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calculus

Calculus Early Transcendentals 7th edition James Stewart - Solutions

Write a polar equation of a conic with the focus at the origin and the given data. a. Ellipse, eccentricity ½, directrix x = 4 b. Parabola, directrix x = - 3 c. Hyperbola, eccentricity 1.5, directrix y = 2
(a) Find the eccentricity and directrix of the conic r = 1/(1 - 2 sin () and graph the conic and its directrix? (b) If this conic is rotated counterclockwise about the origin through an angle 3(/4, write the resulting equation and graph its curve?
Graph the conic r = 4/(5 + 6 cos () and its directrix. Also graph the conic obtained by rotating this curve about the origin through an angle (/3?
Shoat that a conic with focus at the origin, eccentricity e, and directrix x = - d has polar equation
Shoat that a conic with focus at the origin, eccentricity e, and directrix y = - d has polar equation
The orbit of Mars around the sun is an ellipse with eccentricity 0.093 and semimajor axis 2.28 ( 108 km. Find a polar equation for the orbit?
The orbit of Halley's comet, last seen in 1986 and due to return in 2062, 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 sun, about 93 million miles.] Find a polar
The planet Mercury travels in an elliptical orbit with eccentricity 0.206. Its minimum distance from the sum is 4.6 ( 107 km. Find the maximum distance from the sun?
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.)
(a) Find the eccentricity (b) Identify me conic (c) give an equation of the directrix, and (d) sketch the conic. R = 4/5-4 sin θ
(a) What is parametric curve? (b) How do you sketch a parametric curve?
Write an expression for each of the following: (a) The length of a parametric curve. (b) The area of the surface obtained by rotating a parametric curve about the x-axis.
(a) How do you find the slope of a tangent line to a polar curve? (b) How do you find the area of a region bounded by a polar curve? (c) How do you find the length of a polar curve?
(a) Give a definition of an ellipse in terms of foci. (b) Write an equation for the ellipse with foci (( c, 0) and vertices (( a, 0) and vertices (( a, 0)
(a) What is the eccentricity of a conic section?(b) What can you say about the eccentricity if the conic section is an ellipse? A hyperbola? A parabola?(c) Write a polar equation for a conic section with eccentricity e and directrix x = d. what if the directrix is x = - d? y = d? y = - d?
Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.a. x = t2 + 4t, y = 2 - t, 1 - 4 ( t ( 1b. x = cos (, y = sec (, 0 ( ( < (/2
The curve with polar equation r = (sin ()/( is called a cochleoid. Use a graph of as a function ( of in Cartesian coordinates to sketch the cochleoid by hand. Then graph it with a machine to check your sketch?
Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter? a. x = 1n t, y = 1 + t2; t = 1 b. r = e-(; ( = (
Find dy/dx and d2 y/dx2. x = t + sin t, y = t - cos t
Use a graph to estimate the coordinates of the lowest point on the curve x = t3 - 3t, y = t2 + t + 1. Then use calculus to find the exact coordinates.
At what points does the curve x = 2a cos t - a cos 2t y = 2a sin t - a sin 2t Here vertical or horizontal tangents? Use this information to help sketch the curve.
Find the area enclosed by the curve r2 = 9 cos 5( ?
Find the points of intersection of the curves r = 2 and r = 4 cos (?
Find the area of the region that lies inside both of the circles r = 2 sin ( and r = sin ( + cos (?
Write three different sets of parametric equations for the curve y = (x?
(a) Plot the point with polar coordinates (4, 2(/3). Then find its Cartesian coordinates? (b) The Cartesian coordinates of a point are (-3, 3). Find two sets of polar coordinates for the point?
Sketch the polar curve. a. r = 1 - cos ( b. r = 1 + cos 2 ( c. r = 3/1 + 2 sin (
(a) What is a sequence? (b) What does it mean to say that limn ( 8 an? (c) What does it mean to say that limn ( ( an?
Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues.a.b. c.
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. a. an = 3n / 1 + 6n b. an = 1 + (- 1/2)n
Determine whether the sequence converges or diverges. If it converges, find the limit. a. an = 1 - (0.2)n b. an = 3 + 5n2 / n + n2 c. an = e1/n
List the first five terms of the sequence. a. an = 2n / n2 + 1 b. an = (- 1)n-1/5n c. an = 1 / (n + 1)! d.a1 = 1, an+1 = 5an - 3
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 695 for advice on graphing sequences.) a. an = 1 + (- 2/e)n b. an = (3 + 2n2 /
If $1000 is invested at 6% interest, compounded annually, then after n years the investment is an = 1000 (1.06) n worth dollars. (a) Find the first five terms of the sequence {an}. (b) Is the sequence convergent or divergent? Explain.
