1.99
See Answer

** >** Test the series for convergence or divergence.
E (-1)"+'ne

** >** We have seen that the harmonic series is a divergent series whose terms approach 0. Show that is another series with this property.
E In 1 +

** >** Test the series for convergence or divergence.
n? E(-1)**1. n3 + 4 n-1

** >** Test the series for convergence or divergence.
E (-1)" 2n + 3 n-1

** >** Test the series for convergence or divergence.
E (-1)"e ™

** >** If f(0) = t(0) = 0 and f 0 and t0 are continuous, show that
"S9g"(x) dx = f(a)g\(a) – f'(a)g(a) + [' r"(x)g(x) dx %3D

** >** Use a Maclaurin series to obtain the Maclaurin series for the given function.
f(x) V2 + x

** >** Test the series for convergence or divergence.
n? E (-1)": n' п? + п +1 n-1

** >** Test the series for convergence or divergence.
Зп — 1 2 (-1)", 2n + 1

** >** Test the series for convergence or divergence.
(-1)**1 Σ Vn + 1 n-0

** >** Test the series for convergence or divergence.
(-1)* -1 Σ 3 + 5n

** >** Test the series for convergence or divergence.
1 1 1 1 + In 6 1 In 3 In 4 In 5 In 7

** >** Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places.
y = sin x, 0 <I<T

** >** Find the values of x for which the series converges. Find the sum of the series for those values of x.
00 Σ E e" n-0

** >** Use the Root Test to determine whether the series is convergent or divergent.
Σ 1 + 8

** >** Test the series for convergence or divergence.
-중 +8-우 + #-4 +

** >** Test the series for convergence or divergence.
를-3+ 3-3 + 류-.

** >** Prove part (i) of Theorem 8.

** >** (a) What is an alternating series? (b) Under what conditions does an alternating series converge? (c) If these conditions are satisfied, what can you say about the remainder after n terms?

** >** Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
x? + 4 A can be put in the form В x*(x – 4) х — 4

** >** A particle that moves along a straight line has velocity v(t) − t2e2t meters per second after t seconds. How far will it travel during the first t seconds?

** >** Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
x? + 4 A can be put in the form B C x(x² – 4) x + 2 x - 2

** >** Use Simpson’s Rule with n = 6 to estimate the area under the curve y = ex/x from x = 1 to x = 4.

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
4 + 3 + + + 16

** >** Use the Table of Integrals on Reference Pages 6â€“10 to evaluate the integral.
cos x dx sin?x – 9

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Use integration by parts to prove the reduction formula.
x"e*dx = x"e* – n |x" 'e*dx

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Use the Root Test to determine whether the series is convergent or divergent.
5n -2n Σ n + 1, n-

** >** Evaluate the integral.
- tan 0 1 + tan 0

** >** Express the repeating decimal 4.17326326326... as a fraction.

** >** A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is m, the fuel is consumed at rate r, and the exhaust gases are ejected with constant velocity ve (r

** >** Evaluate the integral.
1* In t dt

** >** Find the sum of the series.
e? 1-е + 2! 3! 4! +

** >** Test the series for convergence or divergence.
Σ 3"

** >** Find the sum of the series.
(-1)" 7" 32"(2n)!

** >** Express the number as a ratio of integers.
0.46 = 0.46464646 . . .

** >** Determine whether the sequence converges or diverges. If it converges, find the limit
2 3 + 5n? 1 +n

** >** In Section 4.8 we considered Newtonâ€™s method for approximating a root r of the equation f(x) = 0, and from an initial approximation x1 we obtained successive approximations x2, x3, . . . , where Use Taylorâ€™s Inequality

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
In n In 2n

** >** Evaluate the integral.
dx J31+ |x|

** >** Evaluate the integral.
/2 cos'x sin 2x dx

** >** Determine whether the series is conditionally convergent, absolutely convergent, or divergent.
E(-1)" 'n-3

** >** Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter.
x = t³ + 6t + 1, y=2t – t2; t=-1

** >** Find a polar equation for the curve represented by the given Cartesian equation. x2 + y2 = 2

** >** Use the Root Test to determine whether the series is convergent or divergent.
(-1)“ 1 Σ (In n)" R-2

** >** Find a polar equation for the curve represented by the given Cartesian equation. x + y = 2

** >** Find the values of x for which the series converges. Find the sum of the series for those values of x.
sin"x o 3"

** >** Evaluate the integral.
x? + 2 dx x + 2

** >** Determine whether the series is convergent or divergent.
Σ 5"

** >** Use the Table of Integrals on Reference Pages 6â€“10 to evaluate the integral.
rV4 - x² dx

** >** Evaluate the integral.
/arctan x dx 1 + x?

** >** Determine whether the series is convergent or divergent.
Σ i n' + 1 R-1

** >** Find the area of the surface obtained by rotating the curve in Exercise 9 about the y-axis. Data from Exercise 9: Find the length of the curve
y - i VJi - I di 1<x< 16

