8. One-sided directional derivative in all directions, but no gradient The one-sided directional derivative of f at P(xo, Yo, z0) in the direction u = u¡i + uzj + uzk is the number f (xo + su¡, yo + su2, zo + suz) – f (xo, Yo, zo) lim Show that the one-sided directional derivative of f(x, y, z) = Vx² + y² + z² at the origin equals 1 in any direction but that f has no gradient vector at the origin.

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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8. One-sided directional derivative in all directions, but no
gradient The one-sided directional derivative of f at
P(xo, Yo, z0) in the direction u = u¡i + uzj + uzk is the number
f (xo + su¡, yo + su2, zo + suz) – f (xo, Yo, zo)
lim
Show that the one-sided directional derivative of
f(x, y, z) =
Vx² + y² + z²
at the origin equals 1 in any direction but that f has no gradient
vector at the origin.
Transcribed Image Text:8. One-sided directional derivative in all directions, but no gradient The one-sided directional derivative of f at P(xo, Yo, z0) in the direction u = u¡i + uzj + uzk is the number f (xo + su¡, yo + su2, zo + suz) – f (xo, Yo, zo) lim Show that the one-sided directional derivative of f(x, y, z) = Vx² + y² + z² at the origin equals 1 in any direction but that f has no gradient vector at the origin.
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