CALCULUS (CLOTH)
4th Edition
ISBN: 9781319050733
Author: Rogawski
Publisher: MAC HIGHER
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Chapter 18.2, Problem 34E
To determine
Prove the given equation for two functions.
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3) To find the partial derivative of f (x. y) with respect to y at the point (x-y.). we
A) differentiate both of x and y
B) treat both of x and y as a constants
C) treat x as a constant and differentiate with respect to y
D) treat y as a constant and differentiate with respect to x
Let f be a function of two variables that has continuous partial derivatives and consider the points
A(5, 2), B(13, 2), C(5, 13), and D(14, 14). The directional derivative of f at A in the direction of the vector AB is 4 and the directional derivative at A in the direction of AC is 9.
Find the directional derivative of f at A in the direction of the vector AD.
Show that t, e^t, and sin(t) are linearly independent.
Chapter 18 Solutions
CALCULUS (CLOTH)
Ch. 18.1 - Prob. 1PQCh. 18.1 - Prob. 2PQCh. 18.1 - Prob. 3PQCh. 18.1 - Prob. 4PQCh. 18.1 - Prob. 5PQCh. 18.1 - Prob. 1ECh. 18.1 - Prob. 2ECh. 18.1 - Prob. 3ECh. 18.1 - Prob. 4ECh. 18.1 - Prob. 5E
Ch. 18.1 - Prob. 6ECh. 18.1 - Prob. 7ECh. 18.1 - Prob. 8ECh. 18.1 - Prob. 9ECh. 18.1 - Prob. 10ECh. 18.1 - Prob. 11ECh. 18.1 - Prob. 12ECh. 18.1 - Prob. 13ECh. 18.1 - Prob. 14ECh. 18.1 - Prob. 15ECh. 18.1 - Prob. 16ECh. 18.1 - Prob. 17ECh. 18.1 - Prob. 18ECh. 18.1 - Prob. 19ECh. 18.1 - Prob. 20ECh. 18.1 - Prob. 21ECh. 18.1 - Prob. 22ECh. 18.1 - Prob. 23ECh. 18.1 - Prob. 24ECh. 18.1 - Prob. 25ECh. 18.1 - Prob. 26ECh. 18.1 - Prob. 27ECh. 18.1 - Prob. 28ECh. 18.1 - Prob. 29ECh. 18.1 - Prob. 30ECh. 18.1 - Prob. 31ECh. 18.1 - Prob. 32ECh. 18.1 - Prob. 33ECh. 18.1 - Prob. 34ECh. 18.1 - Prob. 35ECh. 18.1 - Prob. 36ECh. 18.1 - Prob. 37ECh. 18.1 - Prob. 38ECh. 18.1 - Prob. 39ECh. 18.1 - Prob. 40ECh. 18.1 - Prob. 41ECh. 18.1 - Prob. 42ECh. 18.1 - Prob. 43ECh. 18.1 - Prob. 44ECh. 18.1 - Prob. 45ECh. 18.1 - Prob. 46ECh. 18.1 - Prob. 47ECh. 18.1 - Prob. 48ECh. 18.1 - Prob. 49ECh. 18.1 - Prob. 50ECh. 18.1 - Prob. 51ECh. 18.2 - Prob. 1PQCh. 18.2 - Prob. 2PQCh. 18.2 - Prob. 3PQCh. 18.2 - Prob. 4PQCh. 18.2 - Prob. 5PQCh. 18.2 - Prob. 1ECh. 18.2 - Prob. 2ECh. 18.2 - Prob. 3ECh. 18.2 - Prob. 4ECh. 18.2 - Prob. 5ECh. 18.2 - Prob. 6ECh. 18.2 - Prob. 7ECh. 18.2 - Prob. 8ECh. 18.2 - Prob. 9ECh. 18.2 - Prob. 10ECh. 18.2 - Prob. 11ECh. 18.2 - Prob. 12ECh. 18.2 - Prob. 13ECh. 18.2 - Prob. 14ECh. 18.2 - Prob. 15ECh. 18.2 - Prob. 16ECh. 18.2 - Prob. 17ECh. 18.2 - Prob. 18ECh. 18.2 - Prob. 19ECh. 18.2 - Prob. 20ECh. 18.2 - Prob. 21ECh. 18.2 - Prob. 22ECh. 18.2 - Prob. 23ECh. 18.2 - Prob. 24ECh. 18.2 - Prob. 25ECh. 18.2 - Prob. 26ECh. 18.2 - Prob. 27ECh. 18.2 - Prob. 28ECh. 18.2 - Prob. 29ECh. 18.2 - Prob. 30ECh. 18.2 - Prob. 31ECh. 18.2 - Prob. 32ECh. 18.2 - Prob. 33ECh. 18.2 - Prob. 34ECh. 18.2 - Prob. 35ECh. 18.2 - Prob. 36ECh. 18.2 - Prob. 37ECh. 18.2 - Prob. 38ECh. 18.3 - Prob. 1PQCh. 18.3 - Prob. 2PQCh. 18.3 - Prob. 3PQCh. 18.3 - Prob. 4PQCh. 18.3 - Prob. 5PQCh. 18.3 - Prob. 1ECh. 18.3 - Prob. 2ECh. 18.3 - Prob. 3ECh. 18.3 - Prob. 4ECh. 18.3 - Prob. 5ECh. 18.3 - Prob. 6ECh. 18.3 - Prob. 7ECh. 18.3 - Prob. 8ECh. 18.3 - Prob. 9ECh. 18.3 - Prob. 10ECh. 18.3 - Prob. 11ECh. 18.3 - Prob. 12ECh. 18.3 - Prob. 13ECh. 18.3 - Prob. 14ECh. 18.3 - Prob. 15ECh. 18.3 - Prob. 16ECh. 18.3 - Prob. 17ECh. 18.3 - Prob. 18ECh. 18.3 - Prob. 19ECh. 18.3 - Prob. 20ECh. 18.3 - Prob. 21ECh. 18.3 - Prob. 22ECh. 18.3 - Prob. 23ECh. 18.3 - Prob. 24ECh. 18.3 - Prob. 25ECh. 18.3 - Prob. 26ECh. 18.3 - Prob. 27ECh. 18.3 - Prob. 28ECh. 18.3 - Prob. 29ECh. 18.3 - Prob. 30ECh. 18.3 - Prob. 31ECh. 18.3 - Prob. 32ECh. 18.3 - Prob. 33ECh. 18.3 - Prob. 34ECh. 18.3 - Prob. 35ECh. 18.3 - Prob. 36ECh. 18.3 - Prob. 37ECh. 18.3 - Prob. 38ECh. 18.3 - Prob. 39ECh. 18.3 - Prob. 40ECh. 18.3 - Prob. 41ECh. 18.3 - Prob. 42ECh. 18.3 - Prob. 43ECh. 18.3 - Prob. 44ECh. 18 - Prob. 1CRECh. 