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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Determine a reduced fraction that has this decimal expansion.0.222 . . .
Prove the conditional convergence of 1 + 1|2 + 1 - + In + + - 3 100
Find the Taylor series centered at c and the interval on which the expansion is valid. f(x) = 1 3x-2' c = -1
Use Eq. (2) to expand the function in a power series with center c = 0 and determine the interval of convergence. 1-x Σ #=0 for [x] < 1
Compute the Taylor polynomial indicated and use the Error Bound to find the maximum possible size of the error. Verify your result with a calculator.ƒ(x) = x11/2, a = 1; |ƒ(1.2) − T4(1.2)|
Use the Integral Test to determine whether the infinite series converges. 00 n=1 n² (n³ + 1)1.01
Determine a reduced fraction that has this decimal expansion.0.454545 . . .
Use the Limit Comparison Test to prove convergence or divergence of the infinite series. η3 Σ n=2 Vn? + 2n2 + 1
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.dn = √n + 3 − √n
Use the Root Test to determine convergence or divergence (or state that the test is inconclusive). Σ(2+1) n n= -n
Compute the Taylor polynomial indicated and use the Error Bound to find the maximum possible size of the error. Verify your result with a calculator. f(x)=x-¹/2, a = 4; f(4.3) - T3(4.3)|
Differentiate the power series in Exercise 39 to obtain a power series for Data From Exercise 39Use Eq. (2) to expand the function in a power series with center c = 0 and determine the interval of convergence. _g(x) = 3x² (1-x³)²
Use the identity cos2 x = 1/2 (1 + cos 2x) to find the Maclaurin series for ƒ(x) = cos2 x.
Use the Integral Test to determine whether the infinite series converges. 00 n=1 1 (n + 2)(ln(n + 2))
Determine a reduced fraction that has this decimal expansion.0.313131 . . .
Use the Limit Comparison Test to prove convergence or divergence of the infinite series. 3η + 5 | n(n − 1)(n – 2) Σ M=3
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.cn = 1.01n
Show that for |x| tanh-1 x = x+ X 3 5
Prove thatconverges for all exponents a. Show that ƒ(x) = (ln x)a/x is decreasing for x sufficiently large. 00 Σ(−1)n+1 (Inn)a n n=1
Differentiate the power series in Exercise 40 to obtain a power series for Data From Exercise 40Use Eq. (2) to expand the function in a power series with center c = 0 and determine the interval of convergence. g(x): = 4x³ (1-x4)²*
Compute the Taylor polynomial indicated and use the Error Bound to find the maximum possible size of the error. Verify your result with a calculator.ƒ(x) = √1 + x, a = 8; | √9.02 − T3(8.02)|
Use the Integral Test to determine whether the infinite series converges. 00 n=1 n³ 4 ent
Determine a reduced fraction that has this decimal expansion.0.217217217 . . .
Use the Limit Comparison Test to prove convergence or divergence of the infinite series. 00 n=1 en + n e2n -n²
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.bn = e1−n2
We say that {bn} is a rearrangement of {an} if {bn} has the same terms as {an} but occurring in a different order. Show that if {bn} is a rearrangement of {an} and converges absolutely, then also converges absolutely. (This result does not hold if is only conditionally convergent.) Prove that
Prove that diverges. Use 2n2 = (2n)n and n! ≤ nn. 00 n=1 2n² n!
Calculate the Maclaurin polynomial T3 for ƒ(x) = tan−1 x. Compute T3 (1/2) and use the Error Bound to find a bound for the error Ιtan−1 1/2 − T3 (1/2)Ι.Refer to the graph in Figure 8 to find an acceptable value of K. Verify your result by computing Ιtan−1 − T3 (1/2)Ι using a
Use the Direct Comparison or Limit Comparison Test to determine whether the infinite series converges. 00 Σ n=1 1 (n + 1)2
Use the Maclaurin series for ln(1 + x) and ln(1 − x) to show thatfor |x| Data From Exercise 42Show that for |x| In 1+x 1-x =x+ + 3
Determine a reduced fraction that has this decimal expansion.0.123333333 . . .
