*x) x*exp(-3*x) 8. ( pt) set/p-8.pg Let y be the solution of the initial value problem y + y + y =,y() =,y () = The maximum value of y, for t >, is

5 Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

6 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment due 5/8/8 at :59pm MDT Exponential Growth and Decay, Inverse Functions, Circular (Trigonometric) and Hyperbolic functions and their inverses. This assignment will cover the material from Sections ( pt) set/p-.pg In 6 days unknown radioactive substance decay to 44 percent of its size. (a) What is the half life of this substance? t = (days) (b) How long will it take for a sample of mg to decay to 8 mg? T = ( pt) set/p-3.pg A bacteria culture starts with 8 bacteria and grows at a rate propotional to its size. After hours there are 36 bacteria. (a) Find the population after t hours y(t) = (function of t) (b) Find the population after 5 hours. y(5) = (c) When will the population reach 73? T = 8*( ˆ( *t)) ( pt) set/p-4.pg 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 = log (.3P). Find the variation in pressure caused by a rock band at 5 decibels. Answer: pounds per square inch. 4. ( pt) set/p-5.pg The count in a bacteria culture was 5 after minutes and 4 after 3 minutes. What was the initial size of the culture? Find the doubling period in minutes. Find the population after minutes. When (in minutes) will the population reach 4? ( pt) set/p-5a.pg The Hustler Bank Mutual Fund pays interest at a rate of 5.3%, compounded continuously. How much should be invested so as to have thousand dollars in 6 years? ( pt) set/p-6.pg The rat population in a major metropolitan city is given by the formula n(t) = 8e.35t 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? ( pt) set/p-7.pg The half-life of Radium-6 is 59 years. If a sample contains 3 mg, how many mg will remain after 3 years? ( pt) set/p-9.pg Solve the inital value problem for y(x); xy + 7y = x 4 with the initial condition: y() = 8. y(x) = (8 - (/(7+4)))/xˆ{7} + (/(7+4))*xˆ{4}

7 9. ( pt) set/p-.pg Use logarithmic differentiation to find dy/dx, where y = y y = (x 4) 3 (sin(x)) 4 (x x) e x3 ( -3/(x-4) + 4*cos(x)/sin(x) - *(*x-)/(x*x-*x) - 3*x*x 4. () pt) set/p3-.pg Let. ( pt) set/srw 7a.pg (a) If f is one-to-one and f ( 3) =, then f f (x) = 8sin(x)sin () = and (x) ( f ( 3)) =. f (x) = (b) If g is one-to-one and g( 3) = 9, then g (9) = and NOTE: The webwork system will accept arcsin(x) and not (g( 3)) =. sin (x) as the inverse of sin(x) ( pt) set/mec6.pg Let f (x) = + x + 3e x f (5) =. ( pt) set/osu tr 4.pg Simplify the expression answer = tan ( cos (x/4) ) *x*sqrt(4ˆ-xˆ)/(*xˆ-4ˆ) The next three problems deal with two new functions. The first is called the HYPERBOLIC SINE FUNCTION and is denoted as sinh(x). The second is called the HYPERBOLIC COSINE FUNC- TION and is denoted as cosh(x). These two functions are both defined using either the difference or sum of exponential functions and then dividing by : sinh(x) = ex e x 3. ( pt) set/srw4 33.pg sinh() = cosh() = cosh(x) = ex + e x 8*(cos(x)*arcsin(x) + sin(x)/sqrt(-x*x)) 5. ( pt) set/ur in.pg Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list. Enter the appropriate letter (A,B,C,D, or E) in each blank. A. tan(arcsin(x/3)) B. cos(arcsin(x/3)) C. (/) sin( arcsin(x/3)) D. sin(arctan(x/3)) E. cos(arctan(x/3)) x. 9 x x x x x 4. 9 x 9 9 x 5. 3 A E B Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

