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Calculus: Early Transcendental Functions (MindTap Course List)
- 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, 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
- Parametric Representation. In Exercises 7-10, find a parametric representation of the solution set of the linear equation. x+y+z=1arrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (6, 0), and (0, 6) a) 216 b) 72 c) 0 d) 36arrow_forwardA. 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_forward
- A. 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.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_forwardFind the potential function f for the field F.F = (y - z) i + (x + 2y - z) j - (x + y) k a) f(x, y, z) = x(y + y2) - xz - yz + C b) f(x, y, z) = xy + y2 - x - y + C c) f(x, y, z) = xy + y2 - xz - yz + C d) f(x, y, z) = x + y2 - xz - yz + Carrow_forward
- The figure shows a vector field F and two curves C_1 and C_2. Are the line integrals of F over C_1 and C_2 positive, negative, or zero? Explain.arrow_forwardUsing Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. F = (3x + 3y) i + (4x - 9y) j; C is the region bounded above by y = -2x 2 + 45 and below by y=3x2 in the first quadrant a) 90 b) 252 c) -132 d) - 294arrow_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), (8, 0), and (0, 4) a) 0 b) -80/3 c) 112/3 d) 64/3arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning