Exercise · ZW-TANGENT-PLANE-004 · ★★☆ Intermediate
Find the tangent plane to \(z=\sqrt{x^2+y^2}\) at the point \((3,4,5)\).
Choose one option.
Show answer
- B. \(z=\frac{3 x}{5} + \frac{4 y}{5}\)
Explanation
\(f_x=\frac{x}{\sqrt{x^2+y^2}}=\frac35\) and \(f_y=\frac45\). Then \(z=5+\frac35(x-3)+\frac45(y-4)=\frac35x+\frac45y\) (the plane passes through the origin because the cone is homogeneous).
Review the concept: Tangent plane
The tangent plane to the graph of a function \(f(x,y)\) at the point \((a,b,f(a,b))\) is the plane that best approximates the surface near that point and contains the tangent lines of its two coordinate sections.
\[z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\]
More practice problems: tangent plane
- ZW-TANGENT-PLANE-001 Find the tangent plane to z=x^2+y^2 at the point (1,2,5).
- ZW-TANGENT-PLANE-002 Find the tangent plane to z=xy at the point (2,3,6).
- ZW-TANGENT-PLANE-003 Find the tangent plane to z=x^2y at (1,1,1).
- ZW-TANGENT-PLANE-005 The tangent plane to z=x^2+3y at the point (1,0,1) has normal vector:
- ZW-TANGENT-PLANE-006 Use the tangent plane to z=√x^2+y^2 at (3,4) to approximate √3.03^2+3.96^2.
- ZW-TANGENT-PLANE-007 At which point (a,b) is the tangent plane to z=x^2+y^2-2x+4y horizontal?
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