6. Let f(x, y) = ln(2x - y²).(a)(b)(c)Determine the gradient of f at the point (2,-1).Determine the equation of the plane tangent to the graph of f at the point (3,2).Find all unit vectors u such that Duf at (3, 2) is 0.

Answered: Image /qna-images/answer/659ad287-666b-4c0a-a43f-887133ef3182.jpg

Answered: 6. Let f(x, y) = ln(2x - y²). (a) (b)…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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2. Consider the function f(x, y) = a) Find the directional derivative of f in the direction of the vector v (1,2) at the point (0,-2). b) Find a unit vector u in the direction in which f decreases most rapidly at the point (0,-2).
Find the directional derivative of the function at the given point in the direction of the vector v. f(x, у) 7e sin(y), (0, t/3), v = (-6, 8) Df(0, π/3) -
(1n |t – 1], e', vî ) 1. Let 7(t) = (a) Express the vector valued function in parametric form. (b) Find the domain of the function. (c) Find the first derivative of the function. (d) Find T(2). (e) Find the vector equation of the tangent line to the curve when t=2. 2. Complete all parts: (a) Find the equation of the curve of intersection of the surfaces y = x? and z = x3 (b) What is the name of the resulting curve of intersection? (c) Find the equation for B the unit binormal vector to the curve when t= 1. Hint: Instead of using the usual formula for B note that the unit binormal vector is orthogonal to 7 '(t) and 7"(t). In fact, an alternate formula for this vector is ア'(t) × ア"(t) ア(t) ×デ"(t)| B(t) =
#3. Consider the function if (x, y)# (0,0), if (x, y)= (0,0). f (x, y)= x2+y2 Calculate the directional derivative of f at the point (0,0) in the direction of the vector (1,3).
Find the directional derivative of the function f(x, y, z) = x² - y +2z at the point P(1,2,3) in the direction of the vector 4i-2j+k.
4. ) Find the directional derivative of the function f(x, y) = sin²(xy) at the point P (2,7) in the direction of the unit vector ū = −²¹+] √5
Find the directional derivative of the function f(x, y) = In(x +y°) at the point (2, 3) in the direction of the vector (- 3, 3)
Find the directional derivative Duf(1, 1), where f(x, y) = x2 + y2 and u is the unit vector at an angle of π/6 from horizontal.
Find the directional derivative of the function at the given point in the direction of vector v. f(x, y) = 3 + 6x√y, (3, 4), v=(4, -3) D₁f(3, 4) = 6 X
Let 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.

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6. Let f(x, y) = ln(2x - y²). (a) (b) (c) Determine the gradient of f at the point (2, -1). Determine the equation of the plane tangent to the graph of f at the point ( Find all unit vectors u such that Duf at (3, 2) is 0.