The Ultimate Crash Course for STEM Majors multivariable calculus tangent plane free preview

Finding a Tangent Plane to a Surface – Multivariable Calculus Example

This page is nothing more than a free preview of The Ultimate Crash Course for STEM Majors. This multivariable calculus example demonstrates how to find the equation of a tangent plane to a surface using an implicit function, partial derivatives, and the gradient vector.

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[13] Tangent Planes

Find an equation of the tangent plane to the given surface at the specified point.

\[ z=x\sin(x+y), \qquad P(-1,1,0) \]

The easiest formula to use is

\[ f(x,y,z)=0, \qquad P_0=P(x_0,y_0,z_0) \]

\[ f_x(P_0)\Delta x+f_y(P_0)\Delta y+f_z(P_0)\Delta z=0. \]

\[ z=x\sin(x+y) \Rightarrow f(x,y,z)=z-x\sin(x+y)=0 \]

\[ \nabla f = \left\langle -x\cos(x+y)-\sin(x+y), -x\cos(x+y), 1 \right\rangle \]

\[ \nabla f(-1,1,0) = \left\langle -(-1)\cos(-1+1)-\sin(-1+1), -(-1)\cos(-1+1), 1 \right\rangle \]

\[ = \langle 1,1,1\rangle. \]

\[ f_x(x_0,y_0,z_0)(x-x_0) + f_y(x_0,y_0,z_0)(y-y_0) + f_z(x_0,y_0,z_0)(z-z_0) = 0 \]

\[ (1)(x-(-1))+(1)(y-1)+(1)(z-0)=0 \]

\[ x+1+y-1+z=0 \]

\[ x+y+z=0. \]

\[ z=x\sin(x+y), \qquad P(-1,1,0) \]

\[ x+y+z=0 \]

The original lesson includes a 3D visualization of the surface \(z=x\sin(x+y)\) and its tangent plane at \(P(-1,1,0)\).

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