Exercise · ZW-TANGENT-PLANE-005 · ★★☆ Intermediate
The tangent plane to \(z=x^2+3y\) at the point \((1,0,1)\) has normal vector:
Choose one option.
Show answer
- C. \(\langle 2,3,-1\rangle\)
Explanation
The plane is \(z=1+2(x-1)+3y\), that is, \(2x+3y-z=1\), whose normal vector is \(\langle2,3,-1\rangle\). In general the normal is \(\langle f_x,f_y,-1\rangle\); the sign of the third component matters.
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-004 Find the tangent plane to z=√x^2+y^2 at the point (3,4,5).
- 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?
- ZW-TANGENT-PLANE-008 The tangent plane to z=x^2+y^2 at (1,1,2) meets the z-axis at z= …
- ZW-TANGENT-PLANE-009 A tangent plane to z=x^2+y^2 is parallel to the plane z=4x-2y+7. What is the point of tangency (a,b…