6.(a) Find the directional derivative of w = x²y² at the point (1, -3) in the direc-5T5Ttion of the unit vector u = cosi+sinj.(b) What is the maximum value of the directional derivative of w = ²y at(1,-3) and it what direction is it attained?Q SearchA

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Answered: 6. (a) Find the directional derivative…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.5: Polar Coordinates

Problem 97E

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18 > (a) Find the direction for which the directional derivative of the function f (x, y) = 7xy² + 5x² is a maximum at P = (-1,2). (Use symbolic notation and fractions where needed. Give your answer in vector form.) direction: (b) Find the maximum value of the directional derivative. (Use symbolic notation and fractions where needed.) ||Vf(-1,2) || =
(a) Find the directional derivative of z = x²y at (5,5) in the direction of π/2 with the positive x-axis. (b) In which direction is the directional derivative the largest at the point (5, 5)? Enter your answer as a vector whose length is the largest value of the directional derivative.
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f(x, 3) In(x2 + Yof function P(3, 4) at the point of = (9, 12) Directional derivative in the direction of the vector to you.
2. [-/16.66 Points] DETAILS Find the directional derivative of the function at the given point in the direction of the vector v. х, у, 2) %3Dхуz (4, 2, 8), v = -1, –2, 2] Duf(4, 2, 8) =
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(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) =
(8) Subtracting the two equations, find a vector equation for the curve of 3 intersection between y= 4x² +=' and y-1=3x += for x > 0. Find 4 1 and simplify the tangential component of acceleration for your curve. 3 2 cos (2t) + cos 2t a sin? t- sin? (2r) –4 cos?t 3 2sin? (21) - sin 21 Vsin' - sin? t- sin? (2t) –4cos?t 2sin (21)- sin 21 3 sin? t + sin? (21) +4cos?t 3 2 sin (21) +sin 21 /sin?t + sin? (21) + 4 cos²t
5. (a) (b) The surfaces (x − 2)² + z² = 9 and y - Find a vector equation for the curve C. = z² intersect along the curve C. Find parametric equations for the tangent line to this curve at (2, 9, 3).
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
Suppose that the directional derivative of a function f at a point Pin the direction of the vector ū = (-1,0) is - Relate this 4 value to the behaviour of f. A. From the point P, the function f is increasing at a rate of - in the positive x-direction. B. From the point P, the function f is increasing at a rate of - in the positive y-direction. C. From the point P, the function f is decreasing at a rate of - in the negative y-direction.
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6. (a) Find the directional derivative of w = x²y² at the point (1, -3) in the direc- 5T 5T tion of the unit vector u = cos i+sinj. (b) What is the maximum value of the directional derivative of w = ²y at (1,-3) and it what direction is it attained? Q Search A