8 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment 3 due 6/4/8 at :59pm MDT Integration by Substitution, Trigonometric Integrals, Integration by Parts. This assignment will cover the material from Sections ( pt) set3/p3-.pg Let f (x) = f (x) = tan (sin(8x)) 8*cos(8*x)/(+(sin(8*x))**). ( pt) set3/p3-3.pg Evaluate the definite integral x dx [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan(x) for tan x, sin(x) for sinx. ] ( pt) set3/p3-4.pg Evaluate the indefinite integral. 6x x 4 + dx + C [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan(x) for tan x, sin(x) for sinx. ] 3 * arctan(xˆ) 4. ( pt) set3/p3-5.pg Let f (x) = x 4 tan (8x) f (x) = NOTE: The WeBWorK system will accept arctan(x) but not tan (x) as the inverse of tan(x). 4*x**(4-)*arctan(8*x) + x**4*8/(+8*8*x**) 5. ( pt) set3/p3-6.pg Evaluate the integral f (x) = f (x) = arcsin((x-)/8) dx 8 + x x + C 6. ( pt) set3/p3-7.pg Perform the following integration: e x e x e x dx + e x Answer: ( pt) set3/p3-8.pg Perform the following integration: x 4x + 9 dx Answer: + C. arctan((x-)/sqrt(5))/sqrt(5) 8. ( pt) set3/p3-9.pg Perform the following integration: cos 3 x dx Answer: + C. sin(x)-((sin(x))**3)/3 9. ( pt) set3/sc5 5.pg Evaluate the indefinite integral. cos (86x)dx + C *x/ + *sin(*86*x)/(4*86). ( pt) set3/c4s5p3.pg π/3 Find the value of sin(x) sin(x) dx

9 . ( pt) set3/c4s5p4.pg π/ Find the value of cos(x) sin(sin(x))dx ( pt) set3/sc pg Evaluate the definite integral ( pt) set3/osu in 5 7.pg π/4 sin 4 (4x)dx = ( pt) set3/sc5 5 9.pg Evaluate the indefinite integral. + C 8 sin (8x)cos (8x)dx x + 3 x + 6x dx / * ln(abs(xˆ + 6 * x)) 5. ( pt) set3/sc pg Verify that x = ( x ) x + and use this equation to evaluate x dx 6. ( pt) set3/sc5 6 6.pg First make a substitution and then use integration by parts to evaluate the integral. x 3 cos(x 7 )dx +C /7 * xˆ7 * sin(xˆ7) + /7 * cos(xˆ7) 7. ( pt) set3/sc5 6 4.pg A particle that moves along a straight line has velocity v(t) = t e 3t meters per second after t seconds. How many meters will it travel during the first t seconds? - e**(- 3 * t)*(t**/3 + *t/9 + /7) + /7 8. ( pt) set3/osu in 5 3.pg Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the definite integral /8 xsin (8x)dx The first step in evaluating this integral is to apply integration by parts: udv = uv vdu where u = and dv = h(x)dx where h(x) = Note: Use arcsin(x) for sin (x). /8 After integrating by parts, we obtain the integral f (x)dx on the right hand side where /8 vdu = f (x) = The most appropriate substitution to simplify this integral is x = g(t) where g(t) = Note: We are using t as variable for angles instead of θ, since there is no standard way to type θ on a computer keyboard. After making this substitution and simplifying (using trig identities), we obtain the integral k(t) = a = b = b a k(t) dt where

10 After evaluating this integral and plugging back into the integration by parts formula we obtain: /8 xsin (8x)dx = asin(8*x) x 8*xˆ/(*sqrt(-8ˆ*xˆ)) (sin(t))/8 (sin(t))ˆ/(*8ˆ) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

11 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment 4 due 6//8 at :59pm MDT Integration Rational Functions by Substitutions and Partial Fractions, and Strategies for Integration. This assignment will cover the material from Sections ( pt) set4/p4-.pg Evaluate the definite integral. 5sin( π 7 ) x 3 5 x dx 5. ( pt) set4/p4-5.pg Find the integral Answer: xln(x)dx +C x**.*ln(*x)/-x**./4. [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan(x) for tan x, sin(x) for sinx... ] ( pt) set4/p4-.pg Evaluate the definite integral. dx x ( pt) set4/p4-6.pg Use integration by parts to evaluate the integral. 4xcos4xdx +C [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan(x) for tan x, sin(x) for sinx... ] ( pt) set4/p4-3.pg Evaluate the indefinite integral. x x dx +C [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan(x) for tan x, sin(x) for sinx... ] (x-6)*((*x-x**)**.5)/ + 6*6*arcsin((x-6)/6)/ 4. ( pt) set4/p4-4.pg Use integration by parts to evaluate the integral. xe 4x dx 4 *.5 * (x * sin(4 * x) +.5 * cos(4 * x)) 7. ( pt) set4/p4-7.pg Evaluate the following integral cos(lnx) dx Answer: + C. (x/)*( cos(ln(x))+sin(ln(x)) ) 8. ( pt) set4/p4-8.pg Evaluate the integral. (x 3)(x + ) dx +C +C.5 * (x * eˆ(4 * x) -.5 * eˆ(4 * x)) (ln(x + ))/(-5) - (ln(x - 3))/(-5)