18 - Prob. 2CRECh. 18 - Prob. 3CRECh. 18 - Prob. 4CRECh. 18 - Prob. 5CRECh. 18 - Prob. 6CRECh. 18 - Prob. 7CRECh. 18 - Prob. 8CRECh. 18 - Prob. 9CRECh. 18 - Prob. 10CRECh. 18 - Prob. 11CRECh. 18 - Prob. 12CRECh. 18 - Prob. 13CRECh. 18 - Prob. 14CRECh. 18 - Prob. 15CRECh. 18 - Prob. 16CRECh. 18 - Prob. 17CRECh. 18 - Prob. 18CRECh. 18 - Prob. 19CRECh. 18 - Prob. 20CRECh. 18 - Prob. 21CRECh. 18 - Prob. 22CRECh. 18 - Prob. 23CRECh. 18 - Prob. 24CRECh. 18 - Prob. 25CRECh. 18 - Prob. 26CRECh. 18 - Prob. 27CRECh. 18 - Prob. 28CRECh. 18 - Prob. 29CRECh. 18 - Prob. 30CRECh. 18 - Prob. 31CRECh. 18 - Prob. 32CRECh. 18 - Prob. 33CRECh. 18 - Prob. 34CRECh. 18 - Prob. 35CRECh. 18 - Prob. 36CRECh. 18 - Prob. 37CRECh. 18 - Prob. 38CRECh. 18 - Prob. 39CRECh. 18 - Prob. 40CRECh. 18 - Prob. 41CRE
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- Let f(x,y) = x² + y². Which of the following is the minimum directional derivative of the function f(x,y) at the point (x,y) = (1,1)?arrow_forwardLet 2, if (x, Y, z) # (0,0, 0) if (x, y, z) = (0,0, 0) xyz x²+y²+z² > f(x, y, z) = 0, (a) Use the definition to find the directional derivative of the function f at (0,0, 0) in the direction (a, b, c), where a² + 62 + c² = 1. (b) Use Lagrange Multipliers to find the maximal and minimal directional derivatives.arrow_forwardConsider a function f: R? → R°, the derivative of f isa O 3 x 3 matrix O 3 x 2 matrix O 2 x 3 matrix O 2 x 2 matrix O 3 x 5 matrix O 2 x 5 matrix O 5 x 3 matrix O 5 x 2 matrixarrow_forward
- a) Find the directional derivative of f(x, y, z) = x²yz + 4xz² at (1, -2, 1) in the direction 2i - j - 2k.arrow_forwardLet v = (3,4, 12) . Find the directional derivative of f(r, y, z) = x – y? +32³ in the direction of v.arrow_forwardaf Consider the f unction g(t) = (², 2t) + 52(sin3t, t) express se ду g'(t) in terms of partial derivatives of farrow_forward
- a) Find the directional derivative of the function f(x, y, z) = x² y? (2z+1)² at the point P:(1, – 1, 1) and in the direction of a =[3, 3, 0]. b) Find the direction of maximum decrease of f at the point P. Maximum file size: 100MB. maximum number of files: 1arrow_forwardDetermine the directional derivative of f(x, y, z) tan where (-1,2,2) at P(0, 1, 1). Determine also the maximum and minimum directional derivative of f along the unit vector. E. U = 22²arrow_forwardCalculate the directional derivative of the function f(x.y.z)= In(xy+xz+yz) at (1, 1, 1) in the direction from (1, 1, 1) toward the point (-1, -2. 3). V17 -2 V17 V17arrow_forward
- Determine the directional derivative of f(x, y, z) = tan-1 (2²) U = (-1,2,2) at P(0, 1, 1). Determine also the maximum and minimum directional derivative of f along the unit vector. wherearrow_forwardB- Find the directional derivative of the function W = x² + xy + z³ at the point P: (2,1,1) in the direction towards P₂(5,4,2). əz Ju əv B- If Z = 4e* Iny, x = In(u cosv) and y = u sinv find andarrow_forwardExplain how a directional derivative is formed from the two partial derivatives f, and f,. Choose the correct answer below. O A. Form the dot product between the unit direction vector u and the gradient of the function. O B. Form the dot product between the unit direction vectors. O C. Form the sum between the unit direction vector u and the gradient of the function. O D. Form the dot product between the unit coordinate vectors. Click to select your answer.arrow_forward
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