Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Single Variable Calculus: Early Transcendentals
26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49EA cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is 10 cm. (a) Find an equation of the parabola. (b) Find the diameter of the opening |CD|, 11 cm from the vertex.51E52E53E54E55E56E57E58E59E60EFind the area of the region enclosed by the hyperbola x2/a2 y2/b2 = 1 and the vertical line through a focus.62E63E64E65E66E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E1RCC2RCC3RCC4RCC5RCC6RCC7RCC8RCC9RCC10RCC11RCC12RCC1RQ2RQ3RQ4RQ5RQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 6. If cnxn diverges when x = 6, then it diverges when x = 10.7RQ8RQ9RQ10RQ11RQ12RQ13RQ14RQ15RQ16RQ17RQ18RQ19RQ20RQ21RQ22RQ1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23REDetermine whether the series is conditionally convergent, absolutely convergent, or divergent. 24. n=1(1)n1n325RE26RE27REFind the sum of the series. 28. n=11n(n+3)29RE30RE31REExpress the repeating decimal 4.17326326326 as a fraction.33RE34RE35RE36RE37RE38RE39RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE1P2P(a) Show that tan12x=cot12x2cotx. (b) Find the sum of the series n=112ntanx2n4P5P6P7P8P9P10P11P12P13P14PSuppose that circles of equal diameter are packed tightly in n rows inside an equilateral triangle. (The figure illustrates the case n = 4.) If A is the area of the triangle and An is the total area occupied by the n rows of circles, show that limnAnA=23 FIGURE FOR PROBLEM 1516P17P18P19P20P21P22P23P24P25P26P(a) What is a sequence? (b) What does it mean to say that limn an = 8? (c) What does it mean to say that limn an = ?(a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence ? Give two examples.List the first five terms of the sequence. 3. an=2n2n+1List the first five terms of the sequence. 4. an=n21n2+1List the first five terms of the sequence. 5. an=(1)n15nList the first five terms of the sequence. 6. an=cosn2List the first five terms of the sequence. 7. an=1(n+1)!List the first five terms of the sequence. 8. an=(1)nnn!+19EList the first five terms of the sequence. 10. a1 = 6, an+1=annList the first five terms of the sequence. 11. a1 = 2, an+1=an1+anList the first five terms of the sequence. 12. a1 = 2, a2 = 1, an+1 = an an1Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 13. {12,14,16,18,110,}Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 14. {4,1,14,116,164,}Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 15. {3,2,43,89,1627,}Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 16. {5, 8, 11, 14, 17,}17EFind a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. 18. {1, 0, 1, 0, 1, 0, 1, 0,}Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 19. an=3n1+6nCalculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why. 20. an=2+(1)nn21E22EDetermine whether the sequence converges or diverges. If it converges, find the limit. 23. an=3+5n2n+n2Determine whether the sequence converges or diverges. If it converges, find the limit. 24. an=3+5n21+nDetermine whether the sequence converges or diverges. If it converges, find the limit. 25. an=n4n32nDetermine whether the sequence converges or diverges. If it converges, find the limit. 26. an = 2 + (0.86)27EDetermine whether the sequence converges or diverges. If it converges, find the limit. 28. an=3nn+229EDetermine whether the sequence converges or diverges. If it converges, find the limit. 30. an=4n1+9n31EDetermine whether the sequence converges or diverges. If it converges, find the limit. 32. an=cos(nn+1)33E34E35EDetermine whether the sequence converges or diverges. If it converges, find the limit. 36. an(1)n+1nn+n37E38E39E40E41EDetermine whether the sequence converges or diverges. If it converges, find the limit. 42. an = ln(n + 1) ln nDetermine whether the sequence converges or diverges. If it converges, find the limit. 43. an=cos2n2n44E45EDetermine whether the sequence converges or diverges. If it converges, find the limit. 46. an = 2n cos n47EDetermine whether the sequence converges or diverges. If it converges, find the limit. 48. an=nnDetermine whether the sequence converges or diverges. If it converges, find the limit. 49. an = ln(2n2 + 1) ln(n2 + 1)Determine whether the sequence converges or diverges. If it converges, find the limit. 50. an=(lnn)2nDetermine whether the sequence converges or diverges. If it converges, find the limit. 51. an = arctan(ln n)52E53E54E55E56E57E58E59E60E61E62E63E64E65EIf you deposit 100 at the end of every month into an account that pays 3% interest per year compounded monthly, the amount of interest accumulated after n months is given by the sequence In=100(1.0025n10.0025n) (a) Find the first six terms of the sequence. (b) How much interest will you have earned after two years?A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% per month and the farmer harvests 300 catfish per month. (a) Show that the catfish population Pn after n months is given recursively by Pn=1.08Pn1300P0=5000 (a) How many catfish are in the pond after six months?68E69E70E71E72E73EDetermine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 74. an=1n2+n75E76E77E78E79E80EShow that the sequence defined by a1=1an+1=31an is increasing and an 3 for all n. Deduce that {an} is convergent and find its limit.82E(a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer is fn, where {fn} is the Fibonacci sequence defined in Example 3(c). (b) Let an = fn+1/fn and show that an1 = 1 + 1/an 2. Assuming that {an} is convergent, find its limit.84E85E86E87EProve Theorem 7.89E90E91E92EThe size of an undisturbed fish population has been modeled by the formula pn+1=bpna+pn where pn is the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p0 0. (a) Show that if {pn} is convergent, then the only possible values for its limit are 0 and b a. (b) Show that pn+1 (b/a)pn. (c) Use part (b) to show that if a b, then limn pn = 0; in other words, the population dies out. (d) Now assume that a b. Show that if p0 b a, then {pn} is increasing and 0 pn b a. Show also that if p0 b a, then {pn} is decreasing and pn b a. Deduce that if a b, then limn pn = b a.(a) What is the difference between a sequence and a series? (b) What is a convergent series? What is a divergent series?2ECalculate the sum of the series n=1an whose partial sums are given. 3. sn = 2 3(0.8)nCalculate the sum of the series n=1an whose partial sums are given. 4. sn=n214n2+15E6E7E8E9E10E11E12E13E14ELet an=2n3n+1. (a) Determine whether {an} is convergent. (b) Determine whether n=1an is convergent.16EDetermine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 17. 34+163649+18E