[SOLVED] Test the series for convergence or divergence. /

> Find an equation for the conic that satisfies the given conditions. Parabola, vertical axis, passing through (0, 4), (1, 3), and (-2, –6)

> Find an equation for the conic that satisfies the given conditions. Parabola, vertex (3, – 1), horizontal axis, passing through (-15, 2)

> Find an equation for the conic that satisfies the given conditions. Parabola, focus (2, – 1), vertex (2, 3)

> Find an equation for the conic that satisfies the given conditions. Parabola, focus (-4, 0), directrix x 2

> Find an equation for the conic that satisfies the given conditions. Parabola, focus (0, 0), directrix y = 6

> Identify the type of conic section whose equation is given and find the vertices and foci. x? – 2x + 2y? – 8y + 7=0

> Identify the type of conic section whose equation is given and find the vertices and foci. Зх? — 6х — 2у —1

> Identify the type of conic section whose equation is given and find the vertices and foci. y? – 2 = x? – 2.x

> Identify the type of conic section whose equation is given and find the vertices and foci. 4y 2y²

> Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. 9у? — 4x? — 36у — 8х — 4 4.x² 36у - 8x %3D

> Test the series for convergence or divergence. n? + 1 Σ %23 -1 IM

> Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. x? – y? + 2y = 2

> Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. y? – 16x? = 16

> Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. x² – y? = 100 .2 %3|

> Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. .2 y? 36 64

> Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. 2 1 25 9

> Find the vertices and foci of the ellipse and sketch its graph. x² + 3y² + 2x – 12y + 10 = 0

> Find the vertices and foci of the ellipse and sketch its graph. 9x² – 18x + 4y² = 27

> Find the vertices and foci of the ellipse and sketch its graph. 100x? + 36y? = 225

> Find the vertices and foci of the ellipse and sketch its graph. x² + 9y² = 9

> Find the vertices and foci of the ellipse and sketch its graph. .2 y? 1 36 8

> Test the series for convergence or divergence. n! n-

> Find the vertices and foci of the ellipse and sketch its graph. y? .2 2 4

> Find an equation of the parabola. Then find the focus and directrix. 1 2.

> Find an equation of the parabola. Then find the focus and directrix. 1 -2

> Find the vertex, focus, and directrix of the parabola and sketch its graph. 2x2 – 16x - 3y +38 = 0

> Find the vertex, focus, and directrix of the parabola and sketch its graph. y2 + 6y + 2x +1 = 0

> Find the vertex, focus, and directrix of the parabola and sketch its graph. (y – 2)2 = 2x + 1

> Find the vertex, focus, and directrix of the parabola and sketch its graph. (x+2)2 = 8(y - 3)

> Find the vertex, focus, and directrix of the parabola and sketch its graph. 3x2 + 8y = 0

> Find the vertex, focus, and directrix of the parabola and sketch its graph. 2x = -y2

> Find the vertex, focus, and directrix of the parabola and sketch its graph. 2y2 = 5x

> Test the series for convergence or divergence. n sin(1/n) n-1

> Find the vertex, focus, and directrix of the parabola and sketch its graph. x2 = 6y

> Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r > 0 and one with r (a) (1, 7/4) (b) (-2, 37/2) (c) (3, –7/3)

> Using the data from Exercise 29, find the distance traveled by the planet Mercury during one complete orbit around the sun. (If your calculator or computer algebra system evaluates definite integrals, use it. Otherwise, use Simpson’s Rule.) Data from Ex

> The distance from the dwarf planet Pluto to the sun is 4.43 × 109 km at perihelion and 7.37 × 109 km at aphelion. Find the eccentricity of Pluto’s orbit.

> The planet Mercury travels in an elliptical orbit with eccentricity 0.206. Its minimum distance from the sun is 4.6 × 107 km. Find its maximum distance from the sun.

> Comet Hale-Bopp, discovered in 1995, has an elliptical orbit with eccentricity 0.9951. The length of the orbitâs major axis is 356.5 AU. Find a polar equation for the orbit of this comet. How close to the sun does it come? Dean Ket

> The orbit of Halley’s comet, last seen in 1986 and due to return in 2061, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is 36.18 AU. [An astronomical unit (AU) is the mean distance between the earth and the s

> Jupiter’s orbit has eccentricity 0.048 and the length of the major axis is 1.56 × 109 km. Find a polar equation for the orbit

> The orbit of Mars around the sun is an ellipse with eccentricity 0.093 and semi major axis 2.28 × 108 km. Find a polar equation for the orbit.

