## 17Calculus - Applications of The Gradient

Using Vectors

Applications

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There are several applications where the gradient is used and has additional meanings based on what the function represents. Here are three of them.

Temperature

When the function f is a temperature, the gradient can be interpreted as the change in temperature.

Practice

If we have a material where the heat flow (from hot to cold) is given by $$T = x^3 - 3xy^2$$, determine the direction of maximum decrease of temperature at the point $$(1,2)$$.

Problem Statement

If we have a material where the heat flow (from hot to cold) is given by $$T = x^3 - 3xy^2$$, determine the direction of maximum decrease of temperature at the point $$(1,2)$$.

Solution

### 802 video

video by Dr Chris Tisdell

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Tangent Plane

To determine the equation of a plane, we need one point and a normal vector. Since the gradient is normal to the level curves (and also to the tangent plane), we have a normal vector and we are usually given a point in the problem statement. This is enough information to determine the equation of the tangent plane.

Here is a video clip with a great explanation of this idea. He also explains how using the gradient notation simplifies the notation significantly.

### Dr Chris Tisdell - Gradient and Directional Derivative (Part 4) [5mins-39secs]

video by Dr Chris Tisdell

Practice

Calculate the tangent plane and the normal line to the surface $$x^2 + y^2 + z = 9$$ at the point $$(1,2,4)$$.

Problem Statement

Calculate the tangent plane and the normal line to the surface $$x^2 + y^2 + z = 9$$ at the point $$(1,2,4)$$.

Solution

### 805 video

video by Dr Chris Tisdell

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Calculate the gradient, directional derivative and equation of the tangent plane of $$g(x,y,z) = \sin(xyz)$$ at the point $$(\pi, 1/2, 1/2)$$ in the direction $$\vec{v} = \langle \sqrt{3},-2,3 \rangle$$.

Problem Statement

Calculate the gradient, directional derivative and equation of the tangent plane of $$g(x,y,z) = \sin(xyz)$$ at the point $$(\pi, 1/2, 1/2)$$ in the direction $$\vec{v} = \langle \sqrt{3},-2,3 \rangle$$.

Solution

### 806 video

video by Dr Chris Tisdell

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Calculate the equation of the tangent plane to $$x^4 + xy + y^2 = 19$$ at the point $$(2,-3)$$.

Problem Statement

Calculate the equation of the tangent plane to $$x^4 + xy + y^2 = 19$$ at the point $$(2,-3)$$.

Solution

### 809 video

video by Krista King Math

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Calculate the equation of the tangent plane to the surface $$f(x,y)=2y\cos(5x-3y)$$ at the point $$(3,5,10)$$.

Problem Statement

Calculate the equation of the tangent plane to the surface $$f(x,y)=2y\cos(5x-3y)$$ at the point $$(3,5,10)$$.

Solution

### 2201 video

video by MIP4U

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You CAN Ace Calculus

 partial derivatives basics of vectors gradients

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