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.an = 21/n
Use the Limit Comparison Test to prove convergence or divergence of the infinite series. Σ n=1 1 √n + Inn
Let ƒ(x) = ln(x3 − x + 1). The third Taylor polynomial at a = 1 isFind the maximum possible value of |ƒ(1.1) − T3(1.1)|, using the graph in Figure 9 to find an acceptable value of K. Verify your result by computing |ƒ(1.1) − T3(1.1)| using a calculator. 7 T3(x) = 2(x - 1) + (x -
Determine convergence or divergence using any method covered in the text so far. 8 00 2n +4" 7n Σ n=1
In 1829, Lejeune Dirichlet pointed out that the great French mathematician Augustin Louis Cauchy made a mistake in a published paper by improperly assuming the Limit Comparison Test to be valid for nonpositive series. Here are Dirichlet’s two series:Explain how they provide a counterexample to
Determine convergence or divergence using any method covered in the text so far. 00 Σ n=1 n 2n + 1
Use the method of Exercise 45 to expand 1/(4 − x) in a power series with center c = 5. Determine the interval of convergence.Data From Exercise 45Use the equalities to show that for |x − 4| 1 1 - x 1 -3-(x-4) -3-3-3 1 + (34)
The repeating decimal 0.012345678901234567890123456789 . . . can be expressed as a fraction with denominator 1,111,111,111. What is the numerator?
Use the method of Example 10 to show that 00 k=1 1 k1/3 diverges.
Use the Direct Comparison or Limit Comparison Test to determine whether the infinite series converges. 00 3 n=1 10 n¹ + 10" n11 + 11"
Find a power series that converges only for x in [2, 6).
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.
Determine convergence or divergence using any method covered in the text so far. 00 Σ n=1 It 5"
Consider(a) Show that the series converges.(b) Use the inequality in (4) from Exercise 83 of Section 11.3 with M = 99 to approximate the sum of the series. What is the maximum size of the error?Data From Exercise 83 of Section 11.3Data From Exercise 81 00 Σ n=1 n (n² + 1)²
Express the definite integral as an infinite series and find its value to within an error of at most 10−4. S 0 e-x³ dx е
Find a power series
Use Theorem 2 to prove that the ƒ(x) is represented by its Maclaurin series for all x.ƒ(x) = (1 + x)100 THEOREM 2 Let I = (c-R,c +R), where R > 0, and assume that f is infinitely differentiable on 1. Suppose there exists K> 0 such that all derivatives of f are bounded by K on I: |f(k)(x)| ≤
Show, by giving counterexamples, that the assertions of Theorem 1 are not valid if the series n=0 00 an and bn are not convergent. n=0
To estimate the length θ of a circular arc of the unit circle, the seventeenth-century Dutch scientist Christian Huygens used the approximation θ ≈ (8b − a)/3, where a is the length of the chord A̅C̅ of angle θ and b is the length of the chord A̅ B̅ of angle θ/2 (Figure 12).(a) Prove
We estimate integrals using Taylor polynomials. Exercise 72 is used to estimate the error.(a) Compute the sixth Maclaurin polynomial T6 for ƒ(x) = sin(x2) by substituting x2 in P(x) = x − x3/6, the third Maclaurin polynomial for ƒ(x) = sin x.(b) Show that | sin(x2) − T6(x)| ≤|x|10/5!.
