Section 4.4 - Tangent Planes and Linear Approximations in Multivariable Calculus

Section 4.4 - Tangent Planes and Linear Approximations

Tangent Planes

In multivariable calculus, the concept of a tangent plane generalizes the idea of a tangent line to functions of two variables. The tangent plane provides a linear approximation to the surface at a given point.

• Definition: Suppose a surface has equation $z = f(x, y)$, where $f$ has continuous first partial derivatives. Let $(a, b)$ be a point in the domain of $f$, and $P(a, b, f(a, b))$ be the corresponding point on the surface.

• Tangent Plane Equation: The equation of the tangent plane to the surface $z = f(x, y)$ at the point $P(a, b, f(a, b))$ is: $z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$ where $f_x(a, b)$ and $f_y(a, b)$ are the partial derivatives of $f$ with respect to $x$ and $y$ evaluated at $(a, b)$.

• Geometric Interpretation: The tangent plane at $P$ contains the tangent lines to the curves formed by intersecting the surface with planes $x = a$ and $y = b$.

• Example: Find the tangent plane to the elliptic paraboloid $z = 2x^2 + y^2$ at the point $(1, 1, 3)$.

  • Compute partial derivatives:

    • $f_x(x, y) = 4x$

    • $f_y(x, y) = 2y$

  • At $(1, 1)$:

    • $f_x(1, 1) = 4$

    • $f_y(1, 1) = 2$

    • $f(1, 1) = 2(1)^2 + (1)^2 = 3$

  • Equation: $z = 3 + 4(x - 1) + 2(y - 1)$

Linear Approximations

Linear approximation uses the tangent plane to estimate the value of a function near a given point. This is analogous to using the tangent line for functions of one variable.

• Linear Approximation Formula: $L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$ This formula gives the linear approximation of $f(x, y)$ near $(a, b)$.

• Application: The linear approximation is useful for estimating function values when exact computation is difficult.

• Example: Find the linear approximation of $f(x, y) = x^4 + y^2 - 2x^2y$ at $(1, -2)$.

  • Compute partial derivatives:

    • $f_x(x, y) = 4x^3 - 4xy$

    • $f_y(x, y) = 2y - 2x^2$

  • At $(1, -2)$:

    • $f_x(1, -2) = 4(1)^3 - 4(1)(-2) = 4 + 8 = 12$

    • $f_y(1, -2) = 2(-2) - 2(1)^2 = -4 - 2 = -6$

    • $f(1, -2) = (1)^4 + (-2)^2 - 2(1)^2(-2) = 1 + 4 + 4 = 9$

  • Linear approximation: $L(x, y) = 9 + 12(x - 1) - 6(y + 2)$

Differentials

Differentials provide a way to approximate changes in a function based on changes in its variables. For functions of two variables, the differential $dz$ is defined using partial derivatives.

• Definition: For a differentiable function $z = f(x, y)$, the differential $dz$ is: $dz = f_x(x, y)dx + f_y(x, y)dy$ where $dx$ and $dy$ are independent variables representing small changes in $x$ and $y$.

• Interpretation: The differential $dz$ approximates the change in $z$ for small changes in $x$ and $y$.

• Example: If $z = x^2y$, then $f_x(x, y) = 2xy$ and $f_y(x, y) = x^2$. Thus, $dz = 2xy \, dx + x^2 \, dy$

Increment and Differentiability

The increment $riangle z$ represents the change in the value of $z$ as $(x, y)$ changes from $(a, b)$ to $(a + riangle x, b + riangle y)$.

• Increment Formula: $riangle z = f(a + riangle x, b + riangle y) - f(a, b)$

• Differentiability: If $z = f(x, y)$ is differentiable at $(a, b)$, then $riangle z$ can be expressed as: $riangle z = f_x(a, b) riangle x + f_y(a, b) riangle y + ext{error terms}$

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

Summary Table: Tangent Plane and Linear Approximation Formulas

Concept	Formula	Description
Tangent Plane	$z = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$	Equation of the tangent plane to $z = f(x, y)$ at $(a, b)$
Linear Approximation	$L(x, y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)$	Linear estimate of $f(x, y)$ near $(a, b)$
Differential	$dz = f_x(x, y)dx + f_y(x, y)dy$	Approximate change in $z$ for small changes in $x$ and $y$
Increment	$riangle z = f(a + riangle x, b + riangle y) - f(a, b)$	Actual change in $z$ as $(x, y)$ changes

Additional info: The notes also briefly mention the conditions for differentiability and the use of differentials for approximating changes in multivariable functions. These concepts are foundational for understanding error estimation and local linearity in calculus.