Chapter 4: Advanced Applications – Expand Calculus Horizons

Calculus extends beyond single-variable functions to describe complex systems: from 3D shapes to dynamic processes (like population growth) and infinite approximations. This chapter explores three critical advanced topics: multivariable calculus, differential equations, and series—each with deep ties to science, engineering, and economics.

4.1 Multivariable Calculus

Most real-world phenomena depend on multiple variables (e.g., temperature depends on location and time). Multivariable calculus generalizes single-variable concepts to functions with 2, 3, or more inputs.

4.1.1 Functions of Multiple Variables

A function of two variables is a rule $f(x,y)$ that assigns a unique output to each pair $(x,y)$ in its domain (a set of points in the 2D plane). For example:

• $f(x,y) = x^2 + y^2$: Models the height of a paraboloid at $(x,y)$
• $T(x,y,t) = 20 - 0.1x^2 - 0.05y^2 + 0.5t$: Models temperature at position $(x,y)$ and time $t$

Domain and Range

• The domain of $f(x,y)$ is all $(x,y)$ where $f(x,y)$ is defined (e.g., avoid division by zero or square roots of negatives)
• The range is all possible output values $f(x,y)$

Example: Find the domain of $f(x,y) = \sqrt{4 - x^2 - y^2}$

The expression under the square root must be non-negative: $4 - x^2 - y^2 \geq 0 \implies x^2 + y^2 \leq 4$

Thus, the domain is all points $(x,y)$ inside or on a circle of radius 2 centered at the origin (a closed disk).

3D Graphs: Visualizing Multivariable Functions

The graph of $f(x,y)$ is the set of points $(x,y,z)$ where $z = f(x,y)$—a surface in 3D space. Common surfaces:

• Plane: $z = ax + by + c$ (flat surface)
• Sphere: $x^2 + y^2 + z^2 = r^2$ (set of points at distance $r$ from the origin)
• Paraboloid: $z = x^2 + y^2$ (bowl-shaped surface opening upward)

4.1.2 Partial Derivatives

Partial derivatives measure how a multivariable function changes when one variable changes while others are held constant. They generalize the single-variable derivative.

Definition and Notation

For $f(x,y)$, the partial derivative with respect to $x$ (denoted $\frac{\partial f}{\partial x}$, $f_x$, or $\partial_x f$) is:

$$\frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h}$$

Similarly, the partial derivative with respect to $y$ is:

$$\frac{\partial f}{\partial y} = \lim_{h \to 0} \frac{f(x, y + h) - f(x, y)}{h}$$

Calculation: Treat Other Variables as Constants

To compute $\frac{\partial f}{\partial x}$, differentiate $f(x,y)$ with respect to $x$, treating $y$ as a constant. Use standard derivative rules (power rule, chain rule, etc.).

Example 1: Basic partial derivatives

Find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ for $f(x,y) = x^3y^2 + \sin(xy)$.

• For $\frac{\partial f}{\partial x}$ (treat $y$ as constant):

  • Derivative of $x^3y^2$: $3x^2y^2$ (power rule: $y^2$ is constant)
  • Derivative of $\sin(xy)$: $\cos(xy) \cdot y$ (chain rule: derivative of $xy$ with respect to $x$ is $y$)
  • Result: $\frac{\partial f}{\partial x} = 3x^2y^2 + y\cos(xy)$

• For $\frac{\partial f}{\partial y}$ (treat $x$ as constant):

  • Derivative of $x^3y^2$: $x^3 \cdot 2y = 2x^3y$ (power rule: $x^3$ is constant)
  • Derivative of $\sin(xy)$: $\cos(xy) \cdot x$ (chain rule: derivative of $xy$ with respect to $y$ is $x$)
  • Result: $\frac{\partial f}{\partial y} = 2x^3y + x\cos(xy)$

Geometric Meaning

• $\frac{\partial f}{\partial x}(a,b)$ is the slope of the tangent line to the surface $z = f(x,y)$ at $(a,b,f(a,b))$ when sliced parallel to the $xz$-plane (holding $y = b$ fixed)
• $\frac{\partial f}{\partial y}(a,b)$ is the slope of the tangent line when sliced parallel to the $yz$-plane (holding $x = a$ fixed)

4.1.3 Multiple Integrals

Multiple integrals extend definite integrals to 2D or 3D, calculating "total accumulation" over regions (e.g., volume under a surface, mass of a plate with variable density). We focus on double integrals (over 2D regions).