12 9. ( pt) set4/p4-9.pg Write out the form of the partial fraction decomposition of the function appearing in the integral: x 34 x + 5x 84 dx Determine the numerical values of the coefficients, A and B, where A B. A = B = ( pt) set4/p4-.pg Evaluate the integral. A denominator + B denominator ( pt) set4/invtrigs.pg Evaluate the definite integral. 7 x + 5 x + 5x + 6 dx 89 + x dx dx. (64 + x ) 3 dx. (64 x ) 3/ 3. x 64 + x dx 4. x dx 64 x 5. (x 64) 5/ dx A B A B 3. ( pt) set4/p-3.pg Evaluate the following integral..5 sin x dx Note, enter your answer symbolically. Answer: ( pt) set4/p3-.pg Evaluate the following integral: t t + dt Answer: ( pt) set4/ur in 5.pg The form of the partial fraction decomposition of a rational function is given below. [NOTE: Remeber to enter all necessary *, (, and )!! Enter arctan(x) for tan x, sin(x) for sinx... ] ( pt) set4/ur in.pg For each of the indefinite integrals below, choose which of the following substitutions would be most helpful in evaluating the integral. Enter the appropriate letter (A,B, or C) in each blank. DO NOT EVALUATE THE INTEGRALS. A. x = 8tanθ B. x = 8sinθ C. x = 8secθ 5x + 3x 6 (x 5)(x + 9) = A Bx +C + x 5 x + 9 A = B = C = Now evaluate the indefinite integral. 5x + 3x 6 (x 5)(x + 9) dx + C *ln(abs(x+-5))+3*arctan(x/3)/3

13 6. ( pt) set4/osu in 5 9a.pg Consider the integral x 8x 4 + x 7 (x 3 6x + 5x) 3 (x 4 65) dx Enter a T or an F in each answer space below to indicate whether or not a term of the given type occurs in the general form of the complete partial fractions decomposition of the integrand. A,A,A 3... and B,B,B 3,... denote constants. You must get all of the answers correct to receive credit B 4 (x+5) 3 B 6 (x 5) 3 A x+b (x +5) B x+ A 8 x+b 8 (x 5) B 5 (x 5) A 3 x+b 3 (x 5) B 7 (x+5) F T T F F T F T Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

14 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment 5 due 6/8/8 at :59pm MDT Indeterminate Limits and Improper Integrals. This assignment will cover the material from Sections ( pt) set5/p-6.pg Evaluate the definite integral. 4 e 4 dx x lnx. ( pt) set5/p5-.pg Find the indicated limit. Make sure that you have an indeterminate form before you apply l Hopital s Rule. lim x π/ cosx π x =. Instruction: If your answer is, enter INF ; if it is, enter -INF. 3. ( pt) set5/p5-.pg Find the indicated limit. Make sure that you have an indeterminate form before you apply l Hopital s Rule. 5. ( pt) set5/p5-4.pg Find the indicated limit. Make sure that you have an indeterminate form before you apply l Hopital s Rule. lim x + x sinx x =. Instruction: If your answer is, enter INF ; if it is, enter -INF. -INF 6. ( pt) set5/p5-5.pg Find lim x xcos(ax) tan(bx) =. Instruction: If your answer is, enter INF ; if it is, enter -INF. Note that b must be nonzero for the expression to be defined. /b 7. ( pt) set5/p5-6.pg Evaluate the limit 3 + 8x lim x 8x lim x x 3 3x + x x 3 x =. Instruction: If your answer is, enter INF ; if it is, enter -INF ( pt) set5/p5-3.pg Find the indicated limit. Make sure that you have an indeterminate form before you apply l Hopital s Rule. lim x sinx tanx x sinx =. Instruction: If your answer is, enter INF ; if it is, enter -INF ( pt) set5/p5-7.pg Evaluate the limit ( pt) set5/p5-8.pg Evaluate 4x 3 6x 6x lim x 9 x 6x 3 lim x x 4 5x 3 9 7x

