Question: Evaluate the integral. /
Evaluate the integral.
Transcribed Image Text:
( In Va dx
> Find an equation for the ellipse that shares a vertex and a focus with the parabola x2 + y = 100 and that has its other focus at the origin.
> Find an equation of the hyperbola with foci (0, ±4) and asymptotes y = ±3x.
> Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y - e* In(1 + x)
> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. f) — In x, а — 1
> Find an equation of the ellipse with foci (±4, 0) and vertices (±5, 0).
> Find the foci and vertices and sketch the graph. 25x? + 4y? + 50x – 16y = 59
> Evaluate the integral. dx X - 1
> Find the values of x for which the series converges. Find the sum of the series for those values of x. E (-5)"x"
> Find the foci and vertices and sketch the graph. 4x? - у? — 16 %3D
> Find the foci and vertices and sketch the graph. x2 y? 1 8 9
> Determine whether the sequence converges or diverges. If it converges, find the limit. (-3)" an n!
> The curves defined by the parametric equations are called strophoids (from a Greek word meaning “to turn or twist”). Investigate how these curves vary as c varies. (? — с) y = 1? + 1 1? - c 1? + 1
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. у —е", у— 0, х — —1, х — 0; about x— 1 %3D
> Find the area of the surface obtained by rotating the given curve about the x-axis. x = 2 + 3t, y = cosh 31, 0<i<1
> Find the area of the surface obtained by rotating the given curve about the x-axis. x = 4 T, y=+, x = 4 /t, y=- 3 212"
> Find the length of the curve. r= sin'(0/3), 0< 0<
> Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. x = 2 cos 0, y =1 + sin0 = 1 + sin@
> Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y SI sin x
> Express the number as a ratio of integers. 5.71358
> Evaluate the integral.
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1 Σ 1 + (?)"
> 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. 10" a, = 1 + 9"
> Find the length of the curve. r= 1/0, 7< 0 < 27
> Find the length of the curve. x = 2 + 3t, y = cosh 31, 0< t<1
> Find the length of the curve. x = 312, y = 21°, 0<t<2
> Find the area of the region that lies inside both of the circles /
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. у —е", у—е", х — 1;B about the y-аxis
> At what points does the curve have vertical or horizontal tangents? Use this information to help sketch the curve. X= 2a cos t - a cos 21 y = 2a sin t – a sin 2t
> Find the area enclosed by the loop of the curve in Exercise 27. Data from Exercise 27: 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.
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 3 – 4 + – 4 - +
> Determine whether the sequence converges or diverges. If it converges, find the limit. a„ = In(n + 1) – In n
> 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.
> Find the volume obtained by rotating the region bounded by the curves about the given axis. у — sin x, у — 0, п/2 <x< п; about the x-axis
> Find dy/dx and d2y/dx2. x = 1 + t2, y = t – t3
> Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y - sec x
> Find dy/dx and d2y/dx2. x = t + sint. y = t – cos t
> Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. r = e e "; 0 = T
> Populations of birds and insects are modeled by the equations (a) Which of the variables, x or y, represents the bird population and which represents the insect population? Explain. (b) Find the equilibrium solutions and explain their significance. (c) F
> Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. x = In 1, y = 1 + t²; t=1
> Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis. y = cos(Tx/2), y= 0, 0<x< 1; about the y-axis
> Determine whether the sequence converges or diverges. If it converges, find the limit. {{东春 !!!!!!! ..} 3> 4> 6
> Express the number as a ratio of integers. 1.234567
> A tank contains 100 L of pure water. Brine that contains 0.1 kg of salt per liter enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 6 minutes?
> Determine whether the sequence converges or diverges. If it converges, find the limit. (2n – 1)! (2n + 1)!
> The von Bertalanffy growth model is used to predict the length L(t) of a fish over a period of time. If L is the largest length for a species, then the hypothesis is that the rate of growth in length is proportional to L` 2 L, the length yet to be achiev
> Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y = e * cos x
> Determine whether the sequence converges or diverges. If it converges, find the limit. {0, 1, 0, 0, 1, 0, 0, 0, 1, . }
> Express the number as a ratio of integers. 10.135 = 10.135353535 ...
