Differential Calculus II: Taylor Series, Extrema & Jacobians

Unit 3 — Differential Calculus II (8 Hours)

This unit develops advanced calculus techniques for approximating functions and finding extrema of multivariable functions, essential for numerical methods, optimization, and engineering design.

1. What Is Taylor's Theorem?

Taylor's theorem provides a polynomial approximation of a function near a specific point using its derivatives.

Why It Matters

Approximates complex functions with polynomials for computation
Foundation of finite difference methods in numerical analysis
Error estimation in numerical algorithms
Physics: small oscillation approximations

2. Taylor Expansion of Functions of One Variable

2.1 Taylor's Theorem with Lagrange Remainder

If $f(x)$ has continuous derivatives up to order $(n+1)$ in an interval containing $a$ , then for any $x$ in that interval:

f(x) = f(a) + (x-a)f'(a) + \frac{(x-a)^2}{2!}f''(a) + \cdots + \frac{(x-a)^n}{n!}f^{(n)}(a) + R_{n+1}

where the Lagrange form of remainder is:

R_{n+1} = \frac{(x-a)^{n+1}}{(n+1)!} f^{(n+1)}(c), \quad a < c < x

2.2 Taylor's Theorem with Cauchy Remainder

R_{n+1} = \frac{(x-a)^{n+1}(1-\theta)^n}{n!} f^{(n+1)}(a + \theta(x-a)), \quad 0 < \theta < 1

2.3 Taylor Series (Infinite Form)

If $f$ is infinitely differentiable and $R_n \to 0$ as $n \to \infty$ :

f(x) = \sum_{n=0}^{\infty} \frac{(x-a)^n}{n!} f^{(n)}(a)

2.4 Taylor Expansion of Functions of Two Variables

If $f(x, y)$ has continuous partial derivatives up to order $n$ near $(a, b)$ :

f(x, y) = f(a, b) + \left[(x-a)\frac{\partial}{\partial x} + (y-b)\frac{\partial}{\partial y}\right]f(a, b) + \frac{1}{2!}\left[(x-a)\frac{\partial}{\partial x} + (y-b)\frac{\partial}{\partial y}\right]^2 f(a, b) + \cdots + R_n

In operator notation where $h = x-a$ , $k = y-b$ :

f(a+h, b+k) = f(a,b) + \left(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y}\right)f + \frac{1}{2!}\left(h \frac{\partial}{\partial x} + k \frac{\partial}{\partial y}\right)^2 f + \cdots

Expanded to second order:

f(x, y) \approx f(a,b) + (x-a)f_x + (y-b)f_y + \frac{1}{2}\left[(x-a)^2 f_{xx} + 2(x-a)(y-b)f_{xy} + (y-b)^2 f_{yy}\right]

3. Maclaurin's Theorem

3.1 Definition

Maclaurin's theorem is Taylor's theorem expanded about $a = 0$ :

f(x) = f(0) + x f'(0) + \frac{x^2}{2!}f''(0) + \frac{x^3}{3!}f'''(0) + \cdots + \frac{x^n}{n!}f^{(n)}(0) + R_{n+1}

3.2 Maclaurin Series (Standard Expansions)

Function	Maclaurin Series	Valid For
$e^x$	$\displaystyle \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

3.3 Applications in Approximation

For small $x$ (truncating after a few terms):

e^x \approx 1 + x + \frac{x^2}{2}

\sin x \approx x - \frac{x^3}{6}

\cos x \approx 1 - \frac{x^2}{2}

\ln(1+x) \approx x - \frac{x^2}{2}

4. Maxima and Minima of Functions of Several Variables

4.1 Critical Points (Stationary Points)

For $z = f(x, y)$ , critical points occur where:

\frac{\partial f}{\partial x} = 0 \quad \text{and} \quad \frac{\partial f}{\partial y} = 0

Simultaneously solving $f_x = 0$ and $f_y = 0$ gives critical points .

4.2 First Derivative Test

At a critical point $(a, b)$ :

If $f$ changes from increasing to decreasing in all directions: local maximum
If $f$ changes from decreasing to increasing in all directions: local minimum

4.3 Second Derivative Test (Hessian Criterion)

Define the Hessian determinant at $(a, b)$ :

H = \begin{vmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{vmatrix} = f_{xx} f_{yy} - (f_{xy})^2

Condition	Result
$H > 0$ and $f_{xx} < 0$	Local maximum
$H > 0$ and

4.4 Saddle Points

A saddle point is a critical point that is neither a local maximum nor a local minimum. The function increases in some directions and decreases in others.

