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Evaluating a Line
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Multivariable Calculus
- Showing Linear Independence In Exercises 27-30, show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent. {eax,ebx}, abarrow_forwardUsing green's theorem, evaluate the line integral xy^2dx + (1-xy^3)dyarrow_forwardCurve In Exercise 56, sketch the plane curve and find its length over the given interval. 56. r(t) = t 2i + 2tk, [0, 3]arrow_forward
- Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.F = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (7, 0), and (0, 6) a) 0 b) 252 c) 84 d) 42arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.F = xy i + x j; C is the triangle with vertices at (0, 0), (7, 0), and (0, 4)arrow_forwardFind the linearizations of the functions in Exercises 57 and 58 at the given points. 57. ƒ(x, y, z) = xy + 2yz - 3xz at (1, 0, 0) and (1, 1, 0) 58. ƒ(x, y, z) = 22 cos x sin ( y + z) at (0, 0, pai/4) and (pai/4, pai/4, 0)arrow_forward
- A. State the Fundamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = x^2 + 2y^2 + 3z^2 and F = grad f. Find the line integral of F along the line C with parametric equations x = 1 + t, y = 1 + 2t, z = 1 + 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forwardParametric Representation. In Exercises 7-10, find a parametric representation of the solution set of the linear equation. x+y+z=1arrow_forwardFinding a Least Approximation In Exercises 75-78, a find the least squares approximation g(x)=a0+a1xof the function f, and b use a graphing utility to graph fand gin the same viewing window. f(x)=sinxcosx, 0xarrow_forward
- Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. F = xy i + x j; C is the triangle with vertices at (0, 0), (8, 0), and (0, 4) a) 0 b) -80/3 c) 112/3 d) 64/3arrow_forwardA. State the F undamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = xy + 2yz + 3zx and F = grad f. Find the line integral of F along the line C with parametric equations x = t, y = t, z = 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. IF = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (8, 0), and (0, 7) a) 112 b) 392 c) 0 d) 56arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage