∞Calculus IV Unit 4 – Tangent Planes and Linear Approximations

Key Concepts and Definitions

• Tangent plane defined as a plane that touches a surface at a single point and provides the best linear approximation to the surface near that point
• Partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ represent the rates of change of a function $f(x, y)$ with respect to $x$ and $y$, respectively
  • Calculated by treating one variable as constant while differentiating with respect to the other
• Gradient vector $\nabla f(a, b) = \left(\frac{\partial f}{\partial x}(a, b), \frac{\partial f}{\partial y}(a, b)\right)$ represents the direction of steepest ascent on a surface at the point $(a, b)$
• Normal vector to a tangent plane is perpendicular to the plane and can be found using the gradient vector
• Linear approximation estimates the value of a function near a point using the tangent plane equation
• Directional derivative D_\vec{u}f(a, b) measures the rate of change of a function in the direction of a unit vector $\vec{u}$ at the point $(a, b)$
  • Calculated using the dot product of the gradient vector and the unit vector: D_\vec{u}f(a, b) = \nabla f(a, b) \cdot \vec{u}

Geometric Interpretation of Tangent Planes

• Tangent plane provides a flat approximation to a curved surface at a given point
• Analogous to a tangent line for functions of one variable
• Normal vector to the tangent plane is perpendicular to the surface at the point of tangency
  • Represents the direction in which the surface is not changing
• Gradient vector lies in the tangent plane and points in the direction of steepest ascent
• Tangent plane can be visualized as a flat sheet touching the surface at a single point
  • Useful for understanding local behavior and approximating values near the point of tangency
• Helps analyze critical points and classify them as local minima, local maxima, or saddle points

Finding Equations of Tangent Planes

• Equation of a tangent plane at a point $(a, b, f(a, b))$ on a surface defined by $z = f(x, y)$ is given by:
  • $z - f(a, b) = \frac{\partial f}{\partial x}(a, b)(x - a) + \frac{\partial f}{\partial y}(a, b)(y - b)$
• Steps to find the tangent plane equation:
  1. Calculate the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ at the point $(a, b)$
  2. Substitute the values of $a$, $b$, $f(a, b)$, and the partial derivatives into the tangent plane equation
• Normal vector to the tangent plane is given by $\vec{n} = \left(-\frac{\partial f}{\partial x}(a, b), -\frac{\partial f}{\partial y}(a, b), 1\right)$
  • Can be used to write the tangent plane equation in the form $\vec{n} \cdot (\vec{r} - \vec{r}_0) = 0$, where $\vec{r}_0$ is the position vector of the point of tangency
• Tangent plane equation can be used to approximate values of the function near the point of tangency

Linear Approximation in Multiple Variables

• Linear approximation of a function $f(x, y)$ near a point $(a, b)$ is given by:
  • $L(x, y) = f(a, b) + \frac{\partial f}{\partial x}(a, b)(x - a) + \frac{\partial f}{\partial y}(a, b)(y - b)$
• Approximates the value of $f(x, y)$ for points $(x, y)$ close to $(a, b)$
  • Accuracy decreases as the distance from $(a, b)$ increases
• Useful for estimating function values, analyzing local behavior, and solving optimization problems
• Multivariable version of the linear approximation formula for functions of one variable: $L(x) = f(a) + f'(a)(x - a)$
• Can be extended to functions of more than two variables by including additional partial derivative terms
• Approximation error can be quantified using the second-order partial derivatives and the remainder term of the Taylor series expansion

Applications in Optimization and Error Analysis

• Tangent planes and linear approximations help solve optimization problems involving functions of multiple variables
  • Maximize or minimize a function subject to constraints
• Gradient vector points in the direction of steepest ascent, which is useful for finding local maxima and minima
  • Set the gradient vector equal to zero and solve for critical points
• Second-order partial derivatives test classifies critical points as local maxima, local minima, or saddle points
  • Analogous to the second derivative test for functions of one variable
• Linear approximation estimates values of a function near a known point
  • Helps quantify approximation errors and determine the range of validity for the approximation
• Error propagation analysis uses linear approximations to estimate the uncertainty in a calculated quantity based on uncertainties in the input variables
  • Applicable in experimental sciences, engineering, and numerical analysis
• Sensitivity analysis studies how changes in input variables affect the output of a model or system using linear approximations

Limitations and Edge Cases

• Tangent plane approximation is only valid in a small neighborhood around the point of tangency
  • Accuracy decreases as the distance from the point increases
• Linear approximation may not capture the global behavior of a function, especially if the function is highly nonlinear
• Functions with discontinuities or non-differentiable points do not have well-defined tangent planes or linear approximations at those points
  • Example: the absolute value function $f(x, y) = |x| + |y|$ is not differentiable at the origin $(0, 0)$
• Degenerate cases occur when the gradient vector is zero at a point, resulting in an undefined tangent plane
  • Happens at critical points where both partial derivatives are simultaneously zero
• Higher-order approximations (quadratic, cubic, etc.) may be necessary for more accurate estimates, especially near inflection points or saddle points
• Approximation methods may not be suitable for functions with rapid oscillations or highly localized behavior
  • Fourier analysis or wavelets may be more appropriate in such cases

Practice Problems and Examples

• Find the equation of the tangent plane to the surface $z = x^2 + xy + y^2$ at the point $(1, -1, 1)$
  • Solution: $z - 1 = 3(x - 1) - (y + 1)$
• Approximate the value of $f(2.01, 1.99)$ using a linear approximation, given that $f(2, 2) = 5$, $\frac{\partial f}{\partial x}(2, 2) = 3$, and $\frac{\partial f}{\partial y}(2, 2) = -2$
  • Solution: $L(2.01, 1.99) \approx 5.01$
• Classify the critical point $(0, 0)$ of the function $f(x, y) = x^2 - xy + y^2$ using the second-order partial derivatives test
  • Solution: Saddle point
• Find the directional derivative of $f(x, y) = x^2y + xy^2$ at the point $(1, 2)$ in the direction of the unit vector $\vec{u} = \frac{1}{\sqrt{2}}(1, 1)$
  • Solution: D_\vec{u}f(1, 2) = 7\sqrt{2}
• Estimate the maximum error in the linear approximation of $f(x, y) = \sin(xy)$ near the point $(\pi/4, \pi/4)$ for $|x - \pi/4| \leq 0.1$ and $|y - \pi/4| \leq 0.1$
  • Solution: Error $\leq 0.0165$ (using the remainder term of the Taylor series expansion)

Connections to Other Calculus Topics

• Tangent planes and linear approximations are extensions of the concepts of tangent lines and linear approximations for functions of one variable
• Partial derivatives are analogous to ordinary derivatives and are used to analyze the behavior of functions of multiple variables
  • Clairaut's theorem states that mixed partial derivatives are equal under certain conditions, similar to the equality of mixed second derivatives for functions of two variables
• Gradient vector is related to the concept of the derivative as a linear transformation, representing the best linear approximation to a function at a point
• Directional derivatives are related to the chain rule, as they measure the rate of change of a function along a parametric curve
• Double and triple integrals are used to calculate volumes, surface areas, and other quantities related to surfaces and solids, which can be approximated using tangent planes and linear approximations
• Vector fields and conservative vector fields are closely connected to the gradient vector and the concept of path independence
  • Fundamental theorem of line integrals relates the line integral of a gradient vector field to the difference in the potential function at the endpoints
• Optimization problems in multiple variables often involve finding tangent planes, gradient vectors, and critical points, similar to optimization problems in one variable