Example: $f(x, y) = x^2 - y^2$ has a saddle point at $(0, 0)$ .

4.5 Working Procedure for Two Variables

Find $f_x$ and $f_y$
Solve $f_x = 0$ , simultaneously for critical points

4.6 Extension to Three Variables

For $u = f(x, y, z)$ , critical points satisfy:

f_x = 0, \quad f_y = 0, \quad f_z = 0

The nature is determined by the Hessian matrix:

H = \begin{bmatrix} f_{xx} & f_{xy} & f_{xz} \\ f_{yx} & f_{yy} & f_{yz} \\ f_{zx} & f_{zy} & f_{zz} \end{bmatrix}

If all leading principal minors are positive: local minimum
If leading principal minors alternate in sign starting with negative: local maximum

5. Lagrange's Method of Multipliers

5.1 Constrained Optimization

Find extrema of $f(x, y, z)$ subject to constraint $g(x, y, z) = 0$ .

5.2 Formation of Lagrange Multiplier Equations

Construct the Lagrangian function:

\mathcal{L}(x, y, z, \lambda) = f(x, y, z) - \lambda \, g(x, y, z)

The necessary conditions for an extremum are:

\frac{\partial \mathcal{L}}{\partial x} = 0, \quad \frac{\partial \mathcal{L}}{\partial y} = 0, \quad \frac{\partial \mathcal{L}}{\partial z} = 0, \quad \frac{\partial \mathcal{L}}{\partial \lambda} = 0

Which give:

f_x = \lambda g_x, \quad f_y = \lambda g_y, \quad f_z = \lambda g_z, \quad g(x, y, z) = 0

5.3 Multiple Constraints

For $f(x, y, z)$ subject to $g_1 = 0$ and $g_2 = 0$ :

\mathcal{L} = f - \lambda_1 g_1 - \lambda_2 g_2

Conditions:

\frac{\partial \mathcal{L}}{\partial x} = 0, \quad \frac{\partial \mathcal{L}}{\partial y} = 0, \quad \frac{\partial \mathcal{L}}{\partial z} = 0, \quad g_1 = 0, \quad g_2 = 0

5.4 Engineering Applications

Design Optimization: Minimize cost subject to performance constraints
Thermodynamics: Maximize entropy subject to energy conservation
Structural Design: Minimize weight subject to stress limits
Economics: Maximize utility subject to budget constraints
Machine Learning: Regularization as constrained optimization

6. Jacobians

6.1 Definition

For a transformation from $(u, v)$ to $(x, y)$ where $x = x(u, v)$ and $y$ , the is:

J = \frac{\partial(x, y)}{\partial(u, v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix}

For three variables:

\frac{\partial (x, y, z)}{\partial (u, v, w)} = ∣ \begin{array}{ccc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{}{} \end{array}

6.2 Properties of Jacobians

Chain Rule:

\frac{\partial(x, y)}{\partial(u, v)} \cdot \frac{\partial(u, v)}{\partial(x, y)} = 1

Inverse Property:

\frac{\partial(u, v)}{\partial(x, y)} = \frac{1}{\frac{\partial(x, y)}{\partial(u, v)}}

Functional Dependence: If $u$ and $v$ are functionally dependent (i.e., $F(u, v) = 0$ ), then:

\frac{\partial(u, v)}{\partial(x, y)} = 0

6.3 Change of Variables Using Jacobians

When transforming a double integral:

\iint_R f(x, y) \, dx \, dy = \iint_{R'} f(x(u, v), y(u, v)) \left| \frac{\partial(x, y)}{\partial(u, v)} \right| \, du \, dv

The absolute value of the Jacobian serves as the scaling factor for area/volume elements.

Polar Coordinates: $x = r\cos\theta$ , $y = r\sin\theta$

\frac{\partial(x, y)}{\partial(r, \theta)} = \begin{vmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{vmatrix} = r

So $dx \, dy = r \, dr \, d\theta$ .

Cylindrical Coordinates: $x = r\cos\theta$ , $y = r\sin\theta$ , $z = z$

\frac{\partial(x, y, z)}{\partial(r, \theta, z)} = r

Spherical Coordinates: $x = \rho\sin\phi\cos\theta$ , $y = \rho\sin\phi\sin\theta$ , $z = \rho\cos\phi$

\frac{\partial(x, y, z)}{\partial(\rho, \phi, \theta)} = \rho^2 \sin\phi

7. Approximation of Errors

7.1 Absolute Error

If $u = f(x, y)$ and $\Delta x$ , $\Delta y$ are small errors in $x$ and $y$ :

\Delta u \approx \frac{\partial u}{\partial x} \Delta x + \frac{\partial u}{\partial y} \Delta y

