Tangent plane and linear approximation

In one-variable calculus, the tangent line gives the best linear approximation to a curve near a point.

For functions of two variables, the same idea becomes a tangent plane.

1-D tangent line

Let $y=f(x)$ and let $x_0$ be a point in the domain. If $f$ is differentiable at $x_0$ , then the tangent line at the point $(x_0,f(x_0))$ is

\boxed{y = f'(x_0)(x-x_0) + f(x_0).}

This is the line with slope $f'(x_0)$ that passes through $(x_0,f(x_0))$ .

Why this is the best linear approximation

Near $x_0$ , the graph of $f$ behaves almost like a line. The tangent line is the linear function that matches both:

the value of the function at the point, and
the slope of the function at the point.

So if $x$ is close to $x_0$ , then

f(x) \approx f(x_0) + f'(x_0)(x-x_0).

The right-hand side is the tangent-line formula written as a linear approximation.

1-D tangent line: best linear approximation near x₀

For the graph above, the curve is $f(x)=x^3-2x+1$ . At $x_0=1$ we have

f(1)=0, \qquad f'(x)=3x^2-2, \qquad f'(1)=1.

So the tangent line is

y = 1(x-1)+0 = x-1.

That line is the best local linear model for the curve near $x=1$ .

From tangent line to tangent plane

For a function of two variables, $z=f(x,y)$ , a tangent line is not enough because the graph is a surface in 3D, not a curve in 2D.

At a point on a surface, we need a plane that best matches the surface near that point. That plane is the tangent plane.

So the one-variable picture

\text{curve} \rightarrow \text{tangent line}

becomes

\text{surface} \rightarrow \text{tangent plane}.

Plane equation and tangent planes

A general plane in 3D can be written as

ax+by+cz+d=0.

This is the same as saying the plane has a normal vector $(a,b,c)$ .

Why we let $c$ be non-zero

If $c\neq 0$ , we can solve for $z$ :

z = -\frac{a}{c}x - \frac{b}{c}y - \frac{d}{c}.

That form is especially useful for a graph of the form $z=f(x,y)$ , because the tangent plane can then be written as a function of $x$ and $y$ .

If $c=0$ , then the plane is vertical and cannot be written as $z=\text{something}$ .

But the graph of a differentiable function $z=f(x,y)$ has a tangent plane that is not vertical at the point of tangency, so we assume the tangent plane can be written with $c\neq 0$ .

Deriving the tangent plane equation

Let the surface be $z=f(x,y)$ and let the point of tangency be $(x_0,y_0,z_0)$ , where

z_0=f(x_0,y_0).

The tangent plane must:

pass through $(x_0,y_0,z_0)$ ,
have slope $f_x(x_0,y_0)$ in the $x$ -direction,
have slope $f_y(x_0,y_0)$ in the $y$ -direction.

That gives the tangent plane

\boxed{z = f(x_0,y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0).}

This is the multivariable analogue of the tangent line formula.

Example: z = 3x^3 + y^2 at (1,1,4)

Consider

f(x,y)=3x^3+y^2.

At $(1,1)$ ,

f(1,1)=3(1)^3+(1)^2=4.

The partial derivatives are

f_x(x,y)=9x^2, \qquad f_y(x,y)=2y.

So at $(1,1)$ ,

f_x(1,1)=9, \qquad f_y(1,1)=2.

Therefore the tangent plane is

\boxed{z = 4 + 9(x-1) + 2(y-1).}

Expanding gives

z = 9x + 2y - 7.

The graph below shows the surface and its tangent plane at the point $(1,1,4)$ .

Surface z = 3x³ + y² with tangent plane at (1,1,4)

Near $(1,1,4)$ , the plane and the surface are very close. If you zoom in enough, the surface looks almost flat, and the tangent plane captures that local flatness.

Examples of tangent planes

Let

f(x,y)=x^2+y^2.

Then

f_x=2x, \qquad f_y=2y.

At $(1,1)$ ,

f(1,1)=2, \qquad f_x(1,1)=2, \qquad f_y(1,1)=2.

So the tangent plane is

z = 2 + 2(x-1) + 2(y-1) = 2x + 2y - 2.

Let

f(x,y)=xy.

Then

f_x=y, \qquad f_y=x.

