The Central Idea
For a function of one variable,
\[ y=f(x), \]the tangent line gives the best local linear approximation near a point.
For a function of two variables,
\[ z=f(x,y), \]the corresponding local linear object is a tangent plane.
Near \((a,b)\), the surface is approximated by
\[ \boxed{ z \approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) }. \]From Tangent Lines to Tangent Planes
In one-variable calculus, the tangent-line equation at \(x=a\) is
\[ y = f(a)+f'(a)(x-a). \]In two variables, the single slope \(f'(a)\) is replaced by two coordinate slopes:
\[ f_x(a,b) \qquad\text{and}\qquad f_y(a,b). \]Those two slopes determine the local plane.
The Tangent-Plane Formula
If \(f\) is differentiable at \((a,b)\), the tangent plane to
\[ z=f(x,y) \]at the point
\[ (a,b,f(a,b)) \]is
\[ \boxed{ z-f(a,b) = f_x(a,b)(x-a) + f_y(a,b)(y-b) }. \]Linearization Form
Solving the tangent-plane equation for \(z\) gives
\[ \boxed{ L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) }. \]The function \(L(x,y)\) is called the linearization of \(f\) at \((a,b)\).
Why It Is Called Linear
The expression
\[ f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) \]is affine-linear in \(x\) and \(y\).
Its graph is a plane, so it gives the first-order flat approximation to the original surface.
What the Tangent Plane Must Match
A correct tangent plane at \((a,b)\) matches three pieces of local data:
- the function value \(f(a,b)\);
- the \(x\)-direction slope \(f_x(a,b)\);
- the \(y\)-direction slope \(f_y(a,b)\).
These are exactly the first-order data available from the function and its gradient.
Example: A Basic Tangent Plane
Let
\[ f(x,y)=x^2+y^2 \]and find the tangent plane at \((1,2)\).
First,
\[ f(1,2)=1+4=5. \]The partial derivatives are
\[ f_x=2x, \qquad f_y=2y. \]Therefore,
\[ f_x(1,2)=2, \qquad f_y(1,2)=4. \]Complete the Tangent Plane
Substitute into the formula:
\[ z-5 = 2(x-1)+4(y-2). \]Therefore,
\[ \boxed{ z = 5+2(x-1)+4(y-2) }. \]Simplifying,
\[ \boxed{ z=2x+4y-5 }. \]Checking the Point
A quick check is to substitute \(x=1\) and \(y=2\) into the tangent plane:
\[ z = 2(1)+4(2)-5 = 5. \]The plane passes through the correct surface point
\[ (1,2,5). \]Checking the Coordinate Slopes
The plane
\[ z=2x+4y-5 \]has \(x\)-direction slope \(2\) and \(y\)-direction slope \(4\).
These match
\[ f_x(1,2)=2 \]and
\[ f_y(1,2)=4. \]Geometric Interpretation
The tangent plane is the plane that best matches the surface's first-order behavior at the point.
If we zoom in sufficiently near a differentiable point, the difference between the surface and its tangent plane becomes small relative to the size of the input displacement.
Linear Approximation
When \((x,y)\) is close to \((a,b)\), we use
\[ \boxed{ f(x,y)\approx L(x,y) }. \]Explicitly,
\[ \boxed{ f(x,y) \approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) }. \]Increment Form
Write
\[ x=a+\Delta x, \qquad y=b+\Delta y. \]Then
\[ \boxed{ f(a+\Delta x,b+\Delta y) \approx f(a,b) + f_x(a,b)\Delta x + f_y(a,b)\Delta y }. \]Approximate Change
Define the actual change
\[ \Delta f = f(a+\Delta x,b+\Delta y)-f(a,b). \]For small input changes,
\[ \boxed{ \Delta f \approx f_x(a,b)\Delta x + f_y(a,b)\Delta y }. \]The Total Differential
The first-order differential is
\[ \boxed{ df = f_x\,dx+f_y\,dy }. \]When \(dx\) and \(dy\) represent small input changes, \(df\) approximates the actual output change \(\Delta f\).
Gradient Form of the Differential
Since
\[ \nabla f = \langle f_x,f_y\rangle, \]we can write
\[ \boxed{ df = \nabla f \cdot \langle dx,dy\rangle }. \]The gradient is therefore the coefficient vector of first-order change.