The absolute error in $u$ is approximately:

|\Delta u| \approx \left| \frac{\partial u}{\partial x} \right| |\Delta x| + \left| \frac{\partial u}{\partial y} \right| |\Delta y|

7.2 Relative Error

\frac{\Delta u}{u} \approx \frac{1}{u}\left( \frac{\partial u}{\partial x} \Delta x + \frac{\partial u}{\partial y} \Delta y \right)

7.3 Percentage Error

\text{Percentage Error} = \frac{\Delta u}{u} \times 100\%

7.4 Error Propagation Using Differentiation

For a function of $n$ variables $u = f(x_1, x_2, \ldots, x_n)$ :

\Delta u \approx \sum_{i=1}^{n} \frac{\partial u}{\partial x_i} \Delta x_i

Maximum possible absolute error:

|\Delta u|_{max} = \sum_{i=1}^{n} \left| \frac{\partial u}{\partial x_i} \right| |\Delta x_i|

7.5 Special Cases

Product: If $u = xy$ :

\frac{\Delta u}{u} \approx \frac{\Delta x}{x} + \frac{\Delta y}{y}

Quotient: If $u = \frac{x}{y}$ :

\frac{\Delta u}{u} \approx \frac{\Delta x}{x} - \frac{\Delta y}{y}

Power: If $u = x^n$ :

\frac{\Delta u}{u} \approx n \frac{\Delta x}{x}

8. Engineering Applications

Domain	Application
Numerical Analysis	Taylor series forms basis of finite difference methods
Optimization	Lagrange multipliers for constrained design problems
Signal Processing	Maclaurin approximations for filter design
CAD/CAM	Surface approximation using Taylor expansions
Measurement	Error propagation in experimental data
Thermodynamics	Jacobians in thermodynamic potential transformations
Robotics	Jacobian matrix relates joint velocities to end-effector velocity
Statistics	Error propagation in regression models

9. Common Mistakes to Avoid

Mistake	Correction
Forgetting the remainder term in Taylor expansion	Always specify $R_n$ for exact equality
Confusing Taylor and Maclaurin	Maclaurin is Taylor at $a = 0$
Using $f_{xx} > 0$ alone to claim minimum

10. Best Practices & Architecture

Always verify remainder convergence when using infinite Taylor series
Check all critical points — extrema can occur at boundaries too
Use Lagrange multipliers when constraints are equations; use substitution for simple cases
Compute Jacobians symbolically first before numerical evaluation
Propagate errors conservatively using maximum absolute error for safety-critical designs
Validate approximations by comparing with exact values at sample points

11. Practice Problems

Problem 1: Expand $f(x) = \ln(1 + x)$ in Taylor series about $x = 0$ up to the $x^4$ term and estimate .

Problem 2: Find the Taylor expansion of $f(x, y) = e^{x+y}$ about $(0, 0)$ up to second-order terms.

Problem 3: Find and classify the critical points of $f(x, y) = x^3 + y^3 - 3xy$ .

Problem 4: Find the dimensions of a rectangular box with volume $V = 1000$ cm³ that minimize the surface area using Lagrange multipliers.

Problem 5: If $x = u + v$ and $y = uv$ , find $\frac{\partial(u, v)}{\partial(x, y)}$ .

Problem 6: The period of a simple pendulum is $T = 2\pi\sqrt{\frac{L}{g}}$ . If is measured with error and with error, find the maximum percentage error in .

Problem 7: Find the Maclaurin series expansion of $\tan x$ up to the $x^5$ term.

12. Mini Project

Title: Constrained Optimization of a Heat Exchanger Design

Objective: Use Lagrange multipliers and error analysis to optimize the dimensions of a shell-and-tube heat exchanger.

Given:

Heat duty $Q = UA\Delta T_{lm}$ must be satisfied
Total cost $C = \alpha A + \beta V_{shell}$ where is heat transfer area and is shell volume

Tasks:

Formulate the optimization problem with the heat duty as constraint
Set up Lagrangian equations for variables $D$ , $L$ , $N$
Solve for optimal dimensions
Perform error analysis: if $U$ has $\pm 5\%$ uncertainty, compute propagated error in required $A$

Deliverable: A computational report with symbolic derivations, numerical solution, sensitivity analysis, and validation that the critical point is indeed a minimum using the Hessian criterion.

Quick Reference Formula Sheet

Concept	Formula
Taylor series (1D)	$f(x) = \sum_{n=0}^{\infty} \frac{(x-a)^n}{n!}f^{(n)}(a)$

End of Unit 3 — Differential Calculus II