Tangent Planes and Linear Approximations

Main idea

Zoom in far enough on a smooth surface and it looks flat, the way the Earth looks flat from your backyard. That flat piece is the tangent plane, built from the two partial slopes at the point, and near the point it's a great stand-in for the surface.

z = f(a, b) + {\htmlData{sym=0}{f_x(a, b)(x - a)}} + {\htmlData{sym=1}{f_y(a, b)(y - b)}}

What the symbols mean

$f_x(a, b)(x - a)$
Change from moving in x
Slope in the x-direction times the x-distance.
$f_y(a, b)(y - b)$
Change from moving in y
Slope in the y-direction times the y-distance.

Explanation

A tangent plane is simply the flat sheet that best matches a curved surface at one point, like a piece of cardboard resting on top of a ball. It's what engineers use to swap a hard, curved formula for an easy, flat one near a point they care about. The formula $z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$ looks long, but it's three plain pieces: $f(a, b)$ (the height where you start), $f_x(a, b)(x - a)$ (the east slope times how far east you go) and $f_y(a, b)(y - b)$ (the north slope times how far north). Start at the height, add the two changes. Bear with it; you've already done this in one variable.

Take a look at the tangent line formula from Calculus I: $y = f(a) + f'(a)(x - a)$ . Start at the height, add slope times distance. The tangent plane is the same thing with one more direction. And this isn't really new even in this course: it's the differential $dz = f_x\,dx + f_y\,dy$ from the last unit, added onto the starting height.

Why are those the right slopes? Slice the surface along $y = b$ and you get a curve with slope $f_x$ at the point; slice along $x = a$ and you get one with slope $f_y$ . The tangent plane has to contain the tangent lines of both slices, so its slope going east has to be $f_x$ and its slope going north has to be $f_y$ . A plane is fixed once you know a point and two slopes, so that's the plane.

Linear approximation. Near $(a, b)$ , $f(x, y) \approx$ the tangent plane. This is the two-variable version of $f(x) \approx f(a) + f'(a)(x - a)$ , and it works for the same reason: zoomed in, the surface and the plane are nearly the same.

Normal vector. Move everything to one side and the plane reads $f_x(x - a) + f_y(y - b) - (z - z_0) = 0$ . That's the "normal vector dot displacement equals zero" form from the planes lesson, so the normal is sitting in the coefficients: $\langle f_x, f_y, -1\rangle$ .

See it

A small patch of tangent plane riding across the dome z = 2 − 0.3x² − 0.2y² at height y = 0.5. Notice how it tilts as it moves: its x-slope is −0.6x, so it leans one way on the left, lies level in x at the middle, and leans the other way on the right. Close to the point, the patch and the surface are nearly impossible to tell apart.

Worked examples

Example 1

Find the tangent plane to $z = x^2 + y^2$ at $(1, 2)$ and use it to estimate $f(1.1, 1.9)$ .

1
At $(1, 2)$ the height is $f(1, 2) = 5$ , and the slopes are $f_x = 2x = 2$ and $f_y = 2y = 4$ . So the plane is $z = 5 + 2(x - 1) + 4(y - 2)$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Find the tangent plane to $z = x^2 + y^2$ at $(1, 2)$ . Give $z$ in terms of $x$ and $y$ .

Problem 2+5 XP

Use it to estimate $f(1.1, 1.9)$ .

Problem 3+5 XP

Find the tangent plane to $z = xy$ at $(2, 3)$ .

Problem 4+5 XP

Estimate $\sqrt{x^2 + y^2}$ at $(3.1, 3.9)$ by linearizing at $(3, 4)$ .

Problem 5+5 XP

At a point where $f_x = 2$ and $f_y = -1$ , give a normal vector to the tangent plane of $z = f(x, y)$ .

Problem 6+5 XP

A heater's power is $P(V, I) = VI$ . At $V = 120$ V and $I = 10$ A, use the tangent plane (linear approximation) to estimate $P$ when $V = 118$ V and $I = 10.3$ A.

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Find the point on the surface $z = x^2 + y^2$ where the tangent plane is parallel to the plane $2x + 4y - z = 0$ . Give $(x, y, z)$ .

Where it's used

Engineers linearize nonlinear models around an operating point (a plane flying straight and level, a chemical reactor at steady state) to design controllers with linear math.

Small-Signal Power in a ResistorCalc III · Electrical Engineering

Directional Derivatives and the Gradient The Gradient, Tangent Planes and Normal Lines