15 . ( pt) set5/p5-9.pg Determine the infinite limit of the following functions. Enter INF for and -INF for. lim (x π/)tan(x) = x π/ π/ x lim x π/ cos(x) = x lim x + ln(x/) = ln(x/) lim x (x ) = - -INF. ( pt) set5/p5-9a.pg Determine the limit of the following functions. Enter INF for and -INF for. lim x xlnx = lim lnx = x x. ( pt) set5/p5-9b.pg Determine the limit of the following function. Enter INF for and -INF for. lim x + x e x = INF dx x3/ 5. ( pt) set5/p6-4.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. INF 3 dx x.3 6. ( pt) set5/p6-.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as divergent. 9 dx (x + 3) 3/ 3. ( pt) set5/p5-.pg Evaluate the following limit. If needed, enter INF for and -INF for. 5 ( ) lim x + x + x = x + 4. ( pt) set5/p6-.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV ( pt) set5/p6-3.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it is divergent, enter your answer as x + dx

16 8. ( pt) set5/p6-5.pg Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF (without the quotation marks). If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV. 5 3 x 5 dx diverge. ( pt) set5/p6-9.pg Find the area of the region under the curve y = x + x to the right of x =. Answer: ( pt) set5/p6-6.pg Evaluate the following improper integral: x + x dx Answer:. If the integral diverges, enter diverge as answer. diverge. ( pt) set5/p6-7.pg Evaluate the following improper integral: xe x dx Answer:. If the integral diverges, enter diverge as answer ( pt) set5/p6-8.pg Evaluate the following improper integral: dx 4 (π x) /3 Answer:. If the integral diverges, enter diverge as answer. 3. ( pt) set5/p6-.pg Evaluate the following improper integral: e lnx x Answer:. If the integral diverges, enter diverge as answer. diverge dx 4. ( pt) set5/osu in 4.pg Find the indicated integrals (if they exist) x 4x + 5dx = e 5x e x + dx = 5x + 6 4x + x + 5 dx = +C ln(x) x 5 dx = +C +C (/4ˆ3)*((/7)*(4*x+5)ˆ(7/)-(4/5)*5*(4*x+5)ˆ(5/)+(/3)*5ˆ ln(4*x+)/4 + ln(x+5) (xˆ-4/-4)*(ln(x)-/-4) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester 3

17 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment 6 due 6/5/8 at :59pm MDT Sequences and Series. This assignment will cover the material from Sections ( pt) set6/p7-.pg Determine the sum of the following series ( pt) set6/p7-.pg Consider the sequence () n n= 8 n 4. ( pt) set6/p7-4.pg Determine whether the sequences are increasing, decreasing, or not monotonic. If increasing, enter as your answer. If decreasing, enter - as your answer. If not monotonic, enter as your answer.. a n = n 5 n+5. a n = 5n+8 3. a n = cosn 4. a n = 5 n n+5 8n ( pt) set6/p7-5.pg Determine the sum of the following series. 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 = n= ( 4n + 6 n n ) 6. ( pt) set6/p7-6.pg If the following series converges, compute its sum. Otherwise, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, and DIV otherwise. n= 9 n(n + ) (Hint: try breaking the summands up partial fractions-style.) ( pt) set6/p7-3.pg 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 7. ( pt) set6/p7-8.pg Match each of the following with the correct statement. C stands for Convergent, D stands for Divergent.. n= ne n. n= 3. n= 4+3n 7 n 4. 9 n= 9 6nln(n) n 7 +n 6 n 4 n n=

18 8. ( pt) set6/p7-9.pg A ball is dropped from a height of 8 feet. Each time it hits the floor, it rebounds to its previous height. Find the total 3 distance it travels before coming to rest. Answer: feet ( pt) set6/p7-.pg Decide the convergence or divergence of the following series: Answer: convergent k= ( ) 3 k. π (Enter convergent or divergent.) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