At $(1,2)$ ,

f(1,2)=2, \qquad f_x(1,2)=2, \qquad f_y(1,2)=1.

So the tangent plane is

z = 2 + 2(x-1) + 1(y-2) = 2x + y - 2.

Let

f(x,y)=e^{x+y}.

Then

f_x=e^{x+y}, \qquad f_y=e^{x+y}.

At $(0,0)$ ,

f(0,0)=1, \qquad f_x(0,0)=1, \qquad f_y(0,0)=1.

So the tangent plane is

z = 1 + x + y.

Let

f(x,y)=\sqrt{9-x^2-y^2}.

At $(1,2)$ ,

f(1,2)=\sqrt{4}=2.

The partial derivatives are

f_x(x,y)=\frac{-x}{\sqrt{9-x^2-y^2}}, \qquad f_y(x,y)=\frac{-y}{\sqrt{9-x^2-y^2}}.

f_x(1,2)=-\frac12, \qquad f_y(1,2)=-1.

Hence the tangent plane is

z = 2 - \frac12(x-1) - (y-2).

Best linear approximation in two variables

The tangent plane is the best linear approximation to the surface near the point.

If $(x,y)$ is close to $(x_0,y_0)$ , then

f(x,y) \approx L(x,y),

where

L(x,y)=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0).

This is the linear function whose graph is the tangent plane.

So the tangent plane plays exactly the same role for surfaces that the tangent line plays for curves.

Example: approximate a function value

Let

f(x,y)=xe^{xy}.

We want to approximate $f(1.1,1.1)$ using the tangent plane at $(1,1)$ .

First compute the value:

f(1,1)=1\cdot e^1=e.

Now compute the partial derivatives.

f_x(x,y)=e^{xy}+xye^{xy}=e^{xy}(1+xy),

f_y(x,y)=x^2e^{xy}.

At $(1,1)$ ,

f_x(1,1)=2e, \qquad f_y(1,1)=e.

So the linear approximation is

L(x,y)=e+2e(x-1)+e(y-1).

At $(1.1,1.1)$ ,

L(1.1,1.1)=e+2e(0.1)+e(0.1)=1.3e.

Numerically,

L(1.1,1.1)\approx 3.534.

\boxed{f(1.1,1.1) \approx 3.534.}

If you compare this with the exact value, you will see that the tangent plane gives a very good local estimate.

Differentiability

A function $f(x,y)$ is differentiable at $(a,b)$ if, near $(a,b)$ , it behaves like a linear function.

In practice, the most useful test in this course is:

if the partial derivatives $f_x$ and $f_y$ exist near $(a,b)$ and are continuous at $(a,b)$ , then $f$ is differentiable at $(a,b)$ .

For a differentiable function, the natural linear approximation is the tangent-plane formula

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

A function $f(x,y)$ is differentiable at $(a,b)$ if there exists a linear map $L(x,y)$ such that

\lim_{(x,y)\to(a,b)} \frac{f(x,y)-L(x,y)}{\sqrt{(x-a)^2+(y-b)^2}} = 0.

For a differentiable function, the linear map is exactly the tangent-plane approximation

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

So differentiability means the error between the surface and its tangent plane becomes small faster than the distance to the point.

Practical meaning

At a differentiable point:

the surface has a well-defined tangent plane,
the partial derivatives fit together into one linear approximation,
the function is locally well behaved.

In later sections, differentiability will be the condition that makes the chain rule and other multivariable rules work cleanly.

Summary

In one variable, the tangent line is

y = f'(x_0)(x-x_0)+f(x_0).

In two variables, the tangent plane is

z = f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0).

Both are best linear approximations near the point.
Differentiability means the function is well approximated by that linear model.

Tangent Plane and Linear Approximation

Tangent plane and linear approximation

1-D tangent line

Why this is the best linear approximation

1-D tangent line: best linear approximation near x₀

From tangent line to tangent plane

Plane equation and tangent planes

Why we let ccc be non-zero

Deriving the tangent plane equation

Example: z = 3x^3 + y^2 at (1,1,4)

Surface z = 3x³ + y² with tangent plane at (1,1,4)

Examples of tangent planes

Best linear approximation in two variables

Example: approximate a function value

Differentiability

Practical meaning

Summary

Why we let $c$ be non-zero