Find Iz for the lamina z = x2 + y2, 0 ≤ z ≤ 18 with uniform density of 1 . Use a computer algebra system to verify your result. π50(10691+1) π30(10791+1) π60(107(73)3 /2+1) π60(105(73)3 /2+1) π30(106(73)3 /2+1)
Find the Jacobian ∂(x, y)∂(u, v) for the following change of variables: x = 1 5(9u − 6v), y = 15(u − 3v) 51 25 −5 75 −51 25 21 25 −21 25
Suppose the population density of a city is approximated by the model f(x, y) = 6000 e−0.01(x2+y2), x2+y2 ≤ 25, where x and y are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city. Round your answer to the nearest integer.
For a sphere of radius 5 given in parametric form by a vector function r(ϕ,θ) = 5sin(ϕ)cos(θ)i + 5sin(ϕ)sin(θ)j + 5cos(ϕ)k determine the unit normal vector n. n = 15 r(ϕ;θ) n = r(ϕ,θ) n = 5 r(ϕ,θ) n = 125 r(ϕ,θ)
Use the Divergence Theorem to evaluate ∬S(7 x2+14 y+z)dS where S is the sphere x2+y2+z2 = 4.0 2243π 1, 0243π 323π 4483π
Find parametric equations for the surface obtained by rotating the curve x = 2 y, y ≥ 2, about the y-axis. x = 2 ysin(θ), y = y, z = 2 ycos(θ), y ≥ 2, 0 ≤ θ ≤ 2π x = 2 y, y = cos(θ), z = 2 ysin(θ), y ≥ 2, 0 ≤ θ ≤ 2π x = 2 ycos(θ), y = y, z = 2 sin(θ), y ≥ 2, 0 ≤ θ ≤ 2π x = 2 ycos(θ), y = y, z = 2 ysin(θ), y ≥ 2, 0 ≤ θ ≤ 2π x = 2 ycos(θ), y = 2, z = 2 ysin(θ), y ≥ 2, 0 ≤ θ ≤ 2π
Use the Divergence Theorem to find the flux of F across S; that is, calculate ∬SF⋅nds. F(x, y;z) = (3 xy+cosz)i+(x−sinz)j−3 yzk;S is the sphere x2+y2+z2 = 49
Use the Divergence Theorem to calculate the surface integral ∬SF⋅dS where F(x, y, z) = 6 xzi+y3 j+(2 z3−3 z2)k and S is the surface of the solid bounded by the paraboloid x = 9−y2−z2 and the plane x = 5.48π 192π 0 −960π −3,840π
Set up, but do not evaluate, a double integral for the area of the surface with parametric equations x = 5ucosv, y = 7usinv, z = u2, 0 ≤ u ≤ 2, 0 ≤ v ≤ 2π.
Use Stokes' Theorem to evaluate ∬S curl F⋅dS, S consists of the top and the four sides (but not the bottom) of the cube with vertices (±5, ±5, ±5) oriented outward. F(x,y,z) = 6 xyzi + 6xyj + 6x2yzk
Evaluate the surface integral. S is the part of the cylinder z2+y2 = 1 between the planes x = 0 and x = 1 in the first octant. ∬S5(x2 y+z)dS
Find parametric equations for C, if C is the curve of intersection of the hyperbolic paraboloid z = y2−x2 and the cylinder x2+y2 = 4 oriented counterclockwise as viewed from above.
Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone z = x2+y2, 1 ≤ z ≤ 4 if its density function is ρ(x,y,z) = 14−z.
Find the area of the surface S where S is the part of the sphere x2 + y2 + z2 = 81 that lies to the right of the xz-plane and inside the cylinder x2 + z2 = 4.
Use Stoke's theorem to evaluate ∬SF⋅dS. F(x,y,z) = 6.25xi + 25yj + 5(y2+x2)k C is the boundary of the part of the paraboloid z = 6.25−x2−y2 in the first octant. C is oriented counterclockwise as viewed from above.
Use the Shell Method to calculate the volume of rotation about the x-axis. x = y(3−y), x = 0 (Use symbolic notation and fractions where needed. )
Use the Shell Method to find the volume of the solid obtained by rotating the region A in the figure about x = −3. Assume b = 3, a = 8. (Use symbolic notation and fractions where needed. )
Use the Shell Method to find the volume of the solid obtained by rotating the region A in the figure about x = 4. Assume b = 1, a = 4. (Use symbolic notation and fractions where needed. )
Use the Shell Method to compute the volume of a solid obtained by rotating the region enclosed by the graphs of the functions y = x2, y = 18−x2, and x = 2 about the y-axis. (Use symbolic notation and fractions where needed. )
Find the volume of a solid obtained by rotating the region underneath the graph of f(x) = 64 − 4x2 about the y-axis over the interval [0, 4]. (Use symbolic notation and fractions where needed. )
Find the volume of a solid obtained by rotating the region underneath the graph of f(x) = 9x3 about the y-axis over the interval [0, 1]. (Use symbolic notation and fractions where needed. )
The following differential equation models population changes for a harvested, logistically changing species. dPdt = 3P(1 − 1 12 P) − 8. (a) Find the equilibrium population size. (b) By thinking about the sign of the derivative, describe what happens to the population size for each of the initial conditions P(0) = 2, 4, 6, 8, 10.
