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

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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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