Question: Determine whether the geometric series is
Determine whether the geometric series is convergent or divergent. If it is convergent, find its
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> 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
> Evaluate the integral. ( In Va dx
> 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?
> 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
> Evaluate the integral. dx e*/1 – e-2x
> (a) A direction field for the differential equation y’ = x2 – y2 is shown. Sketch the solution of the initial-value problem Use your graph to estimate the value of y(0.3). (b) Use Euler’s method wit
> Find a polar equation for the curve represented by the given Cartesian equation. y =x
> Evaluate the integral. dx 4x + e
> Four bugs are placed at the four corners of a square with side length a. The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach the center of the square along spiral paths. (a) Find
> Evaluate the integral. Vx + 1 dx Vx - 1
> Use Exercise 52 to find Data from Exercise 52: Use integration by parts to prove the reduction formula. fx*e* dx. x"e*dx = x"e* – n |x" 'e*dx
> Determine whether the sequence converges or diverges. If it converges, find the limit. an sin(1/n)
> Use series to evaluate the limit. sin x – x + a lim .3
> Evaluate the integral. 2 dx X1
> Evaluate the integral. |x sin x cos x dx
> Evaluate the integral. 3x – x? + 6x – 4 dx (x² + 1)(x² + 2)
> Barbara weighs 60 kg and is on a diet of 1600 calories per day, of which 850 are used automatically by basal metabolism. She spends about 15 cal/kg/day times her weight doing exercise. If 1 kg of fat contains 10,000 cal and we assume that the storage of
> Suppose the model of Exercise 22 is replaced by the equations (a) According to these equations, what happens to the insect population in the absence of birds? (b) Find the equilibrium solutions and explain their significance. (c) The figure shows the pha
> The transport of a substance across a capillary wall in lung physiology has been modeled by the differential equation where h is the hormone concentration in the bloodstream, t is time, R is the maximum transport rate, V is the volume of the capillary, a
> Find a polar equation for the curve represented by the given Cartesian equation. y = 2
> Determine whether the series is convergent or divergent by expressing sn as a telescoping sum. If it is convergent, find its sum. 1 Σ -2 n
> The Brentano-Stevens Law in psychology models the way that a subject reacts to a stimulus. It states that if R represents the reaction to an amount S of stimulus, then the relative rates of increase are proportional: where k is a positive constant. Find
> Determine whether the sequence converges or diverges. If it converges, find the limit. 1+3. +3n an
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 2 3**'4 R-I
> (a) Sketch a direction field for the differential equation y = x/y. Then use it to sketch the four solutions that satisfy the initial conditions / / (b) Check your work in part (a) by solving the differential equation explicitly. What type of curve is ea
> One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the we
> Use Exercise 51 to find Data from Exercise 51: Use integration by parts to prove the reduction formula. S (In x)³ dx. | (In x)*dx = x(In x)" – n (In x)* ' dx
> (a) The population of the world was 6.1 billion in 2000 and 6.9 billion in 2010. Find an exponential model for these data and use the model to predict the world population in the year 2020. (b) According to the model in part (a), when will the world popu
> Use series to evaluate the limit. 1 lim 1-0 1 e* cos x + x -
> (a) Write the solution of the initial-value problem and use it to find the population when t = 20. (b) When does the population reach 1200? dP 0.1P(1 dt P P(0) = 100 2000
> Find the orthogonal trajectories of the family of curves. y = ek*
> Find the orthogonal trajectories of the family of curves. y = ke"
> Solve the initial-value problem y’ = 3x2 ey, y(0) = 1 and graph the solution.
> Determine whether the series is convergent or divergent by expressing sn as a telescoping sum. If it is convergent, find its sum. 1/n 1/(n+ R-1
> Identify the curve by finding a Cartesian equation for the curve. r?sin 20 = 1
> Solve the initial-value problem. ху' — у — х In х, у(1) — 2
> Solve the initial-value problem.
> (a) A direction field for the differential equation y’ = y(y-2) (y-4) is shown. Sketch the graphs of the solutions that satisfy the given initial conditions. (b) If the initial condition is y(0) = c, for what values of c is / finite? W
> Find the length of the curve y - i VJi - I di 1<x< 16
> Determine whether the series is absolutely convergent or conditionally convergent. n E (-1)"-1 п? + 4 n-1
> Use Simpson’s Rule with n = 10 to estimate the area of the surface obtained by rotating the sine curve in Exercise 7 about the x-axis. Data from Exercise 7: Use Simpson’s Rule with n = 10 to estimate the length of the sine curve /
> Use series to evaluate the limit. - In(1 + x) lim .2
> (a) The curve y = x2, 0 < x < 1, is rotated about the y-axis. Find the area of the resulting surface. (b) Find the area of the surface obtained by rotating the curve in part (a) about the x-axis.
> Determine whether the sequence converges or diverges. If it converges, find the limit. {sin n}
> Determine whether the series is convergent or divergent by expressing sn as a telescoping sum. If it is convergent, find its sum. Σ n + 1
> Let C be the arc of the curve y = 2/(x + 1) from the point (0,2) to (3, 1 2 ). Use a calculator or other device to find the value of each of the following, correct to four decimal places. (a) The length of C (b) The area of the surface obtained by rot
> Identify the curve by finding a Cartesian equation for the curve. r?cos 20 = 1
> (a) Find the length of the curve (b) Find the area of the surface obtained by rotating the curve in part (a) about the y-axis. 1 y = 16 2x? 1<x< 2