> Use a graph to find approximate x coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = x у — х Iп(x + 1), у3 Зх — .2
> Solve the initial-value problem. dr + 2tr = r, r(0) = 5 dt
> Determine whether the series converges or diverges. 1/m Σ 1 п
> Solve the differential equation. x?y' – y = 2x'e-1/½
> Solve the differential equation. 2ye "'y' = 2x + 3 /x %3D
> Solve the differential equation. dx 1 - t + x - tx dt
> Determine whether the sequence converges or diverges. If it converges, find the limit. а, — п — уп +1 ул + 3
> Use the Table of Integrals on the Reference Pages to evaluate the integral. cot x - dx 1 + 2 sin x
> Use the Table of Integrals on the Reference Pages to evaluate the integral. cos x /4 + sin²x dx
> Express the number as a ratio of integers. 2.516 = 2.516516516 ...
> Use the Table of Integrals on the Reference Pages to evaluate the integral. | csc't dt
> Use the series in Example 13(b) to evaluate We found this limit in Example 4.4.4 using l’ Hospital’s Rule three times. Which method do you prefer? tan x - x lim
> Graph the function f(x) = cos2x sin3x and use the graph to guess the value of the integral / . Then evaluate the integral to confirm your guess.
> Use a graph to find approximate x­ coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = arcsin n(4x), у — 2 — х?
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). dx /x² + 1
> Determine whether the sequence converges or diverges. If it converges, find the limit. an arctan(In n)
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). In(x² + 2x + 2) dx
> Evaluate the integral or show that it is divergent. tan 'x - dx x?
> Solve the differential equation. sin x y' = xe y cos x
> Test the series for convergence or divergence. E n’e E n²e n-
> Evaluate the integral or show that it is divergent. dx 4x2 + 4x + 5
> Evaluate the integral or show that it is divergent. dx x? 2x
> Evaluate the integral or show that it is divergent. - : dx
> Evaluate the integral or show that it is divergent. ах 2 - Зх · dx
> Use series to evaluate the limit. x3 – 3x + 3 tan 'x lim
> Evaluate the integral or show that it is divergent. In x
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 2" + 4" Σ e"
> Find a polar equation for the curve represented by the given Cartesian equation. x? – y? = 4
> Find the area of the region bounded by the given curves. y - x'e *, y- xe * хе
> Evaluate the integral or show that it is divergent. y dy - 2 /y – 2
> Evaluate the integral or show that it is divergent. dx ½ x In x
> A sequence of terms is defined by aj = 1 а, — (5 — п)а, I Calculate Σa.
> Evaluate the integral or show that it is divergent. In x dx a.
> Evaluate the integral or show that it is divergent. 1 · dx. (2x + 1)'
> Evaluate the integral. Ls Vtan o Ju/4 sin 20 *w OP
> (a) Use Euler’s method with step size 0.2 to estimate y(0.4), where y(x) is the solution of the initial-value problem (b) Repeat part (a) with step size 0.1. (c) Find the exact solution of the differential equation and compare the value
> Evaluate the integral. 2x xe 1/2 dx Jo (1 + 2x)²
> Determine whether the sequence converges or diverges. If it converges, find the limit. an Vn
> Evaluate the integral. ·dx
> Evaluate the integral. (cos x + sin x)? cos 2x dx
> Evaluate the integral. arctan(1/x) dx
> Use series to evaluate the limit. Vī +x - 1 - }x lim x?
> Find the area of the region bounded by the given curves. y = у — х' In x, у— 4 lnx y у — 4 Inx
> 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?
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its (-3)"-| 4"
> Evaluate the integral. dx
> Evaluate the integral. 1 dx Vx + x3/2
> Evaluate the integral. | (arcsin x)° dx
> Determine whether the sequence converges or diverges. If it converges, find the limit. an 1 +
> Evaluate the integral. (4 – x²)/2
> Evaluate the integral. *w/4 X sin x dx cos'x CoS X
> Evaluate the integral. e*Ve* – 1 CIn 10 dx e* + 8