Problem #4: For which of the following points does Theorem 1.2.1 guarantee that the differential equation y′ = ex2 36 − y2 possesses a unique solution through the given point? (i) (−3, 6) (ii) (0, 2) (iii) (4, 9) (A) (i) and (iii) only (B) (ii) and (iii) only (C) (ii) only (D) all of them (E) none of them (F) (i) and (ii) only (G) (i) only (H) (iii) only
Use the ratio test to determine whether ∑n = 17∞n+7 n! converges or diverges. (a) Find the ratio of successive terms. Write your answer as a fully simplified fraction. For n ≥ 17, limn→∞|an+1 an| = limn→∞ (b) Evaluate the limit in the previous part. Enter ∞ as infinity and −∞ as -infinity. If the limit does not exist, enter DNE. limn→∞|an+1 an| = 0 (c) By the ratio test, does the series converge, diverge, or is the test inconclusive? Converges
Determine whether the integral is convergent or divergent. ∫e∞ 37 x(ln(x))3 dx convergent divergent If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES. )
Consider the series ∑n = 1∞xn = ∑n = 1∞3+3 n26 n10−3 Give some real constants a, b > 0 and the unique power p > 0 such that xn satisfies anp ≤ xn ≤ bnp for all n∈N. Hence determine whether the series converges or diverges. We may take a = , b = and p = It follows that the series (No answer given)
Use the remainder term to estimate the absolute error in approximating the following quantity with the nth-order Taylor polynomial centered at 0. cos(0, 38), n = 3 Select the correct choice below and fill in the answer box to complete your choice. (Use scientific notation. Use the multiplication symbol in the math palette as needed. Do not round until the final answer. Then round to two decimal places as needed. ) A. Errors for M = 0.38 B. Errors for M = 1 C. Errors for M = 0 D. Error ≤ for M = 1π
Find the second Taylor polynomial T2(x) for the function f(x) = cos(x) based at b = π6. T2(x) = Let a be a positive real number and let J be the closed interval [π6−a, π6+a]. Use the Quadratic Approximation Error Bound to verify that |f(x)−T2(x)| ≤ a33! for all x in J Use this error bound to find a value of a so that |f(x) − T2(x)| ≤ 0.01 for all x in J. (Round your answer to six decimal places. ) a =
a. Approximate the given quantity using Taylor polynomials with n = 3. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. cos(−0.04) a. p3(−0.04) = (Do not round until the final answer. Then round to six decimal places as needed. )
Suppose that ∑n = 0∞an(x+1)n converges at x = −2. At which of the following points must the series also converge? Use the fact that if ∑an(x−c)n converges at x, then it converges at any point closer to c than x. a. x = 2 b. x = −1 c. x = −3 d. x = 0 e. x = 0.99 f. x = 0.000001
Given the series: ∑k = 1∞2 k does this series converge or diverge? diverges converges If the series converges, find the sum of the series: ∑k = 1∞2 k = (If the series diverges, just leave this second box blank. )
Use the Root Test to determine whether the series converges absolutely or diverges. ∑k = 1 ∞(k k+5)4 k2 Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer in simplified form. ) A. The series diverges because ρ = . B. The series converges absolutely because ρ = . C. The Root Test is inconclusive because ρ =
Does the alternating series ∑k = 2 ∞ (−1)k ln(k3) k+7 converge absolutely, converge conditionally, or diverge? Be sure to give all the details for each convergence test you used to get your answer.
Find T5(x) : Taylor polynomial of degree 5 of the function f(x) = cos(x) at a = 0. T5(x) = Using the Taylor Remainder Theorem, find all values of x for which this approximation is within 0.000624 of the right answer. Assume for simplicity that we limit ourselves to |x| ≤ 1 |x| ≤
In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f ? Choose the correct answer below. A. The Taylor series for a function f converges to f on an interval if, for all nonzero x in the interval, lim Rn(x) = 0, where Rn(x) is the remainder at x. B. The Taylor series for a function f converges to f on an interval if, for all positive x in the interval, limn→∞ Rn(x) = 0, where Rn(x) is the remainder at x. C. The Taylor series for a function f converges to f on an interval if, for all x in the interval, limn→∞ Rn(x) = 0, where Rn(x) = 0 and Rn(x) is the remainder at x. D. The Taylor series for a function f converges to f on an interval if, for all x in the interval, limn→∞ Rn(x) = 0, where Rn(x) is the remainder at x.
1. (a) (15) Write down the Taylor series expansion of the function f(x) = 1 − 4x, about the point x0 = −3. Neglect terms of order four or higher. (b) (15) Write down the Taylor series expansion of the function f(x, y) = e−(x2 + y2), about the point (x0, y0) = (1, 2). Neglect terms of order three or higher.