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mathematics
calculus 4th

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

Evaluate using the appropriate method or combination of methods. dx (x²+5)3/2
Use the Error Bound to find a value of N for which Error(SN) ≤ 10−9. 4 So xe dx x
Use the Error Bound to find a value of N for which Error(SN) ≤ 10−9. So NO xp.
For which values of a does ∫∞0 eax dx converge?
Evaluate using the appropriate method or combination of methods. f(x + 1)e+-3x dx
Use the Error Bound to find a value of N for which Error(SN) ≤ 10−9. Ši sin(In x) dx
Show that ∫10 dx xp converges if p < 1 and diverges if p ≥ 1.
Evaluate using the appropriate method or combination of methods. fx x² tan tan-xdx
Evaluate using the appropriate method or combination of methods. x³ cos(x²) dx
Show that 1√x4 + 1 ≤ 1 x2 for all x, and use this to prove that converges. 00€ S dx √x + 1 X
Evaluate using the appropriate method or combination of methods. x²(In x)² dx
Show thatconverges by comparing with 700 J1 dx x³ +4 43
Let ƒ(x) = sin(x2) and I = ∫10 ƒ(x) dx. (a) Check that f'(x) = 2 cos(x²) - 4x² sin(x²). Then show that f"(x)| ≤ 6 for x = [0, 1]. 12 cos(x²)| ≤ 2 and 14x² sin(x²)| ≤ 4 for x € [0, 1]. 1 (b) Show that Error(MN) is at most 4N² (c) Find an N such that I - MN| ≤ 10-³.
Evaluate using the appropriate method or combination of methods. Sx xtanhxdx
Evaluate using the appropriate method or combination of methods. tan-¹t dt 1 + 1² Star
Show thatconverges by comparing with J2 00 dx +3 x³-4
Evaluate using the appropriate method or combination of methods. fin In(x² + 9) dx
The Error Bound for MN is proportional to 1/N2, so the Error Bound decreases by 1/4 if N is increased to 2N. Compute the actual error in MN for ∫π0 sin x dx for N = 4, 8, 16, 32, and 64. Does the actual error seem to decrease by 1/4 as N is doubled?
Prove thatdx converges by comparing with dx (Figure 12). 00% J ·00 e-R
Evaluate using the appropriate method or combination of methods. f(sir (sin x)(cosh x) dx
Observe that the Error Bound for TN (which has 12 in the denominator) is twice as large as the Error Bound for MN (which has 24 in the denominator). Compute the actual error in TN for ∫π0 sin x dx for N = 4, 8, 16, 32, and 64 and compare it with the calculations of Exercise 56. Does the actual
Evaluate using the appropriate method or combination of methods. Sc cosh 2t dt
Explain why the Error Bound for SN decreases by 1/16 if N is increased to 2N. Compute the actual error in SN for ∫π0 sin x dx for N = 4, 8, 16, 32, and 64. Does the actual error seem to decrease by 1/16 as N is doubled?
Show that x2 dx converges. noo 1 - sin x x²
Evaluate using the appropriate method or combination of methods. ’sinh x cosh xdx f sink
Verify that S2 yields the exact value of ∫10 (x − x3) dx.
Verify that S2 yields the exact value of ∫ba (x − x3) dx for all a < b.
The method of partial fractions shows thatThe computer algebra system Mathematica evaluates this integral as −tanh−1 x, where tanh−1 x is the inverse hyperbolic tangent function. Can you reconcile the two answers? dx S₂²³²x₁ = 2₂ m|x-1|-2₂ mm|x +1| +C с
Evaluate using the appropriate method or combination of methods. Scott coth² (1-4t) dt
Use the Comparison Test to determine whether or not the integral converges. 00 1 √x³ +2 dx
Evaluate using the appropriate method or combination of methods. 0.3 dx -0.3 1-x²
Show that if ƒ(x) = rx + s is a linear function (r, s constants), then TN = ∫ba ƒ(x) dx for all N and all endpoints a, b.
Use the Comparison Test to determine whether or not the integral converges. 00 S dx (x³ + 2x + 4)¹/2
Evaluate using the appropriate method or combination of methods. 3√3/2 dx √9 - x²
Use the Comparison Test to determine whether or not the integral converges. J3 00 dx √x-1
For N even, divide [a, b] into N subintervals of width Δx = b − a/N. Set xj = a + j Δx, yj = ƒ(xj), and s2= b-a 3N (V2j + 4y2j+1 + y2j+2)
Evaluate using the appropriate method or combination of methods. √x² + 1dx x²
Use the substitution u = tanh t to evaluate S dt cosh’t+ sinh?t 2
Use the Comparison Test to determine whether or not the integral converges. S dx x1/3 + x3
Use the Comparison Test to determine whether or not the integral converges. 00 S e-(x+x¹) dx
Use the Error Bound for SN to obtain another proof that Simpson’s Rule is exact for all cubic polynomials.
