Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)
6th Edition
ISBN: 9781524908102
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 13.4, Problem 53PS
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To find: The prove the
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3. Let fe L(X,Y) and let dim X> dim Y .Show that there exists nonzero vector x, in X
such that f(x)=0
Let f: R²R be defined by f(x, y) = |x|+|y|. Suppose u = (u₁, u₂) € R² be any unit vector and let
(0,0). Determine whether the following exist or not.
C=
(a) first partial derivatives of f at c;
(b) directional derivative of f at c in the direction of u.
Is the function continuous and differentiable at c?
Let f : R2 → R differentiable with f(a, b) = 5, ∂f/∂x (a, b) ≠ 0 and ∂f/∂y (a, b) ≠ 0. Let C be the intersection curve of the equation f(x, y) = 5 and u→ a nonzero vector tangent to C at the point (a, b).
Detail whether each of the following statements is true or false:
I. ∂f/du→ ≠ 0 necessarily.
II. ∇f(a, b)·u→ = 0, necessarilyIII. If v→ is any vector and w→ = ∇f(a, b), then we necessarily have ∂f/dw→(a,b)≥ df/dv→(a,b).
IV. r(t) = (a + df/dx (a,b) t, b - df/dy (a,b) t), t ∈ R, is the equation of the line normal to C at (a,b).
V. If γ(t) = (x(t), y(t)), t ∈ I, where I is some open interval, is a parametrization of C and h(t) = f(γ(t)), then h'(α) = 0, where α ∈ I is such that γ(α) = (a, b).
Chapter 13 Solutions
Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)
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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Prob. 40PSCh. 13.7 - Prob. 41PSCh. 13.7 - Prob. 42PSCh. 13.7 - Prob. 43PSCh. 13.7 - Prob. 44PSCh. 13.7 - Prob. 45PSCh. 13.7 - Prob. 46PSCh. 13.7 - Prob. 47PSCh. 13.7 - Prob. 48PSCh. 13.7 - Prob. 49PSCh. 13.7 - Prob. 50PSCh. 13.7 - Prob. 51PSCh. 13.7 - Prob. 52PSCh. 13.7 - Prob. 53PSCh. 13.7 - Prob. 54PSCh. 13.7 - Prob. 55PSCh. 13.7 - Prob. 56PSCh. 13.7 - Prob. 57PSCh. 13.7 - Prob. 58PSCh. 13.7 - Prob. 59PSCh. 13.7 - Prob. 60PSCh. 13 - Prob. 1PECh. 13 - Prob. 2PECh. 13 - Prob. 3PECh. 13 - Prob. 4PECh. 13 - Prob. 5PECh. 13 - Prob. 6PECh. 13 - Prob. 7PECh. 13 - Prob. 8PECh. 13 - Prob. 9PECh. 13 - Prob. 10PECh. 13 - Prob. 11PECh. 13 - Prob. 12PECh. 13 - Prob. 13PECh. 13 - Prob. 14PECh. 13 - Prob. 15PECh. 13 - Prob. 16PECh. 13 - Prob. 17PECh. 13 - Prob. 18PECh. 13 - Prob. 19PECh. 13 - Prob. 20PECh. 13 - Prob. 21PECh. 13 - Prob. 22PECh. 13 - Prob. 23PECh. 13 - Prob. 24PECh. 13 - Prob. 25PECh. 13 - Prob. 26PECh. 13 - Prob. 27PECh. 13 - Prob. 28PECh. 13 - Prob. 29PECh. 13 - Prob. 30PECh. 13 - 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