Q. 11 - Calculus

Question

Let $f\left(x\right)$ be a function of a single variable. Define the directional derivative of $f$ in the direction of the unit vector $\mathbf{u}=⟨\alpha ⟩$ at a point $c$. What are the only possible values for $\alpha$?

Step-by-Step Solution

Answer

The possible value of $\alpha$ is equal to $1.$

Step 1 Introduction

Directional Derivative of a function of single variables:

Let $f\left(x\right)$ be a function of single variables defined on an open set containing the point $\left(x_0\right)$, and let $u=\left(\alpha \right)$ be a unit vector at the point $c$. The directional derivative of $f$ at $\left(x_0\right)$ in the direction of $u$, denoted by $D_{u}f\left(x_0\right)$.

Step 2 Equation

The Equation is,

$D_{u}f\left(x_0,y_0\right)={Lim}_{h\to 0}\frac{f\left(x_0+\alpha +h\right)-f\left(x_0\right)}{h}$

Provided that limit is exists.

$u$ is the unit vector, hence the only possible value of $\alpha$ is equal to $1.$