Use the Comparison Test to determine whether or not the integral converges. Jo | sin x| √x dx
Find the volume obtained by rotating the region enclosed by y = ln x and y = (ln x)2 about the y-axis.
Let In = xn dx/x2 + 1.(a) Prove that In = xn−1 − 1 − In−2.(b) Use (a) to calculate In for 0 ≤ n ≤ 5.(c) Show that, in general, 12n+1 = 12n = x²n 2n x2n-2 2n - 2 12 + (−1)²-¹ + (−1)¹¹1 In(x² + 1) + C 2 x2n-1 x2n-3 2n - 1 2n-3 + (-1)"¹x + (-1)" tan¹ x + C
Calculate M10 and S10 for the integral ∫10 √1 − x2 dx, whose value we know to be π/4 (one-quarter of the area of the unit circle).(a) We usually expect SN to be more accurate than MN. Which of M10 and S10 is more accurate in this case?(b) How do you explain the result of part (a)? The
Use the Comparison Test to determine whether or not the integral converges. S ex x² dx
Use the Comparison Test to determine whether or not the integral converges. ~00 S J 1 x4 + ex xp -
Let Jn = ∫xn e−x2/2 dx.(a) Show that J1 = −e−x2/2.(b) Prove that Jn = −xn−1e−x2/2 + (n − 1)Jn−2.(c) Use (a) and (b) to compute J3 and J5.
Use the Comparison Test to determine whether or not the integral converges. Jo 1 x + √√√x Vx dx
Determine whether the improper integral converges and, if so, evaluate it. 00 So dx (x + 2)²
Use the Comparison Test to determine whether or not the integral converges. roo S In x sinh x dx
Determine whether the improper integral converges and, if so, evaluate it. 00 S 4 dx x2/3
Use the Comparison Test to determine whether or not the integral converges. 00% J5 1 : x² ln x dx
Determine whether the improper integral converges and, if so, evaluate it. So dx x2/3
Use the Comparison Test to determine whether or not the integral converges. 00 S dx 1 √x1/3 + x³ x3
Determine whether the improper integral converges and, if so, evaluate it. x12/5 xp S 00%
Use the Comparison Test to determine whether or not the integral converges. dx S Jo (8x² + x4)1/3
Determine whether the improper integral converges and, if so, evaluate it. میر dx x² + 1
Use the Comparison Test to determine whether or not the integral converges. roo S dx (x + x²)1/3
Determine whether the improper integral converges and, if so, evaluate it. S e4x dx
Use the Comparison Test to determine whether or not the integral converges. So dx xex+r
Determine whether the improper integral converges and, if so, evaluate it. -m/2 cot Ꮎ dᎾ
Use the Comparison Test to determine for what values of p this integral converges: noo S 1 xP In - dx. x
Determine whether the improper integral converges and, if so, evaluate it. 00 S. dx (x + 2)(2x + 3)
Determine whether the improper integral converges and, if so, evaluate it. 00€ (5 + x)-1/³ dx
Define as the sum of the two improper integrals Use the Comparison Test to show that J converges. = 50° 10 J = dx x1/2(x + 1)
Determine whether (defined as in Exercise 77) converges.Data From Exercise 77Define as the sum of the two improper integrals Use the Comparison Test to show that J converges. مر J = 0 dx x³/2(x + 1)
Determine whether the improper integral converges and, if so, evaluate it. L (5- 2 (5-x)-1/3 dx
Use the Comparison Test to determine whether the improper integral converges or diverges. 00 S 8/ dx x²-4
An investment pays a dividend of $250/year continuously forever. If the interest rate is 7%, what is the present value of the entire income stream generated by the investment?
Use the Comparison Test to determine whether the improper integral converges or diverges. 00 (sin² x)e* dx
An investment is expected to earn profits at a rate of 10,000e0.01t dollars per year forever. Find the present value of the income stream if the interest rate is 4%.
Use the Comparison Test to determine whether the improper integral converges or diverges. 700 3 dx x4 + cos²x
Compute the present value of an investment that generates income at a rate of 5000te0.01t dollars per year forever, assuming an interest rate of 6%.
