## 17Calculus Directional Derivative Application - Normal & Tangent Lines & Planes

##### 17Calculus

This page covers one application of gradient vectors and directional derivatives, to find normal and tangent lines and planes.

To determine the equation of a plane, we need one point and a normal vector. Since the gradient vector 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

Find the tangent plane to the surface $$P(x,y,z) = x^2+y^2-z^3-9$$ at the point $$(3,0,0)$$

Problem Statement

Find the tangent plane to the surface $$P(x,y,z) = x^2+y^2-z^3-9$$ at the point $$(3,0,0)$$

Solution

### Thomas Wernau - 4355 video solution

video by Thomas Wernau

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Find the equation of the tangent plane and the symmetric equations of the normal line to the surface $$2(x-2)^2 + (y-1)^2 + (x-3)^2 = 10$$ at the point $$(3,3,5)$$.

Problem Statement

Find the equation of the tangent plane and the symmetric equations of the normal line to the surface $$2(x-2)^2 + (y-1)^2 + (x-3)^2 = 10$$ at the point $$(3,3,5)$$.

Solution

### Krista King Math - 1482 video solution

video by Krista King Math

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Determine a normal vector and the equation of the tangent plane to the surface $$z = x^2 + 2y^2$$ at the point $$A(2,-1,6)$$.

Problem Statement

Determine a normal vector and the equation of the tangent plane to the surface $$z = x^2 + 2y^2$$ at the point $$A(2,-1,6)$$.

Solution

### Dr Chris Tisdell - 1828 video solution

video by Dr Chris Tisdell

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Find the tangent plane to the surface $$x=u^2, y=u-v^2, z=v^2$$ for $$u,v \geq 0$$ at the point $$(1,0,1)$$.

Problem Statement

Find the tangent plane to the surface $$x=u^2, y=u-v^2, z=v^2$$ for $$u,v \geq 0$$ at the point $$(1,0,1)$$.

Solution

### Michael Hutchings - 2529 video solution

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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

### Dr Chris Tisdell - 805 video solution

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

### Dr Chris Tisdell - 806 video solution

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

### Krista King Math - 809 video solution

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

### MIP4U - 2201 video solution

video by MIP4U

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Find the normal line to the surface $$f(x,y)=x^2-5xy+2$$ at the point $$(1,1,-1)$$

Problem Statement

Find the normal line to the surface $$f(x,y)=x^2-5xy+2$$ at the point $$(1,1,-1)$$

Solution

### Thomas Wernau - 4356 video solution

video by Thomas Wernau

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Find a set of symmetric equations for the normal line to the surface $$xyz=12$$ at $$(2,-2,-3)$$

Problem Statement

Find a set of symmetric equations for the normal line to the surface $$xyz=12$$ at $$(2,-2,-3)$$

Solution

### Thomas Wernau - 4357 video solution

video by Thomas Wernau

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Find a set of symmetric equations for the normal line to the surface $$F(x,y)=x^2-y^2+1$$ at the point $$(1,1,1)$$.

Problem Statement

Find a set of symmetric equations for the normal line to the surface $$F(x,y)=x^2-y^2+1$$ at the point $$(1,1,1)$$.

Solution

### Thomas Wernau - 4358 video solution

video by Thomas Wernau

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Find the set of symmetric equations for the normal line to the surface $$x^2+2y^2 = 20$$ at $$(0,1,3)$$.

Problem Statement

Find the set of symmetric equations for the normal line to the surface $$x^2+2y^2 = 20$$ at $$(0,1,3)$$.

Solution

### Thomas Wernau - 4359 video solution

video by Thomas Wernau

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