Math Problem Statement

Find the directional derivative and gradient of given functions as described in the image.

Solution

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img_path = '/mnt/data/17337028185964o1d7tky.jpg' img = Image.open(img_path) img.show()I have opened the image, but I need to analyze its content to help with any math-related queries. I will extract the relevant information from the image now. pytesseract

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text = pytesseract.image_to_string(img) textIt appears that the image contains a math problem related to calculus, specifically the directional derivative and gradient. The content mentions:

Directional Derivative: You are asked to find the directional derivative of the function $f(r, s) = \ln(3r + 6s + 90)$ at the point $(2, 2, 2)$ in the direction of the vector $\mathbf{v}$ , which is given as $\mathbf{v} = 5\hat{i} + 10\hat{j}$ .
Gradient: The second problem involves finding the gradient of the function $f(x, y) = x \ln(y)$ at the point $P(4, 1)$ , and the vector $\mathbf{u} = -\hat{i} + 2\hat{j}$ is provided.

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Math Problem Analysis

Mathematical Concepts

Directional Derivatives
Gradient of a Function
Vectors and Unit Vectors

Formulas

Directional Derivative: D_uf = ∇f ⋅ u
Gradient of f: ∇f = (∂f/∂x, ∂f/∂y, ...)
Unit Vector: u = v / |v|

Theorems

Gradient points in the direction of steepest ascent
Directional derivative uses the dot product of the gradient and direction vector

Suitable Grade Level

Undergraduate (Calculus Level)

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