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 months is given recursively by Pn = 1.08Pn-1 - 300 Po = 5000 (b) How many catfish are in the pond after six months?
For what values of r is the sequence {nrn} convergent?
Suppose you know that {an} is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can you say about the value of the limit?
Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a. an = 1 / 2n + 3 b. an = n(- 1)n c. an = n / n2 + 1
Find the limit of the sequence {√2, √2 √2, √2 √2 √2.....}
Show that the sequence defined by a1 = 1, an+1 = 3 - 1/an Is increasing and an < 3 for all n. Deduce that {an} is convergent and find its limit.
(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
(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 N of that correspond to ( = 0.1 and in Definition 2.
Prove Theorem 6.If |an| = 0Then an = 0.
Prove that if lim n → ( an = 0 and {bn} is bounded, then lim n → ( (anbn) = 0.
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 an > an+1 > bn+1 > bn (b) Deduce that both {an} and {bn} are convergent. (c) Show that lim n†’( an =
The size of an undisturbed fish population has been modeled by the formulaWhere is the fish population after years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is Po > 0.(a) Show that if {Pn} is convergent, then the only
If for all x, write a formula for b8.
Find the Taylor series for f(x) centered at the given value of a [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] Also find the associated radius of convergence. a. f(x) = x4 - 3x2 + 1. a = 1 b. f(x) = sin (x c. f(x) = 2x
Prove that the series obtained in Exercise 7 represents sin (x for all x?
Prove that the series obtained in Exercise 11 represents sinh x for all x?
Use the binomial series to expand the function as a power series. State the radius of convergence?
Use a Maclaurin Series in Table 1 to obtain the Maclaurin series for the given function?a.f(x) = sin (xb. f(x) = ex + e2x
If f(n) (0) = (n + 1)! for n = 0, 1, 2, ...., find the Maclaurin series for f and its radius of convergence.
Find the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and f? a. f(x) = cos (x2) b. f(x) = xe - x
Use the Maclaurin series for cos x to compute cos 5O correct to five decimal places?
(b) Use part (a) to find the Maclaurin series for sin- 1 x.
Evaluate the indefinite integral as an infinite series.a.b.
Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] also find the associated radius of convergence.a. f(x) = (1 - x)-2b. f(x) = sin (xc. f(x) = 2x
Use series to approximate the definite integral to within the indicated accuracy.
Use series to evaluate the limit.(a).(b).
Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function.a. y = e-x2 cos xb. y = x/sin x
Find the sum of the series.a.b.c.
Show that if p is an nth-degree polynomial, then
Prove Taylor's Inequality for n = 2, that is, prove that if | f'''(x) | ( M for | x - a | ( d, then|f'''(x)| ( M/6 for |x - a|3 for |x - a| ( d
Use the following steps to prove 17.(b) Let h(x) = (1 + x) -k g(x) and show that h'(x) = 0. (c) Deduce that g(x) = (1 + x)k.
(a)Find the Taylor polynomials up to degree 6 for f(x) = cos x centered at a = 0, Graph f and these polynomials on a common screen. (b) Evaluate f and these polynomials at x = (/4 (/2, and (. (c) Comment on how the Taylor polynomials converge to f(x).
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) = cot x, a = ( / 4
(a) Approximate f by a Taylor polynomial with degree n at the number a.(b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ( Tn (x) when x line in the given interval.(c) Check your result in part (b) by graphing |Rn (x)|.1. f(x) = (x, a = 4, n = 2, 4 ( x ( 4.22. f(x) =
Use the information from Exercise 5 to estimate cos 80O correct to five decimal places?
Use Taylor's Inequality to determine the number of terms of the Maclaurin series for ex that should be used to estimate e0.1 to within 0.00001?