Apply the Root Test to determine convergence or divergence, or state that the Root Test is inconclusive. 00 n=1 COS n n3
Use Eq. (3) to find the length of the path over the given interval, and verify your answer using geometry.(3t − 1, 2 − 2t), 0 ≤ t ≤ 5 / T = S= x(1)² + y'(1)² di
Use Eq. (3) to find the length of the path over the given interval, and verify your answer using geometry.(1 + 5t, t − 5), −3 ≤ t ≤ 3 ob. = f* x(1)² + y(t)² di a S=
Use Eq. (3) to find the length of the path over the given interval.(2t2, 3t2 − 1), 0 ≤ t ≤ 4 = S= b. x(1)² + y²(1)² di
Use Eq. (3) to find the length of the path over the given interval.(3t, 4t3/2), 0 ≤ t ≤ 1 = S= ·b. DI x(1)² + y'(1)² di
Use Eq. (3) to find the length of the path over the given interval.(3t2, 4t3), 1 ≤ t ≤ 4 = S= ·b. DI x(1)² + y(1)² di
Use Eq. (3) to find the length of the path over the given interval.(t3 + 1, t2 − 3), 0 ≤ t ≤ 1 = S= ·b. DI x(1)² + y²(1)² di
Use Eq. (3) to find the length of the path over the given interval.(sin 3t, cos 3t), 0 ≤ t ≤ π = S= ·b. DI x(1)² + y²(1)² di
Use Eq. (3) to find the length of the path over the given interval.(sin θ − θ cos θ, cos θ + θ sin θ), 0 ≤ θ ≤ 2 = S= ·b. DI x(1)² + y(1)² di
Find the length of the path. The following identity should be helpful:(2 cos t − cos 2t, 2 sin t − sin 2t), 0 ≤ t ≤ π/2 1 - cost 2 sin² 2
Find the length of the path. The following identity should be helpful:(5(θ − sin θ), 5(1 − cos θ)), 0 ≤ θ ≤ 2π 1 - cost 2 sin² 2
Find the length of the spiral c(t) = (t cos t, t sin t) for 0 ≤ t ≤ 2π to three decimal places (Figure 7). S √₁ + 1² dt = 2/1 V₁ + 1² + ⁄2 ln(t + V1 +t²) -10 5 -10 y t=0 t = 2π 17. 10 X
Show that one arch of a cycloid generated by a circle of radius R has length 8R.
Find the length of the parabola given by c(t) = (t, t2) for 0 ≤ t ≤ 1. See the hint for Exercise 12.Data From Exercise 12Find the length of the spiral c(t) = (t cos t, t sin t) for 0 ≤ t ≤ 2π to three decimal places (Figure 7). S √₁ + 1² dt = 2/1 V₁ + 1² + 1⁄2 ln(t + V1 +
Find a numerical approximation to the length of c(t) = (cos 5t, sin 3t) for 0 ≤ t ≤ 2π (Figure 8). -X
Determine the speed ds/dt at time t (assume units of meters and seconds).(t3, t2), t = 2
Determine the speed ds/dt at time t (assume units of meters and seconds).(3 sin 5t, 8 cos 5t), t = π/4
Determine the speed ds/dt at time t (assume units of meters and seconds).(5t + 1, 4t − 3), t = 9
Determine the speed ds/dt at time t (assume units of meters and seconds).(ln(t2 + 1), t3), t = 1
Determine the speed ds/dt at time t (assume units of meters and seconds).(t2, et), t = 0
Determine the speed ds/dt at time t (assume units of meters and seconds).(sin−1 t, tan−1 t), t = 0
Find the minimum speed of a particle with trajectory c(t) = (t3 − 4t, t2 + 1) for t ≥ 0. It is easier to find the minimum of the square of the speed.
Find the minimum speed of a particle with trajectory c(t) = (t3, t−2) for t ≥ 0.5.
Find the speed of the cycloid c(t) = (4t − 4 sin t, 4 − 4 cos t) at points where the tangent line is horizontal.
Calculate the arc length integral s(t) for the logarithmic spiral c(t) = (et cos t, et sin t).