Vector Form of Linearization
Let
\[ \mathbf{a}=\langle a,b\rangle \]and
\[ \mathbf{x}=\langle x,y\rangle. \]Then the linearization can be written compactly as
\[ \boxed{ L(\mathbf{x}) = f(\mathbf{a}) + \nabla f(\mathbf{a}) \cdot (\mathbf{x}-\mathbf{a}) }. \]Example: Estimating a Nearby Value
Let
\[ f(x,y)=\sqrt{x^2+y^2} \]and estimate
\[ f(3.02,3.96) \]using the nearby point \((3,4)\).
At \((3,4)\),
\[ f(3,4)=5. \]Compute the Partial Derivatives
We have
\[ f_x = \frac{x}{\sqrt{x^2+y^2}} \]and
\[ f_y = \frac{y}{\sqrt{x^2+y^2}}. \]Therefore,
\[ f_x(3,4)=\frac35, \qquad f_y(3,4)=\frac45. \]Complete the Estimate
The input changes are
\[ \Delta x=0.02, \qquad \Delta y=-0.04. \]Thus,
\[ \Delta f \approx \frac35(0.02) + \frac45(-0.04). \]Therefore,
\[ \Delta f \approx 0.012-0.032 = -0.020. \]Hence,
\[ \boxed{ f(3.02,3.96)\approx4.98 }. \]Why Nearby Base Points Matter
Linear approximation is usually best when the target input is close to the base point.
A good base point should ideally have:
- a known or easily computed function value;
- easy partial derivatives;
- small differences from the target input.
Example: Linearization of a Square Root
Let
\[ f(x,y)=\sqrt{4+x-y} \]and linearize at \((0,0)\).
First,
\[ f(0,0)=2. \]The partial derivatives are
\[ f_x = \frac{1}{2\sqrt{4+x-y}} \]and
\[ f_y = -\frac{1}{2\sqrt{4+x-y}}. \]At \((0,0)\),
\[ f_x=\frac14, \qquad f_y=-\frac14. \]Complete the Linearization
Therefore,
\[ L(x,y) = 2 + \frac14x - \frac14y. \]Thus,
\[ \boxed{ L(x,y) = 2+\frac{x-y}{4} }. \]Using the Linearization
To estimate
\[ \sqrt{4.03}, \]we can take
\[ x=0.03, \qquad y=0. \]Then
\[ L(0.03,0) = 2+\frac{0.03}{4} = 2.0075. \]So
\[ \boxed{ \sqrt{4.03}\approx2.0075 }. \]Tangent Plane Through a Known Point
For
\[ z=f(x,y), \]the tangent plane at \((a,b,f(a,b))\) must pass through that point.
The point-normal form
\[ z-z_0 = m_x(x-a)+m_y(y-b) \]becomes the tangent-plane formula when
\[ z_0=f(a,b), \quad m_x=f_x(a,b), \quad m_y=f_y(a,b). \]Implicit Surfaces
Tangent planes also arise for surfaces given implicitly by
\[ F(x,y,z)=c. \]The gradient
\[ \nabla F \]is normal to the level surface.
This gives another form of the tangent-plane equation.
Tangent Plane to an Implicit Surface
At a point
\[ (x_0,y_0,z_0) \]on the level surface
\[ F(x,y,z)=c, \]where \(\nabla F\neq\mathbf{0}\), the tangent plane is
\[ \boxed{ F_x(x_0,y_0,z_0)(x-x_0) + F_y(x_0,y_0,z_0)(y-y_0) + F_z(x_0,y_0,z_0)(z-z_0) = 0 }. \]Example: Tangent Plane to a Sphere
Consider the sphere
\[ x^2+y^2+z^2=9. \]Define
\[ F(x,y,z)=x^2+y^2+z^2. \]Then
\[ \nabla F = \langle2x,2y,2z\rangle. \]At the point \((1,2,2)\),
\[ \nabla F(1,2,2) = \langle2,4,4\rangle. \]Complete the Sphere Tangent Plane
Using the normal vector \(\langle2,4,4\rangle\),
\[ 2(x-1)+4(y-2)+4(z-2)=0. \]Dividing by \(2\),
\[ \boxed{ (x-1)+2(y-2)+2(z-2)=0 }. \]Equivalently,
\[ \boxed{ x+2y+2z=9 }. \]Normal Lines
For an implicit surface
\[ F(x,y,z)=c, \]the gradient also gives the direction of the normal line.