** >** Evaluate the integral.
sin(In t) dt

** >** Evaluate the integral.
dt J 21? + 3t + 1

** >** Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.
n sin n an n2 + 1

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
Σ

** >** Find the values of x for which the series converges. Find the sum of the series for those values of x.
2" Σ o x"

** >** Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.
3. n an 1 + .2 re

** >** Use the Root Test to determine whether the series is convergent or divergent.
п? + 1 2n? + 1

** >** Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.
9n+1 an 10"

** >** Calculate the average value of f(x) = x sec2x on the interval /

** >** Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral.
(4x? – 3 dx; entry 39

** >** Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.
2 + n3 an 1+ 2n3

** >** What does your result from Problem 1 say about the areas A1 and A2 shown in the figure? Data from Problem 1: Suppose the cups have height h, cup A is formed by rotating the curve x = fs/d about the y-axis, and cup B is formed by rotating the same curv

** >** All solutions of the differential equation y’ = -1 – y4 are decreasing functions.

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
2 arctan n n-1

** >** Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
x(x? + 4) x² – 4 A can be put in the form x + 2 B x - 2

** >** Find the values of x for which the series converges. Find the sum of the series for those values of x.
E (-4)"(x – 5)"

** >** Find a formula for the area of the surface obtained by rotating C about the line y = mx + b.

** >** If f is continuous, then /

** >** Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
- dx is convergent.

** >** Use the Ratio Test to determine whether the series is convergent or divergent.
2" n! E (-1)" - 5. 8· 11 · · (3n + 2)

** >** Evaluate the integral.
M dM eM

** >** Calculate the volume generated by rotating the region bounded by the curves y = ln x, y = 0, and x = 2 about each axis. (a) The y axis (b) The x axis

** >** Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
x? - 4 B A + x? + 4' can be put in the form x(x² + 4)

** >** Determine whether the series is absolutely convergent or conditionally convergent.

** >** Show that any tangent line to a hyperbola touches the hyperbola halfway between the points of intersection of the tangent and the asymptotes.

** >** Use the CAS to graph f near x = 0. Does it appear that f has a limit as x→ 0?

** >** Use your computer algebra system to evaluate f(x) for x = 1, 0.1, 0.01, 0.001, and 0.0001. Does it appear that f has a limit as x→ 0?

** >** In view of the answers to Problems 4 and 5, how do you explain the results of Problems 1 and 2? Data from Problems 1: Use your computer algebra system to evaluate f(x) for x = 1, 0.1, 0.01, 0.001, and 0.0001. Does it appear that f has a limit as x→ 0?

** >** More complicated shapes can be represented by piecing together two or more Bézier curves. Suppose the first Bézier curve has control points P0, P1, P2, P3 and the second one has control points P3, P4, P5, P6. If we want these two pieces to join together

** >** Some laser printers use Bézier curves to represent letters and other symbols. Experiment with control points until you find a Bézier curve that gives a reasonable representation of the letter C.

** >** Find dy/dx. x = tet, y = t + sin t

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.

** >** Try to produce a Bézier curve with a loop by changing the second control point in Problem 1. Data from Problem 1: Graph the Bézier curve with control points P0(4, 1), P1(28, 48), P2(50, 42), and P3(40, 5). Then, on the same screen, graph the line segme

** >** Use the Ratio Test to determine whether the series is convergent or divergent.

** >** Use multiplication or division of power series to find the firstÂ three nonzero terms in the Maclaurin series for each function.

** >** Evaluate the integral.

** >** From the graph in Problem 1, it appears that the tangent at P0 passes through P1 and the tangent at P3 passes through P2. Prove it. Data from Problem 1: Graph the Bézier curve with control points P0(4, 1), P1(28, 48), P2(50, 42), and P3(40, 5). Then, o

** >** Graph the BÃ©zier curve with control points P0(4, 1), P1(28, 48), P2(50, 42), and P3(40, 5). Then, on the same screen, graph the line segments P0P1, P1P2, and P2P3. (Exercise 10.1.31 shows how to do this.) Notice that the middle control point

** >** Based on your own measurements and observations, suggest a value for h and an equation for x = f(y) and calculate the amount of coffee that each cup holds.

** >** Use Pappusâ€™s Theorem to explain your result in Problems 1 and 2. Data from Problem 1: Suppose the cups have height h, cup A is formed by rotating the curve x = fs/d about the y-axis, and cup B is formed by rotating the same curve abou

** >** Suppose the cups have height h, cup A is formed by rotating the curve x = f(y) about the y-axis, and cup B is formed by rotating the same curve about the line x = k. Find the value of k such that the two cups hold the same amount of coffee.

** >** Use the Ratio Test to determine whether the series is convergent or divergent.

** >** Evaluate the integral.

** >** Find the values of x for which the series converges. Find the sum of the series for those values of x.

** >** (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylorâ€™s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (c) Check your result in part (b) by graphing /