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Given w f(z,y, 2) = zy²z. Find: Ograd/(1,3,-2) 3. (b) equation of the tangent plane to the surface zys = 12 at the point (1,3, -2). (c) directional derivative f-(1,3, -2) in the direction of a unit vector making angle of 60" with grad f(1, 3, -2).

Given w f(z,y, 2) = zy²z. Find: Ograd/(1,3,-2) 3. (b) equation of the tangent plane to the surface zys = 12 at the point (1,3, -2). (c) directional derivative f-(1,3, -2) in the direction of a unit vector making angle of 60" with grad f(1, 3, -2).

Calculus: Early Transcendentals
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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Question
### Problem 3 Given the function \( w = f(x, y, z) = xy^2 z \). **Tasks:** (a) **Find \( \text{grad} f(1, 3, -2) \):** - Calculate the gradient of the function at the point \((1, 3, -2)\). (b) **Equation of the tangent plane to the surface \( xy^2 z = 12 \) at the point \((1, 3, -2)\):** - Derive the equation for the tangent plane using the given surface and point. (c) **Directional derivative \( f_{\mathbf{u}}(1, 3, -2) \) in the direction of a unit vector making an angle of \( 60^\circ \) with \( \text{grad} f(1, 3, -2) \):** - Compute the directional derivative in the specified direction from the gradient at the point \((1, 3, -2)\). ### Explanation: 1. **Gradient Calculation:** - Find the partial derivatives \(\frac{\partial w}{\partial x}, \frac{\partial w}{\partial y}, \frac{\partial w}{\partial z} \). - Evaluate these partial derivatives at \((1, 3, -2)\) to find the gradient vector. 2. **Tangent Plane:** - Use the point and gradient information to construct the tangent plane equation. 3. **Directional Derivative:** - Use the formula for the directional derivative involving the dot product of the gradient vector and the unit vector in the given direction. This problem helps in understanding how to work with multivariable calculus concepts like gradient, tangent planes, and directional derivatives.
Transcribed Image Text:### Problem 3 Given the function \( w = f(x, y, z) = xy^2 z \). **Tasks:** (a) **Find \( \text{grad} f(1, 3, -2) \):** - Calculate the gradient of the function at the point \((1, 3, -2)\). (b) **Equation of the tangent plane to the surface \( xy^2 z = 12 \) at the point \((1, 3, -2)\):** - Derive the equation for the tangent plane using the given surface and point. (c) **Directional derivative \( f_{\mathbf{u}}(1, 3, -2) \) in the direction of a unit vector making an angle of \( 60^\circ \) with \( \text{grad} f(1, 3, -2) \):** - Compute the directional derivative in the specified direction from the gradient at the point \((1, 3, -2)\). ### Explanation: 1. **Gradient Calculation:** - Find the partial derivatives \(\frac{\partial w}{\partial x}, \frac{\partial w}{\partial y}, \frac{\partial w}{\partial z} \). - Evaluate these partial derivatives at \((1, 3, -2)\) to find the gradient vector. 2. **Tangent Plane:** - Use the point and gradient information to construct the tangent plane equation. 3. **Directional Derivative:** - Use the formula for the directional derivative involving the dot product of the gradient vector and the unit vector in the given direction. This problem helps in understanding how to work with multivariable calculus concepts like gradient, tangent planes, and directional derivatives.
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