When a capacitor of capacitance C is charged by a source of voltage V, the power expended at time t iswhere R is the resistance in the circuit. The total energy stored in the capacitor isShow that W = 1/2 CV2. P(t) = 12 R 5(e-1/RC-e-21/RC)
Use the Comparison Test to determine whether the improper integral converges or diverges. 00€ S dx x1/3 + x2/3
Find the volume of the solid obtained by rotating the region below the graph of y = e−x about the x-axis for 0 ≤ x < ∞.
Use the Comparison Test to determine whether the improper integral converges or diverges. Jo dx x1/3 + x2/3
Use the Comparison Test to determine whether the improper integral converges or diverges. 00 S e-³ dx ex
Let ƒ(x) = e−0.05x(1 + sin x).(a) Obtain a plot of ƒ(x) for 0 ≤ x ≤ 20, and discuss the behavior of the function for positive and increasing x.(b) ∫∞ 0 ƒ(x) dx is the area above the positive x-axis and under the infinitely many “humps” of the graph of ƒ. Compute this area.
Compute the volume of the solid obtained by rotating the region below the graph of y = e−|x|/2 about the x-axis for −∞ < x < ∞.
For which integers p does 1/2 dx x(In x)p converge?
Calculate the volume of the infinite solid obtained by rotating the region under y = (x2 + 1)−2 for 0 ≤ x < ∞ about the y-axis.
Conservation of Energy can be used to show that when a mass m oscillates at the end of a spring with spring constant k, the period of oscillation iswhere E is the total energy of the mass. Show that this is an improper integral with value T = 2π √m/k. T = 4√m √2E/k dx √2E-kx²
Let R be the region under the graph of y = (x + 1)−1 for 0 ≤ x < ∞. Which of the following quantities is finite?(a) The area of R(b) The volume of the solid obtained by rotating R about the x-axis(c) The volume of the solid obtained by rotating R about the y-axis
Use Eq. (6) to calculate (ln x)k dx for k = 2, 3. [(In x)* dx = x(In x)* — k [(In.x)*-1 dx -
Show that ∫∞0 xne−x2 dx converges for all n > 0. First observe that xne−x2 < xne−x for x > 1.Then show that xne−x < x−2 for x sufficiently large.
Estimate ∫52 ƒ(x) dx by computing T2, M3, T6, and S6 for a function ƒ taking on the values in the following table: 2 1 f(x) // 2 X 2.5 3 3.5 4 2 1 0 3 نانا 2 4.5 5 -4 -2
State whether the approximation MN or TN is larger or smaller than the integral. (a) S sin sin x dx dx •St x² (c) (b) (d) 27 S sin x dx In x dx
The rainfall rate (in inches per hour) was measured hourly during a 10-h thunderstorm with the following results:Use Simpson’s Rule to estimate the total rainfall during the 10-h period. 0, 0.41, 0.49, 0.32, 0.3, 0.23, 0.09, 0.08, 0.05, 0.11, 0.12
When a radioactive substance decays, the fraction of atoms present at time t is ƒ(t) = e−kt, where k > 0 is the decay constant. It can be shown that the average life of an atom (until it decays) is A = − ∫ ∞0 t ƒ(t) dt. Use Integration by Parts to show that A = ∫∞0 ƒ(t) dt and
Let Jn = ∫∞0 xn e−αx dx, where n ≥ 1 is an integer and α>0. Prove thatand J0 = 1/α. Use this to compute J4. Show that Jn = n!/αn+1. Jn n -J₁-1 a
Compute the given approximation to the integral. L'e e-dx, Ms
Let a > 0 and n > 1. Define ƒ(x) = xn/eax − 1 for x ≠ 0 and ƒ(0) = 0.(a) Use L’Hôpital’s Rule to show that ƒ is continuous at x = 0.(b) Show that ∫∞0 ƒ(x) dx converges. Show that ƒ(x) ≤ 2xne−ax if x is large enough. Then use the Comparison Test and Exercise 93.Data
Compute the given approximation to the integral. S 2 √6t³+1 dt, T3
Compute the given approximation to the integral. T/2 Jx/4 Vsin ede, M4
Let I =∫10 xp ln x dx.(a) Show that I diverges for p = −1.(b) Show that if p ≠ −1, then(c) Use L’Hôpital’s Rule to show that I converges if p > −1 and diverges if p [xnxdx=(x-₁)+c XP+1 In p+1 1 p+1) +C
Compute the given approximation to the integral. Si dx x³ + 1' T6
Let F(x) = ∫x2 dt/ln t and G(x) = x/ln xVerify that L’Hôpital’s Rule applies to the limit and evaluate L. F(x) L = lim x-00 G(x)
Compute the given approximation to the integral. So e-² dx, S4

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