At
\[ P=(x_0,y_0,z_0), \]a normal line can be written
\[ \boxed{ \mathbf{r}(t) = \mathbf{P} + t\nabla F(P) }. \]Tangent Plane and Directional Derivatives
The tangent plane predicts every first-order directional rate.
For a unit direction
\[ \mathbf{u} = \langle u_1,u_2\rangle, \]its predicted slope is
\[ f_x(a,b)u_1+f_y(a,b)u_2. \]This is exactly
\[ D_{\mathbf{u}}f(a,b). \]Differentiability and Linear Approximation
The existence of partial derivatives alone does not always guarantee a good tangent-plane approximation.
Differentiability means that the function can be approximated by one linear map whose error becomes negligible compared with the size of the input displacement.
Formal Error Statement
If \(f\) is differentiable at \((a,b)\), then
\[ f(a+\Delta x,b+\Delta y) = f(a,b) + f_x(a,b)\Delta x + f_y(a,b)\Delta y + E, \]where the error satisfies
\[ \boxed{ \frac{E} {\sqrt{(\Delta x)^2+(\Delta y)^2}} \to0 } \]as
\[ (\Delta x,\Delta y)\to(0,0). \]What the Error Condition Means
The error does not merely approach zero.
It becomes small faster than the distance from the base point.
That is what makes the tangent plane a genuine first-order approximation.
A Practical Differentiability Criterion
A standard sufficient condition is:
if \(f_x\) and \(f_y\) exist in a neighborhood of \((a,b)\) and are continuous at \((a,b)\), then \(f\) is differentiable there.
This criterion is widely used because continuity of the first partials is often easy to verify.
Local Flatness
Differentiability is the multivariable version of local flatness.
A differentiable curve looks like its tangent line at small scales.
A differentiable surface looks like its tangent plane at small scales.
Relative Error Intuition
Linear approximation is a first-order method.
If the target point is moved farther from the base point, curvature effects become more important and the approximation usually becomes less accurate.
Higher-order methods can improve this, but the tangent plane remains the fundamental local model.
Approximation and Measurement Error
Suppose a measured quantity
\[ Q=f(x,y) \]depends on measured inputs \(x\) and \(y\).
Small measurement errors \(dx\) and \(dy\) produce an approximate output error
\[ \boxed{ dQ = f_x\,dx+f_y\,dy }. \]Worst-Case First-Order Error Bound
Taking absolute values gives
\[ |dQ| \le |f_x||dx| + |f_y||dy|. \]Thus a practical first-order error estimate is
\[ \boxed{ |dQ| \lesssim |f_x||dx| + |f_y||dy| }. \]Example: Error in Rectangle Area
Let
\[ A=xy \]be the area of a rectangle.
Then
\[ dA = y\,dx+x\,dy. \]If
\[ x=10,\qquad y=6, \]with measurement errors
\[ |dx|\le0.1, \qquad |dy|\le0.05, \]then
\[ |dA| \lesssim 6(0.1)+10(0.05) = 1.1. \]So the first-order area error is at most approximately
\[ \boxed{1.1} \]square units.
Relative Differential
For a nonzero quantity \(Q\), the ratio
\[ \frac{dQ}{Q} \]approximates the relative change.
Multiplying by \(100\%\) gives an approximate percentage change.
Example: Relative Change in a Product
If
\[ Q=xy, \]then
\[ dQ=y\,dx+x\,dy. \]Divide by \(Q=xy\):
\[ \frac{dQ}{Q} = \frac{dx}{x} + \frac{dy}{y}. \]Thus,
\[ \boxed{ \text{relative change in }Q \approx \text{relative change in }x + \text{relative change in }y }. \]Functions of Three Variables
For
\[ w=f(x,y,z), \]the first-order approximation at
\[ (a,b,c) \]is
\[ \boxed{ L(x,y,z) = f(a,b,c) + f_x(a,b,c)(x-a) + f_y(a,b,c)(y-b) + f_z(a,b,c)(z-c) }. \]Three-Variable Differential
The total differential becomes
\[ \boxed{ df = f_x\,dx + f_y\,dy + f_z\,dz }. \]In vector notation,
\[ \boxed{ df = \nabla f\cdot \langle dx,dy,dz\rangle }. \]Higher Dimensions
For a differentiable scalar function
\[ f:\mathbb{R}^n\to\mathbb{R}, \]the first-order approximation at \(\mathbf{a}\) is
\[ \boxed{ f(\mathbf{x}) \approx f(\mathbf{a}) + \nabla f(\mathbf{a}) \cdot (\mathbf{x}-\mathbf{a}) }. \]The geometry may be harder to draw, but the algebraic structure remains the same.
