Calculus
6th Edition
ISBN: 9781465208880
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 13.1, Problem 48PS
To determine
To prove: The property
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Chapter 13 Solutions
Calculus
Ch. 13.1 - Prob. 1PSCh. 13.1 - Prob. 2PSCh. 13.1 - Prob. 3PSCh. 13.1 - Prob. 4PSCh. 13.1 - Prob. 5PSCh. 13.1 - Prob. 6PSCh. 13.1 - Prob. 7PSCh. 13.1 - Prob. 8PSCh. 13.1 - Prob. 9PSCh. 13.1 - Prob. 10PS
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- No. 6 (a, b, c)arrow_forward(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of -Vf(x). Let 8 be the angle between Vf(x) and unit vector u. Then Duf = |Vf||---Select--- . Since the minimum value of ---Select--- is direction of u is ---Select--- ✓the direction of Vf (assuming Vf is not zero). (b) Use the result of part (a) to find the direction in which the function f(x, y) = x¹y x²y³ decreases fastest at the point (1, -1). occurring, for 0 ≤ 0 < 2π, when 8 = the minimum value of Duf is -|Vf, occurring when thearrow_forwardShow that t, e^t, and sin(t) are linearly independent.arrow_forward
- 2.31 Let A and B be vector functions of position vector x with continuous first and second derivatives, and let F and G be scalar functions of position x with continuous first and second derivatives. Show that: (a) V. (V x A) = 0. (b) ▼x (VF) = 0. (c) V (VF x VG) = 0. (d) V. (FA) = A·VF+ FV.A.arrow_forwardLet u(t) = 2t'i+ (P -7)j-8k and v(t) = e'i+3 ej- ek. Compute the derivative of the following function. 3t L u(t) • v(t) Select the correct choice below and fill in the answer box(es) to complete your choice. O A. The derivative is the scalar function 訓 這 O B. The derivative is the vector-valued function (i+ (Dj+ ( k.arrow_forwardpls write legibly for the solutionsarrow_forward
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