19 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment 7 due 6/3/8 at :59pm MDT Convergence Tests. This assignment will cover the material from Sections ( pt) set7/p8-.pg Use the Integral Test to decide the convergence or divergence of the following series: Answer: converge. ( pt) set7/p8-.pg k k= e k. (Enter converge or diverge.) Use the Integral Test to decide the convergence or divergence of the following series: Answer: divergent k k= + k 3. (Enter convergent or divergent.) 3. ( pt) set7/p8-3.pg Determine the sum of the following series () n n= 8 n 4. ( pt) set7/p8-4.pg Determine the convergence or divergence of the following series. A. convergent B. divergent A n + n= n 5. ( pt) set7/p8-5.pg Determine whether the following series is n= ( ) n+ A. absolutely convergent B. conditionally convergent. divergent A 5n. 6. ( pt) set7/p8-6.pg Match each of the following with the correct statement. C stands for Convergent, D stands for Divergent.. n= 4+ n 3. n= + 9+6n n 3. 5 n= n(n+7) ln(n) 3n 5 n n= 5. n= 7. ( pt) set7/p8-7.pg Select the FIRST correct reason on the list why the given series converges. A. Geometric series. B. Comparison with a convergent p series. C. Integral test. D. Ratio test. E. Alternating series test.. n=. n= 3. n= 4. n= en (cos(nπ) ln(4n) sin (4n) n (n+)(4 ) n 4 n n! n + n n n= 6. n= ( e π )n E B B A

20 8. ( pt) set7/p8-8.pg Select the FIRST correct reason on the list why the given series converges. A. Geometric series. B. Comparison with a convergent p series. C. Integral test. D. Ratio test. E. Alternating series test n= n= n= n= n= n(ln(n)) sin (7n) n (n + )(5) n 4(4) n 6 n 4 n n + n n 4 5 ( ) n ln(e n ) n 4 cos(nπ) 6. n= B A B B 9. ( pt) set7/p8-9.pg Select the FIRST correct reason why the given series diverges. A. Diverges because the terms don t have limit zero B. Divergent geometric series C. Divergent p series D. Integral test E. Comparison with a divergent p series F. Diverges by limit comparison test G. Diverges by alternating series test. n= n. n= ( )n (n)! 3. n= (n!) nln(n) ln(n) n 4. n= 5. n= 4n+3 ( ) n cos(nπ) ln(5) 6. n= A A A. ( pt) set7/p8-.pg Here are some series and sequences. Enter the letter C if there is convergence, and the letter D if not n!. lim n (n)! n. lim n n 3n n + 3. lim n 4n 3 + n! 4. n= (n)! n 5. n n= n= n= n= n(n )(n 3) n n(ln(n)) n 3 + (n + )(n + ) Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

21 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment 8 due 7//8 at :59pm MDT Power and Taylor Series. This assignment will cover the material from Sections ( pt) set8/p9-.pg Find the interval of convergence for the given power series. (x ) n n= n( 6) n The series is convergent from x =, left end included (Y,N): to x =, right end included(y,n): -5 N 7 Y. ( pt) set8/p9-.pg Match each of the power series with its interval of convergence n= n= n= n= (x 8) n (n!)8 n n!(x 8) n 8 n (x 8) n (8) n (x) n n 8 A. (,) B. {8/} C. [, ] D. (,6) A B 3. ( pt) set8/p9-3.pg 7x Suppose that (+x) = n= c nx n. Find the first few coefficients. c = c = c = c 3 = c 4 = Find the radius of convergence R of the power series. R = ( pt) set8/p9-3a.pg The function f (x) = ( 8x) is represented as a power series f (x) = n= c nx n. Find the first few coefficients in the power series. c = c = c = c 3 = c 4 = Find the radius of convergence R of the series. R = ( pt) set8/p9-4.pg Find a formula for the sum of the series for 5 < x < 5. /((5-x)ˆ) (n + )x n 5 n+

22 6. ( pt) set8/p9-5.pg Find the power series representation for f (x) = ( + x) and specify the radius of convergence. (Note: To determine e n uniquely, we require a n is positive and e n is either n or n. ) 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- 7. ( pt) set8/p9-6.pg Find the power series representation for f (x) = xe x. f (x) = n= a n! xp n, where a n = and p n =. n *n+ 8. ( pt) set8/p9-7.pg Find the power series representation for x tan t f (x) = dt. t f (x) = n= ( ) e n a n x p n, where e n =, and a n =, and p n =. n- /((*n-)**) *n- 9. ( pt) set8/p9-8.pg Find the sum of n= n(n + )xn = for < x <. *x/((-x)**3) -. ( pt) set8/p9-9.pg Find the terms through x 5 in the Maclaurin series for f (x) = (x 3 x + )e x. f (x) = +O(x 6 ). -x + (3/)*xˆ + (/3)*xˆ3 - (9/4)*xˆ4 + (9/)*xˆ5. ( pt) set8/p-.pg x Let F(x) = sin(t ) dt. Find the MacLaurin polynomial of degree 7 for F(x). Use this polynomial to estimate the value of.78 sin(x ) dx. * xˆ3 / 3 - ˆ3 * xˆ7 / Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

23 Hsiang-Ping Huang Math -9, Summer 8 WeBWorK Assignment 9 due 7/9/8 at :59pm MDT Conics. This assignment will cover the material from Sections..4.. ( pt) set9/p-4.pg The parabola y = x + x has its focus at the point (b,c) where b = c = ( pt) set9/p-5.pg The ellipse 6x + x + y = has its center at the point (b,c) where b = c = The length of the major diameter of this ellipse is ( pt) set9/p-6.pg Determine the distance D between the vertices of 9x + 8x + 4y + 4y 9 =. G. Hyperbola, EQUATIONS. 4x 4y + 8x + y 5 =. 9x + 4y + 7x 6y + 4 = 3. 9x + 4y + 7x 6y + 6 = 4. 3x + 3y 6x + y + 6 = 5. x + y x + y + = 6. y 5x 4y 6 = 7. x + xy + y 6 = 8. 4x 3xy 8 = F B E A B G 5. ( pt) set9/p-8.pg A bridge underpass in the shape of an elliptical arch, that is, half of an ellipse, is 3 feet wide and 4 feet high. An eight foot wide rectangular truck is to drive (safely) underneath. How high can it be? h = ( pt) set9/p-.pg Match each polar equation below to the best description. Each answer should be C,F,I,L,M,O,or T. D = 6 4. ( pt) set9/p-7.pg Match each equation below to the curve it represents. answer should be A, B, C, D, E, F, or G. CURVES A. Circle, B. Ellipse, C. Point, D. Parabola, E. Empty Set, F. Intersecting lines, Each DESCRIPTIONS C. Cardioid, F. Rose with four petals, I. Inwardly spiraling spiral, L. Lemacon, M. Lemniscate, O. Outwardly spiraling spiral, T. Rose with three petals POLAR EQUATIONS. r = 4 + 4cosθ. r = 6 6sinθ 3. r = cosθ 4. r = 4sinθ 5. r = 6cos3θ 6. r = 6θ,r > M F T O

24 Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester

25 WeBWorK demonstration assignment The main purpose of this WeBWorK set is to familiarize yourself with WeBWorK. Here are some hints on how to use WeBWorK effectively: After first logging into WeBWorK change your password. Find out how to print a hard copy on the computer system that you are going to use. Print a hard copy of this assignment. Get to work on this set right away and answer these questions well before the deadline. Not only will this give you the chance to figure out what s wrong if an answer is not accepted, you also will avoid the likely rush and congestion prior to the deadline. The primary purpose of the WeBWorK assignments in this class is to give you the opportunity to learn by having instant feedback on your active solution of relevant problems. Make the best of it! 5. ( pt) setdemo/demo pr5.pg Enter here Enter here a + + b a + b c + d the expression the expression 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 pr6.pg Enter here the expression a + b. ( pt) setdemo/demo pr.pg Evaluate the expression 9(9 5) =. 36 Enter here Enter here a a + b a + b a + b the expression the expression. ( pt) setdemo/demo pr.pg Evaluate the expression /(8 + 5) =. 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 pr3.pg Let r = 7. Evaluate 4/π r =. Next, enter the expression 4/(π r) = WorK compute the result ( pt) setdemo/demo pr4.pg Enter here the expression a + b. Enter here the expression a+b. /a+/b /(a+b) and let WeB- sqrt(a+b) a/sqrt(a+b) (a+b)/sqrt(a+b) 7. ( pt) setdemo/demo pr7.pg Enter here Enter here Enter here 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

26 8. ( pt) setdemo/demo pr8.pg 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) 9. ( pt) setdemo/demo pr9.pg Simplify the expression sin (x) (cos(x))ˆ. ( pt) setdemo/demo pr.pg Evaluate the expression ( + ) sqrt() Generated by the WeBWorK system c WeBWorK Team, Department of Mathematics, University of Rochester