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(a) Find the directional derivative of f(x, y, z) = xy In(x? + y² + z?) at (1, 1, 1) in the direction given by the vector from the point (3, 2, 1) to the point (-1, 1, 2). (b) Determine also the direction of maximal increase from the point (1, 1, 1). 6. Suppose that f (x, y) has continuous partial derivatives. Suppose too that the maximum direc- tional derivative of f at (0, 0) is equal to 4, and that it is attained in the direction given by the vector from the origin to the point (1, 3). Find Vf(0, 0). 7. Let b = (b1, ..., bn) be a given vector and define-the function f(x) = f(x1, . . . , Xn) = b x. Show that the derivative of f along the vector a = (a),..., an) is b a. 8. Let f (v) = f(v1, ..., Vn) denote a positive valued differentiable function of n variables defined whenever vi > 0, i = 1, 2, ..., n. The directional elasticity of f at the point v along the vector v/|v = a, hence in the direction from the origin to v, is denoted by El, f (v) and is, by definition, %3D = (A)f "13 (A)S All %3D where f'(v) is the directional derivative of f in the direction given by a. Use (8) to show that

(a) Find the directional derivative of f(x, y, z) = xy In(x? + y² + z?) at (1, 1, 1) in the direction given by the vector from the point (3, 2, 1) to the point (-1, 1, 2). (b) Determine also the direction of maximal increase from the point (1, 1, 1). 6. Suppose that f (x, y) has continuous partial derivatives. Suppose too that the maximum direc- tional derivative of f at (0, 0) is equal to 4, and that it is attained in the direction given by the vector from the origin to the point (1, 3). Find Vf(0, 0). 7. Let b = (b1, ..., bn) be a given vector and define-the function f(x) = f(x1, . . . , Xn) = b x. Show that the derivative of f along the vector a = (a),..., an) is b a. 8. Let f (v) = f(v1, ..., Vn) denote a positive valued differentiable function of n variables defined whenever vi > 0, i = 1, 2, ..., n. The directional elasticity of f at the point v along the vector v/|v = a, hence in the direction from the origin to v, is denoted by El, f (v) and is, by definition, %3D = (A)f "13 (A)S All %3D where f'(v) is the directional derivative of f in the direction given by a. Use (8) to show that

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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ad,Ame%%20Strom%20(z-lib.org).pdf erstad,Ar. 31 / 310 %097 SMO 5. (a) Find the directional derivative of f(x, y, z) = xy In(x² + y² + z²) .2 %3D at (1, 1, 1) in the direction given by the vector from the point (3, 2, 1) to the point (-1, 1, 2). (b) Determine also the direction of maximal increase from the point (1, 1, 1). EM 6. Suppose that f (x, y) has continuous partial derivatives. Suppose too that the maximum direc- tional derivative of f at (0, 0) is equal to 4, and that it is attained in the direction given by the vector from the origin to the point (1, 3). Find V f (0, 0). 7. Let b = (bị,. Show that the derivative of f along the vector a = (aj, ., an) is b · a. ba) be a given vector and define-the function f (x) = ƒ(x1,..., Xn) = b · x. 8. Let f(v) = f(v1,., Vn) denote a positive valued differentiable function of n variables defined whenever vi > 0, i = 1, 2, ..., n. The directional elasticity of ƒ at the point v along the vector v/||v| = a, hence in the direction from the origin to v, is denoted by El, f (v) and is, by definition, El, f(v) = |A|| || A|| where f'(v) is the directional derivative of ƒ in the direction given by a. Use (8) to show that BT321 Sp21_PS2.pdf 20215-QF202-quiz. pdf 20215 QF202 qui.Rmd CAPM Notes.pdf 2-58 PM 3/12/2021 ily BANG & OLUFS dn 6d d prt sc delete home pua 144 トト wnu lock + backspace 8. 6.
Transcribed Image Text:ad,Ame%%20Strom%20(z-lib.org).pdf erstad,Ar. 31 / 310 %097 SMO 5. (a) Find the directional derivative of f(x, y, z) = xy In(x² + y² + z²) .2 %3D at (1, 1, 1) in the direction given by the vector from the point (3, 2, 1) to the point (-1, 1, 2). (b) Determine also the direction of maximal increase from the point (1, 1, 1). EM 6. Suppose that f (x, y) has continuous partial derivatives. Suppose too that the maximum direc- tional derivative of f at (0, 0) is equal to 4, and that it is attained in the direction given by the vector from the origin to the point (1, 3). Find V f (0, 0). 7. Let b = (bị,. Show that the derivative of f along the vector a = (aj, ., an) is b · a. ba) be a given vector and define-the function f (x) = ƒ(x1,..., Xn) = b · x. 8. Let f(v) = f(v1,., Vn) denote a positive valued differentiable function of n variables defined whenever vi > 0, i = 1, 2, ..., n. The directional elasticity of ƒ at the point v along the vector v/||v| = a, hence in the direction from the origin to v, is denoted by El, f (v) and is, by definition, El, f(v) = |A|| || A|| where f'(v) is the directional derivative of ƒ in the direction given by a. Use (8) to show that BT321 Sp21_PS2.pdf 20215-QF202-quiz. pdf 20215 QF202 qui.Rmd CAPM Notes.pdf 2-58 PM 3/12/2021 ily BANG & OLUFS dn 6d d prt sc delete home pua 144 トト wnu lock + backspace 8. 6.
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