> Test the series for convergence or divergence. E tan(1/n) n-1

> Show that a conic with focus at the origin, eccentricity e, and directrix y = -d has polar equation ed r = 1- e sin0

> Show that a conic with focus at the origin, eccentricity e, and directrix y = d has polar equation ed r = 1 + e sin 0

> Show that a conic with focus at the origin, eccentricity e, and directrix x = -d has polar equation ed 1- e cos 0

> (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. 4 r = 2 + 3 cos 0

> (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. 3 r = 4 - 8 cos 0

> (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. 1 3 – 3 sin0

> (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. 6 + 2 cos 0

> (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. 5 r = 2 – 4 cos 0

> (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. r = 3 + 3 sin 0

> (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. 1 r = 2 + sin 0

> (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. 4 r = 5 - 4 sin 0

> Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity 2, directrix r = -2 sec0

> Write a polar equation of a conic with the focus at the origin and the given data. Parabola, vertex (3, 7/2)

> Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity 0.6, directrix r = 4 csc 0

> Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity, vertex (2, 1) TT

> Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity 3, directrix x 3

> Test the series for convergence or divergence. E(-1)" cos(1/n²) n-

> Write a polar equation of a conic with the focus at the origin and the given data. Hyperbola, eccentricity 1.5, directrix y = 2

> Write a polar equation of a conic with the focus at the origin and the given data. Parabola, directrix x = -3

> Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity , directrix x= 4

> The size of an undisturbed fish population has been modeled by the formula 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 / then / is convergent and / (b) If a1 = 1 and / find the first eight terms of the sequence /. Then use part (a) to show that /This gives the continued fraction expansion lima-a Aza L and lim,o arn+1 = L,

> Let a and b be positive numbers with a > b. Let a1 be their arithmetic mean and b1 their geometric mean: Repeat this process so that, in general, (a) Use mathematical induction to show that (b) Deduce that both / are convergent. (c) Show that / Gauss

> Test the series for convergence or divergence. K – 1 Σ k(VR + 1) k-1

> (a) Use a graph to guess the value of the limit (b) Use a graph of the sequence in part (a) to find the smallest values of N that correspond to / = 0.1 and / = 0.001 in Definition 2. n' lim 0 n!

> (a) Let a1 = a, a2 = f(a), a3 = f(a2) = f(f(an)) ,……. , = an+1 = f(an), where f is a continuous function. If / (b) Illustrate part (a) by taking / and estimating the value of L to five decimal places.

> (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

> Show that the sequence defined by satisfies 0 1 aj = 2 an+1 3 - an

> Show that the sequence defined by is increasing and an aj = 1 An+1 = 3 an

> Find the limit of the sequence 2, /2, v2 /2 /2

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? а, — п3 — Зп + 3 n

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? а, — 3 — 2пе 2ne

> Test the series for convergence or divergence. In n E(-1)":

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? (-1)" а, — 2 + n

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? а, — п(-1)"

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 1-n an 2 + n

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? 1 an 2n + 3

> Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? а, cos n

> Find the first 40 terms of the sequence defined by and a1 = 11. Do the same if a1 = 25. Make a conjecture about this type of sequence. if a, is an even number | 3a, + 1 if a, is an odd number

> 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) that the catfish population Pn after n months is given recursively by (b) How many catfish are in the pond afte

> Test the series for convergence or divergence. (-1)"-1 Vn - 1 R-2

> If you deposit $100 at the end of every month into an accountÂ that pays 3% interest per year compounded monthly, the amount of interest accumulated after n months is given by the sequence (a) Find the first six terms of the sequence. (b) How

> If $1000 is invested at 6% interest, compounded annually, then after n years the investment is worth an = 1000(1.06)n dollars. (a) Find the first five terms of the sequence /. (b) Is the sequence convergent or divergent? Explain.

> (a) Determine whether the sequence defined as follows is convergent or divergent: (b) What happens if the first term is a1 = 2? aj = 1 An+1 = 4 - a, for n> 1

> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 699 for advice on graphing sequences

> Test the series for convergence or divergence. • 3. 5. .... (2n – 1) 2.5.8... (Зп — 1) (Зл — R-1

> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. f(x) = tan-'x, а—1

> Explain why species that overlap a great deal in their fundamental niches have a high probability of competing. Now explain why species that overlap a great deal in their realized niches and live in the same area probably do not compete significantly.

> Researchers have characterized the niches of Darwin’s finches by beak size (which correlates with diet) and the niches of salt marsh grasses by position in the intertidal zone. How would you characterize the niches of sympatric canid species such as red

> Explain how self-thinning in field populations of plants can be used to support the hypothesis that intraspecific competition a common occurrence among natural plant populations is.

> How can the results of greenhouse experiments on competition help us understand the importance of competition among natural populations? How can a researcher enhance the correspondence of results between greenhouse experiments and the field situation?

Question: Test the series for convergence or divergence. /

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