Common Error: Forgetting the Function Value
The tangent plane is not merely
\[ f_x(a,b)(x-a)+f_y(a,b)(y-b). \]It must include the base height:
\[ \boxed{ f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) }. \]Common Error: Using \(x\) and \(y\) Instead of Increments
The tangent-plane formula uses
\[ x-a \]and
\[ y-b, \]not simply \(x\) and \(y\), unless the base point happens to be the origin.
Common Error: Evaluating at the Target Instead of the Base Point
Linearization uses
\[ f(a,b), \qquad f_x(a,b), \qquad f_y(a,b) \]evaluated at the chosen base point.
The nearby target point is inserted only after the linearization has been constructed.
Common Error: Confusing \(\Delta f\) and \(df\)
The actual change is
\[ \Delta f = f(a+\Delta x,b+\Delta y)-f(a,b). \]The differential is
\[ df = f_x\,dx+f_y\,dy. \]For small changes,
\[ \Delta f\approx df, \]but they are not generally identical.
Common Error: Assuming Every Surface Has a Tangent Plane
A tangent plane in the first-order approximation sense requires appropriate differentiability.
A surface with a corner, cusp, or sufficiently irregular local behavior may fail to have one well-defined tangent plane.
Common Error: Treating Approximation as Equality
Linearization gives
\[ f(x,y)\approx L(x,y), \]not necessarily
\[ f(x,y)=L(x,y). \]Equality occurs everywhere only when the original function is itself affine-linear.
A Reliable Tangent-Plane Procedure
- Identify the base point \((a,b)\).
- Compute \(f(a,b)\).
- Compute \(f_x\) and \(f_y\).
- Evaluate \(f_x(a,b)\) and \(f_y(a,b)\).
- Substitute into the tangent-plane formula.
- Check that the plane passes through \((a,b,f(a,b))\).
A Reliable Linear-Approximation Procedure
- Choose a convenient base point near the target input.
- Construct the linearization \(L\).
- Compute the small input changes.
- Evaluate \(L\) at the target point.
- Interpret the result as an approximation.
- Keep in mind that accuracy generally decreases as the target moves farther from the base point.
Example: Full Tangent-Plane Workflow
Let
\[ f(x,y)=x^2y+y^3 \]and find the tangent plane at \((1,1)\).
First,
\[ f(1,1)=2. \]The partial derivatives are
\[ f_x=2xy \]and
\[ f_y=x^2+3y^2. \]At \((1,1)\),
\[ f_x=2, \qquad f_y=4. \]Complete the Full Workflow
The tangent plane is
\[ z-2 = 2(x-1)+4(y-1). \]Therefore,
\[ \boxed{ z = 2+2(x-1)+4(y-1) }. \]Simplifying,
\[ \boxed{ z=2x+4y-4 }. \]Practice 1: Basic Tangent Plane
Find the tangent plane to
\[ f(x,y)=x^2+y^2 \]at \((2,1)\).
Practice 2: Linearization
Find the linearization of
\[ f(x,y)=xy+x \]at \((1,2)\).
Practice 3: Estimate a Nearby Value
Use linear approximation at \((1,1)\) to estimate
\[ f(1.02,0.97) \]for
\[ f(x,y)=x^2+y^2. \]Practice 4: Differential
Find \(df\) for
\[ f(x,y)=x^3y+2y^2. \]Practice 5: Approximate Change
For
\[ f(x,y)=x^2y, \]estimate \(\Delta f\) near \((2,3)\) when
\[ \Delta x=0.01, \qquad \Delta y=-0.02. \]Practice 6: Implicit Tangent Plane
Find the tangent plane to
\[ x^2+y^2+z^2=14 \]at \((1,2,3)\).
Practice 7: Normal Line
Find a vector equation for the normal line to
\[ x^2+y^2+z^2=14 \]at \((1,2,3)\).