Use the Alternating Series Estimation Theorem or Taylor's Inequality to estimate the range of values of for which the given approximation is accurate to within the stated error. Check your answer graphically.a. sin x ( x - x3/6 (| error | < 0.01)b. arctan x ( x - x3/3 + x5/5 (| error | < 0.05)
Find the Taylor polynomial T3 (x) for the function f centered at the number a. Graph f and T3 on the same screen. a. f(x) = 1/x. a = 2 b. f(x) = cos x, a = (/2 c. f(x) 1n x, a = 1
A car is moving with speed 20 m/s and acceleration 2 m/s2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled during the next minute?
An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are q and - q are and located at a distance d from each other, then the electric field E at the point P in the figure isBy expanding this expression for E as a series in powers of d/D, show that
If a water wave with length L movies with velocity v across a body of water with depth d, as in the figure on page 776. Than(c) Use the Alternating Series Estimation Theorem to show that if L > 10d, then the estimate v2 ( gd is accurate to within 0.014gL.
If a surveyor measures differences in elevation when making plans for a highway across a desert, corrections must be made for the curvature of the earth.(a) If R is the radius of the earth and L is the length of the highway, show that the correction isC = R sec (L/R) - R(b) Use a Taylor polynomial
In Section 4.8 we considered Newton's method for approximating a root r of the equation f(x) = 0, and from an initial approximation we obtained successive approximations x2, x3, ...., whereUse Taylor's Inequality with n = 1, a = xn, and x = r to show that if f'' (x) exists on an interval I
(a) What is the difference between a sequence and a series? (b) What is a convergent series? What is a divergent series?
Let an = 2n / 3n +1. (a) Determine whether {an}is convergent. (b) Determine whether ∑(n-1 an is convergent.
Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.a. 3 - 4 + 16/3 - 64/9 + ...b. 10 - 2 + 0.4 - 0.08 +...c. 6(0.9) n-1
Determine whether the series is convergent or divergent. If it is convergent, find its sum.a. 1/3 + 1/6 + 1/9 + 1/12 + 1/15 + ....b.c.
Calculate the sum of the series whose partial sums ∑(n=1 an whose partial sums are given. Sn = 2 - 3(0.8) n
Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Example 7). If it is convergent, find its sum.a.b.c.
Let x = 0.99999 ....(a) Do you think that x < 1 or x = 1?(b) Sum a geometric series to find the value of x.(c) How many decimal representations does the number 1 have?(d) Which numbers have more than one decimal representation?
Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?a.b.
Express the number as a ratio of integers.a. 0. = 0.8888. . .b. 2. 516 = 2.516516516. . .c. 1.53
Find the values of for which the series converges. Find the sum of the series for those values of x.a.b. c.
Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using the CAS to sum the series directly. $Use the partial fraction command on your CAS to find$
If the nth partial sum of a seriesIs Find an and
A patient takes 150 mg of a drug at the same time every day. Just before each tablet is taken, 5% of the drug remains in the body.(a) What quantity of the drug is in the body after the third tablet? After the nth tablet?(b) What quantity of the drug remains in the body in the long run?
When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government
Find the value of c if (1 + c) - n = 2
In Example 8We showed that the harmonic series is divergent. Here we outline another method, making use of the fact that ex > + x for any x > 0. If Sn is the nth partial sum of the harmonic series, show that eSn >n + 1. Why does this imply that the harmonic series is divergent?
The figure shows two circles C and D of radius 1 that touch at P.T is a common tangent line; C1 is the circle that touches C, D, and T; C2 is the circle that touches C, D, and C1; C3 is the circle that touches C, D, and C2. This procedure can be continued indefinitely and produces an infinite
Prove part (i) of Theorem 8.If ˆ‘ an and ˆ‘ bn are convergent series, then so are the series ˆ‘ can (where c is a constant), ˆ‘(an + bn) and ˆ‘ (an - bn), and(i)
If ∑ an is divergent and c ( 0, show that ∑ can is divergent.
Suppose that a series ∑ an has positive terms and its partial sums Sn satisfy the inequality Sn ≤ 1000 for all n. Explain why ∑ an must be convergent.
The Cantor set, named after the German mathematician Georg Cantor (1845-1918), is constructed as follows. We start with the closed interval [0, 1] and remove the open interval (1/3, 2/3). That leaves the two intervals [0, 1/3] and [2/3, 1] and we remove the open middle third of each. Four intervals
Consider the series(a) Find the partial sums s1, s2, s3, and s4. Do you recognize the denominators? Use the pattern to guess a formula for sn. (b) Use mathematical induction to prove your guess. (c) Show that the given infinite series is convergent, and find its sum.
Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.a.b. c.

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