Tangent Planes and Linear Approximation

Quick answer

Near a point where f is differentiable, the surface z = f(x, y) looks flat, and the plane it resembles is the tangent plane: z = f(a, b) + f_x(a, b)(x − a) + f_y(a, b)(y − b). The two partial derivatives are its slopes in the x and y directions. The same formula, read as a function, is the linearization L(x, y), which approximates f near (a, b). Written with small changes, it becomes the differential dz = f_x dx + f_y dy, used to estimate how errors in measured inputs affect a computed output.

What you'll learn

Write the tangent plane to a surface z = f(x, y)
Linearize a function at a point and use it to approximate
Estimate changes and errors with differentials
Explain the tangent plane as the plane with matching slopes

The plane that fits

Zoom in on a smooth surface and it looks flat, the way a smooth curve looks straight under a microscope. In one variable, the straight line it resembles is the tangent line, and the linear approximation $f(a) + f'(a)(x - a)$ uses it to estimate values. In two variables the flat object is a plane. The tangent plane to $z = f(x, y)$ at the point $(a, b, f(a, b))$ is

z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

Why this is the right plane

Every non-vertical plane through $(a, b, f(a, b))$ has the form $z = f(a, b) + m(x - a) + n(y - b)$ , where $m$ is its slope in the $x$ direction and $n$ its slope in the $y$ direction. The surface has slopes of its own at the point: the partial derivatives $f_x(a, b)$ and $f_y(a, b)$ are the slopes of its two slices. The tangent plane is the one whose slices match both: $m = f_x(a, b)$ and $n = f_y(a, b)$ . Two slopes pin down a plane, so matching the two partial derivatives picks out the tangent plane. Each slice of the plane is then the tangent line of the matching slice of the surface.

surface slice: z = 2x² + 1
tangent plane slice: z = 4x − 1

In the slice y = 1, the tangent plane touches the surface

Linearization and differentials

The tangent plane formula, used as a function, is the linearization:

L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) \qquad f(x, y) \approx L(x, y) \text{ near } (a, b)

Writing $dx = x - a$ and $dy = y - b$ for small changes in the inputs, the predicted change in the output is the total differential

dz = f_x\,dx + f_y\,dy

Each input contributes its own rate times its own change, and the contributions add.

Worked examples

Example 4 — Error in a measured volume

A cylinder’s radius is measured as $3$ cm and its height as $10$ cm, each with possible errors of $0.05$ cm and $0.1$ cm. Estimate the largest error in the computed volume $V = \pi r^2h$ .

$dV = V_r\,dr + V_h\,dh = 2\pi rh\,dr + \pi r^2\,dh$ . At the measured values, with the errors taken in the worst direction,

|dV| \le 2\pi(3)(10)(0.05) + \pi(9)(0.1) = 3\pi + 0.9\pi = 3.9\pi \approx 12.25

cubic centimeters, compared with a volume of $90\pi \approx 282.7$ . The radius error contributes more than three times as much as the height error.

Common mistakes

Practice problems

Find the tangent plane to $z = x^2 + y^2$ at $(1, 2, 5)$ .

Answer

$z = 2x + 4y - 5$

Full solution

$f_x = 2x = 2$ and $f_y = 2y = 4$ at the point, so $z = 5 + 2(x - 1) + 4(y - 2)$ .
Linearize $f(x, y) = xy$ at $(2, 3)$ , and use it to estimate $f(2.1, 2.9)$ .

Answer

$L = 6 + 3(x - 2) + 2(y - 3)$ ; about $6.1$

Full solution

$f_x = y = 3$ and $f_y = x = 2$ . $L(2.1, 2.9) = 6 + 0.3 - 0.2 = 6.1$ , and the exact value is $6.09$ .
Linearize $f(x, y) = e^{x - y}$ at $(1, 1)$ .

Answer

$L = 1 + (x - 1) - (y - 1)$

Full solution

$f(1, 1) = 1$ , $f_x = e^{x - y} = 1$ and $f_y = -e^{x - y} = -1$ at the point.
Find the total differential of $z = x^3y^2$ .

Answer

$dz = 3x^2y^2\,dx + 2x^3y\,dy$

Full solution

The two partial derivatives multiply their own changes, and the contributions add.
A rectangle measures $10$ cm by $20$ cm, each with a possible error of $1\%$ . Estimate the largest error in its area.

Answer

$4$ square centimeters

Full solution

$A = xy$ with $dx = 0.1$ and $dy = 0.2$ . Then $|dA| \le y\,|dx| + x\,|dy| = 20(0.1) + 10(0.2) = 4$ , which is $2\%$ of the area $200$ .
For $f(x, y) = \sqrt{x^2 + y^2}$ , why is $L(3.1, 3.9) = 4.98$ so close to the true value?

Answer

The point is close to $(3, 4)$ , where the surface and its tangent plane touch.

Full solution

The inputs moved by only $0.1$ each. The error in a linearization shrinks like the square of the distance moved, so small steps give very small errors.
Find the tangent plane to $z = 4 - x^2 - 2y^2$ at $(1, 1, 1)$ .

Answer

$z = 7 - 2x - 4y$

Full solution

$f_x = -2x = -2$ and $f_y = -4y = -4$ , so $z = 1 - 2(x - 1) - 4(y - 1)$ .
The tangent plane to a surface at $(0, 0, 5)$ is $z = 5 + 3x - 2y$ . Find $f_x(0, 0)$ and $f_y(0, 0)$ .

Answer

$3$ and $-2$

Full solution

The coefficients of $x - a$ and $y - b$ in the tangent plane are the partial derivatives.
For $V = \pi r^2h$ at $r = 3$ , $h = 10$ , which measurement error matters more: $0.1$ cm in $r$ or $0.1$ cm in $h$ ?

Answer

The error in $r$ , by more than six times

Full solution

$V_r\,dr = 60\pi(0.1) = 6\pi$ and $V_h\,dh = 9\pi(0.1) = 0.9\pi$ .
A student writes the tangent plane to $z = 2x^2 + y^2$ at $(1, 1, 3)$ as $z = 3 + 4x(x - 1) + 2y(y - 1)$ . What went wrong?

Hint

Is the student’s answer a plane?

Answer

The partial derivatives must be evaluated at $(1, 1)$ . The plane is $z = 3 + 4(x - 1) + 2(y - 1)$ .

Full solution

With $4x$ and $2y$ left unevaluated, the right side contains $x^2$ and $y^2$ , so it describes a curved surface. The slopes of a plane are constants, here $f_x(1, 1) = 4$ and $f_y(1, 1) = 2$ .

Frequently asked questions

What is the equation of a tangent plane?

z = f(a, b) + f_x(a, b)(x − a) + f_y(a, b)(y − b), the plane through (a, b, f(a, b)) with the same slopes as the surface in the x and y directions.

What is a linearization?

The function L(x, y) given by the tangent plane formula. Near (a, b), L(x, y) is close to f(x, y) and much easier to compute.

What is the total differential?

dz = f_x dx + f_y dy. It estimates the change in z caused by small changes dx and dy in the inputs.

How are differentials used to estimate error?

If the inputs are measured with small errors dx and dy, then |f_x dx| + |f_y dy| bounds the resulting error in f, to first order.

Does every surface have a tangent plane?

No. A cone has none at its tip. A surface has a tangent plane where f is differentiable, which is guaranteed when the partial derivatives are continuous nearby.

What to learn next

The Multivariable Chain RuleDifferentiate a function of several variables along a path, or through a change of variables, by adding one term per route, and differentiate implicit curves.

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