Plot the curve and use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate its length.c(t) = (cos t, esin t) for0≤ t ≤ 2π
Plot the curve and use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate its length. The ellipse ( + - ( 3 )² = 1
Plot the curve and use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate its length.c(t) = (t − sin 2t, 1 − cos 2t) for0≤ t ≤ 2π
Let a > b and set Use a parametric representation to show that the ellipse has length L = 4aG(π/2, k), whereis the elliptic integral of the second kind. k = 1- 62 ม
If you unwind thread from a stationary circular spool, keeping the thread taut at all times, then the endpoint traces a curve C called the involute of the circle (Figure 9). Observe that PQ has length Rθ. Show that C is parametrized by c(θ) = R(cos θ + θ sin θ), R(sin θ − θ cos θ) Then
Plot the curve and use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate its length.x = sin 2t, y = sin 3t for 0 ≤ t ≤ 2π
Use Eq. (4) to compute the surface area of the given surface.The cone generated by revolving c(t) = (t, mt) about the x-axis for 0 ≤ t ≤ A [°* S = 2π y(t)√√x'(t)² +y' (t)² dt
Use Eq. (4) to compute the surface area of the given surface.A sphere of radius R [°* y(t)√√x'(t)² + y' (t)² dt S = 2π
Use Eq. (4) to compute the surface area of the given surface.The surface generated by revolving the curve c(t) = (t2, t) about the x-axis for 0 ≤ t ≤ 1 [°* y(t)√√x'(t)² + y' (t)² dt S = 2π
Use Eq. (4) to compute the surface area of the given surface.The surface generated by revolving the curve c(t) = (t, et) about the x-axis for 0 ≤ t ≤ 1 S = 2π * y(t)√√x'(t)² + y'(t)² dt
Use Eq. (4) to compute the surface area of the given surface.The surface generated by revolving the curve c(t) = (sin2 t, cos2 t) about the x-axis for 0 ≤ t ≤ π/2 [°* y(t)√√x'(t)² + y' (t)² dt S = 2π
Use Eq. (4) to compute the surface area of the given surface.The surface generated by revolving the curve c(t) = (t, sin t) about the x-axis for 0 ≤ t ≤ π [°* y(t)√√x'(t)² + y'(t)² dt S = 2π
Use Eq. (4) to compute the surface area of the given surface.The surface generated by revolving one arch of the cycloid c(t) = (t − sin t, 1 − cos t) about the x-axis [°* y(t)√√x'(t)² + y'(t)² dt S = 2π
The surface generated by revolving the astroid c(t) = (cos3 t, sin3 t) about the x-axis for 0 ≤ t ≤ π/2
Use Simpson’s Rule and N = 30 to approximate the surface area of the surface generated by revolving c(t) = (t2, e−t), 0 ≤ t ≤ 2 about the x-axis.
Let b(t) be the “Butterfly Curve”:(a) Use a computer algebra system to plot b(t) and the speed s'(t) for 0 ≤ t ≤ 12π.(b) Approximate the length b(t) for 0 ≤ t ≤ 10π. x(t) = sint eco cost-2 cos 4t - sin (1/2)) (12 ª(12)²) ( y(t) = cost ecost - 2 cos 4t - sin ost(ecosi
Use Simpson’s Rule and N = 50 to approximate the surface area of the surface generated by revolving c(t) = ((t + 1)3, ln t), 1 ≤ t ≤ 5 about the x-axis.
The acceleration due to gravity on the surface of the earth isUse Exercise 43(b) to show that a satellite orbiting at the earth’s surface would have period Te = 2π √Re/g ≈ 84.5 minutes. Then estimate the distance Rm from the moon to the center of the earth. Assume that the period of the
Let a ≥ b > 0 and set k = 2√ab/a − b. Show that the trochoid x = at − b sin t, y = a − b cos t, 0 ≤ t ≤ T has length 2(a − b)G(T/2, k), with G(θ, k) as in Exercise 30.Data From Exercise 30 Let a b and set k= Use a parametric representation to show that the ellipse ()+(² = 1 has
Find the coordinates at times t = 0, 2, 4 of a particle following the path x = 1 + t3, y = 9 − 3t2.
Use the table of values to sketch the parametric curve (x(t), y(t)), indicating the direction of motion. t X y -3 -2 -1 -15 0 3 5 0 12 3 15 5 0-30 0 -3 -4 -3 0
Find the coordinates at t = 0, π/4, π of a particle moving along the path c(t) = (cos 2t, sin2 t).
Graph the parametric curves. Include arrows indicating the direction of motion. (a) (t, t), -∞o < t < ∞ (t,t), (b) (sin t, sint), 0≤t≤ 2π (c) (e¹,e¹), -∞ < t < ∞ (d) (t³, ³), -1≤t≤1
Give two different parametrizations of the line through (4, 1) with slope 2.

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