Practice 8: Measurement Error
The area of a rectangle is
\[ A=xy. \]At \(x=8\), \(y=5\), estimate the maximum first-order area error if
\[ |dx|\le0.02, \qquad |dy|\le0.03. \]Practice 9: Three-Variable Linearization
Find the linearization of
\[ f(x,y,z)=x^2+y^2+z^2 \]at \((1,1,1)\).
Practice 10: Identify the Error
A student writes the tangent plane to \(z=f(x,y)\) at \((a,b)\) as
\[ z = f_x(a,b)x + f_y(a,b)y. \]Explain what is missing.
Practice Answers
Practice 1
We have
\[ f(2,1)=5, \] \[ f_x=2x, \qquad f_y=2y. \]Therefore,
\[ f_x(2,1)=4, \qquad f_y(2,1)=2. \]The tangent plane is
\[ \boxed{ z-5=4(x-2)+2(y-1) }. \]Practice 2
For
\[ f(x,y)=xy+x, \]we have
\[ f(1,2)=3, \] \[ f_x=y+1, \qquad f_y=x. \]At \((1,2)\),
\[ f_x=3, \qquad f_y=1. \]Thus,
\[ \boxed{ L(x,y) = 3+3(x-1)+(y-2) }. \]Practice 3
At \((1,1)\),
\[ f=2, \qquad f_x=2, \qquad f_y=2. \]Therefore,
\[ L(x,y)=2+2(x-1)+2(y-1). \]At \((1.02,0.97)\),
\[ L = 2+2(0.02)+2(-0.03) = 1.98. \]Hence,
\[ \boxed{ f(1.02,0.97)\approx1.98 }. \]Practice 4
The partial derivatives are
\[ f_x=3x^2y \]and
\[ f_y=x^3+4y. \]Therefore,
\[ \boxed{ df = 3x^2y\,dx + (x^3+4y)\,dy }. \]Practice 5
We have
\[ f_x=2xy, \qquad f_y=x^2. \]At \((2,3)\),
\[ f_x=12, \qquad f_y=4. \]Therefore,
\[ \Delta f \approx 12(0.01)+4(-0.02) = 0.04. \]Thus,
\[ \boxed{ \Delta f\approx0.04 }. \]Practice 6
Let
\[ F(x,y,z)=x^2+y^2+z^2. \]Then
\[ \nabla F = \langle2x,2y,2z\rangle. \]At \((1,2,3)\),
\[ \nabla F = \langle2,4,6\rangle. \]Hence,
\[ \boxed{ 2(x-1)+4(y-2)+6(z-3)=0 }. \]Practice 7
A normal direction is
\[ \langle2,4,6\rangle. \]Therefore,
\[ \boxed{ \mathbf{r}(t) = \langle1,2,3\rangle + t\langle2,4,6\rangle }. \]Practice 8
Since
\[ A=xy, \]we have
\[ dA=y\,dx+x\,dy. \]Thus,
\[ |dA| \lesssim 5(0.02)+8(0.03) = 0.34. \]Therefore,
\[ \boxed{ |dA|\lesssim0.34 } \]square units.
Practice 9
At \((1,1,1)\),
\[ f=3, \qquad f_x=2, \qquad f_y=2, \qquad f_z=2. \]Therefore,
\[ \boxed{ L(x,y,z) = 3 + 2(x-1) + 2(y-1) + 2(z-1) }. \]Practice 10
The formula is missing both the base function value and the shifted input differences.
The correct tangent-plane formula is
\[ \boxed{ z = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) }. \]What to Remember
Tangent planes and linearization extend the central local idea of calculus into several variables.
- The tangent plane matches the function value and first partial derivatives at the base point.
- The linearization is \(L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\).
- Near the base point, \(f(x,y)\approx L(x,y)\).
- The differential \(df=f_x\,dx+f_y\,dy\) approximates small output changes.
- The gradient packages the coefficients of the linear approximation.
- For implicit surfaces, the gradient supplies the normal vector to the tangent plane.
- Linear approximation is fundamentally a first-order model, so nearby inputs matter.
The central relationship is
\[ \boxed{ \text{local surface} \approx \text{tangent plane} = \text{first-order linear model} }. \]Where We Go Next
Tangent planes describe local first-order behavior at individual points. The next major task is accumulation across two- and three-dimensional regions.
The next lesson, Multiple Integrals, extends definite integration to functions of several variables and develops double and triple integrals for area, volume, mass, averages, probability, and other accumulated quantities.