Find the vertex, focus, and directrix of the parabola and sketch its graph. (y – 2)2 = 2x + 1
> Determine the type of curve represented by the equation in each of the following cases: (d) Show that all the curves in parts (a) and (b) have the same foci, no matter what the value of k is. y² = 1 k'k – 16 (a) k> 16 (b) ) < k< 16 (c) k < 0
> Find an equation for the ellipse with foci (1, 1) and (21, 21) and major axis of length 4.
> Show that the function defined by the upper branch of the hyperbola y2/a2 - x2/b2 = 1 is concave upward.
> Use the definition of a hyperbola to derive Equation 6 for a hyperbola with foci (±c, 0) and vertices (±a, 0).
> The LORAN (LOng RAnge Navigation) radio navigation system was widely used until the 1990s when it was superseded by the GPS system. In the LORAN system, two radio stations located at A and B transmit simultaneous signals to a ship or an aircraft located
> A 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. C A 5 cm
> The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude 110 km and apolune altitude 3
> Find an equation for the conic that satisfies the given conditions. Hyperbola, foci (2, 0), (2, 8), asymptotes y = 3 + }x and y = 5 – x
> Find an equation for the conic that satisfies the given conditions. Hyperbola, vertices (±3, 0), asymptotes y = ±2x
> Test the series for convergence or divergence. 1/n .2
> Find an equation for the conic that satisfies the given conditions. Нуperbola, vertices (— 1, 2), (7, 2), foci (-2, 2), (8, 2)
> Find an equation for the conic that satisfies the given conditions. Нурerbola, vertices (-3, —4), (-3, 6), foci (-3, –7), (-3, 9)
> Find an equation for the conic that satisfies the given conditions. Hyperbola, vertices (0, ±2), foci (0, ±5)
> Find an equation for the conic that satisfies the given conditions. Hyperbola, vertices (±3, 0), foci (±5, 0)
> Find an equation for the conic that satisfies the given conditions. Ellipse, foci (+4, 0), passing through (-4, 1.8)
> Find an equation for the conic that satisfies the given conditions. Ellipse, center (-1,4), vertex (-1, 0), focus (-1,6)
> Find an equation for the conic that satisfies the given conditions. Ellipse, foci (0, –1), (8, – 1), vertex (9, – 1)
> Find an equation for the conic that satisfies the given conditions. Ellipse, foci (0, 2), (0, 6), vertices (0, 0), (0, 8)
> Find an equation for the conic that satisfies the given conditions. Ellipse, foci (0, ±/2), vertices (0, ±2)
> Find an equation for the conic that satisfies the given conditions. Ellipse, foci (±2, 0), vertices (±5, 0)
> Test the series for convergence or divergence. k In k (k + 1)° k-1 3
> Use a computer algebra system to find the Taylor polynomials Tn centered at a for n = 2, 3, 4, 5. Then graph these polynomials and f on the same screen. f(x) — сot x, а—п/4
> 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. (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 orbit’s 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
> Test the series for convergence or divergence. 1 A 2 + sin k
> (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)"
