What Is the Maximum Value of Dv f(1,2) and Its Corresponding Direction v?

In summary, the maximum value of the directional derivative D_{\vec{v}}f(1,2) occurs when the unit vector \vec{v} is parallel to the gradient vector \vec{\nabla}f(1,2), which is the direction of greatest increase of f at the point (1,2). This maximum value is equal to the magnitude of the gradient vector at that point. The symbol Dv is used to represent the directional derivative in the direction of a unit vector \vec{v}, while Du is used to represent the directional derivative in the direction of a unit vector \vec{u}.
  • #1
team31
10
0
when u=-0.6i+0.8j, Du f(1,2)=? gradient f=(18,9)
I use unit vector dot (-0.6,0.8) dotted with the gradient f, which turns out to be -3.6
so here is the question. A: the maximum value of Dv f(1,2) is=?
B: this maximum occurs when v=(?,?)
I'm confued with the Dv thing. For my guess is that I should use (1,2) to creat a unit vector which is (1/sqrt5, 2/sqrt5), and use that unit vetor dotted with the gradient f, which gives me the maximum value of Dv f(1,2)
then i will find a vector that its dot product with ( 1/sqrt5, 2/sqrt5) is 0, and tha should be the direction of V which is the answer for part B

Am I doing this right? plaese help me out. I've been looking into my textbook but couldn't find the symbol Dv, they all used Du. Thank u
 
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  • #2
Directional Derivatives

Directional Derivatives

Directional derivatives of $f(x,y)$ are, for unit vectors [tex]\vec{u}[/tex], are defined by:

[tex]D_{\vec{u}}f(x,y) = \vec{u}\cdot \vec{\nabla}f (x,y)[/tex]​

where [tex]\cdot[/tex] denotes the dot product. To see when the maximum value of the directional derivatives occurs, consider the magnitude of the directional derivative, recalling that [tex]\left|\vec{u}\cdot\vec{v} \right| = \left|\vec{u}\right|\cdot \left|\vec{v} \right|\cos \theta[/tex] where [tex]\theta[/tex] is the acute angle between the vectors we have that

[tex]\left| D_{\vec{u}}f(x,y)\right| = \left| \vec{u}\cdot \vec{\nabla}f (x,y)\right| = \left| \vec{u}\right|\cdot\left| \vec{\nabla}f (x,y)\right|\cos \theta = 1\cdot\left| \vec{\nabla}f (x,y)\right|\cos \theta = \left| \vec{\nabla}f (x,y)\right|\cos \theta[/tex]​

since [tex]\left| \vec{u}\right| =1[/tex] since [tex]\vec{u}[/tex] is a unit vector; consider that the quantity [tex] \left| \vec{\nabla}f (x,y)\right|\cos \theta[/tex] is maximized when [tex]\cos \theta =1[/tex], i.e. when [tex]\theta =0[/tex] namely when the vectors [tex]\vec{u}[/tex] and [tex]\vec{\nabla}f (x,y)[/tex] are paralell. This occurs if [tex]\vec{u} = \frac{\vec{\nabla}f (x,y)}{\left| \vec{\nabla}f (x,y)\right|}[/tex] which is a unit vector parallel to [tex]\vec{\nabla}f (x,y)[/tex]. The maximum value is then

[tex]D_{\frac{\vec{\nabla}f (x,y)}{\left| \vec{\nabla}f (x,y)\right|}}f(x,y) = \frac{\vec{\nabla}f (x,y)}{\left| \vec{\nabla}f (x,y)\right|}\cdot \vec{\nabla}f (x,y) = \frac{\left| \vec{\nabla}f (x,y)\right|^{2}}{\left| \vec{\nabla}f (x,y)\right|} = \left| \vec{\nabla}f (x,y)\right|[/tex]​
 
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Related to What Is the Maximum Value of Dv f(1,2) and Its Corresponding Direction v?

What is the maximum value of gradient?

The maximum value of gradient refers to the steepest slope or change in a function or curve at a particular point. It is the highest rate of change in the function, and it is represented by the derivative of the function at that point.

How is the maximum value of gradient calculated?

The maximum value of gradient is calculated by finding the derivative of a function at a specific point. This can be done using calculus, specifically the first derivative test, or by graphing the function and identifying the point with the highest slope.

Why is the maximum value of gradient important?

The maximum value of gradient is important because it helps us understand the behavior of a function at a particular point. It can tell us the direction in which the function is increasing or decreasing the fastest, and it is often used in optimization problems to find the maximum or minimum value of a function.

Can the maximum value of gradient be negative?

Yes, the maximum value of gradient can be negative. This occurs when the slope of a function is decreasing at a particular point. It is important to note that the maximum value of gradient can be positive, negative, or zero, depending on the behavior of the function at that point.

How is the maximum value of gradient used in real-world applications?

The maximum value of gradient has various real-world applications, such as in physics, engineering, and economics. For example, in physics, it can be used to calculate the maximum velocity of an object at a given point. In economics, it can be used to determine the optimal production level for a company. In general, it is used to optimize and improve processes and systems by finding the point with the steepest slope.

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