Definition: Double Integrals as Volume

For a non-negative function $f(x,y)$ over a region $R$ in the $xy$-plane, the double integral $\iint_R f(x,y) dA$ gives the volume of the solid under the surface $z = f(x,y)$ and above $R$.

It is defined as a limit of Riemann sums (summing volumes of small rectangular prisms):

$$\iint_R f(x,y) dA = \lim_{m,n \to \infty} \sum_{i=1}^m \sum_{j=1}^n f(x_{ij}^*, y_{ij}^*) \Delta A$$

where $\Delta A$ is the area of each subrectangle, and $(x_{ij}^*, y_{ij}^*)$ is a sample point in the $ij$-th subrectangle.

Calculation: Iterated Integrals

Double integrals are computed as iterated integrals: integrate with respect to one variable first, then the other.

• For a region $R$ bounded by $x = a$ to $x = b$ and $y = g_1(x)$ to $y = g_2(x)$ (Type I region):

  $$\iint_R f(x,y) dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y) dy dx$$

• For a region bounded by $y = c$ to $y = d$ and $x = h_1(y)$ to $x = h_2(y)$ (Type II region):

  $$\iint_R f(x,y) dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y) dx dy$$

Example: Compute the volume under $z = x + y$ over the rectangle $R: 0 \leq x \leq 1, 0 \leq y \leq 2$

Step 1: Set up the iterated integral (Type I region):

$$\text{Volume} = \int_0^1 \int_0^2 (x + y) dy dx$$

Step 2: Integrate with respect to $y$ first (treat $x$ as constant):

$$\int_0^2 (x + y) dy = \left[xy + \frac{y^2}{2}\right]_0^2 = (2x + \frac{4}{2}) - 0 = 2x + 2$$

Step 3: Integrate the result with respect to $x$:

$$\int_0^1 (2x + 2) dx = \left[x^2 + 2x\right]_0^1 = (1 + 2) - 0 = 3$$

The volume is 3 cubic units.

4.2 Differential Equations

Differential equations relate a function to its derivatives, modeling dynamic systems (e.g., population growth, radioactive decay, or electric circuits).

4.2.1 Basic Concepts

• A differential equation (DE) is an equation containing a function and its derivatives
• Order: The highest derivative in the equation (e.g., $y' = 2x$ is first-order; $y'' + 3y' = 0$ is second-order)
• Solution: A function $y(x)$ that satisfies the DE. A general solution includes all possible solutions (often with constants); a particular solution fixes these constants (using initial conditions)

4.2.2 First-Order Differential Equations

These involve $y'$ (the first derivative) but no higher derivatives. We focus on two solvable types: separable and linear.

1. Separable Equations

A DE is separable if it can be written as $\frac{dy}{dx} = \frac{f(x)}{g(y)}$, where $x$ and $y$ terms can be separated:

$g(y) dy = f(x) dx$

Integrate both sides to solve.

Example: Solve $\frac{dy}{dx} = \frac{x}{y}$ with $y(0) = 2$

Step 1: Separate variables: $y dy = x dx$

Step 2: Integrate both sides: $\int y dy = \int x dx \implies \frac{y^2}{2} = \frac{x^2}{2} + C$

Step 3: Simplify (general solution): $y^2 = x^2 + K$ (where $K = 2C$)

Step 4: Apply initial condition $y(0) = 2$: $2^2 = 0^2 + K \implies K = 4$

Particular solution: $y^2 = x^2 + 4$ (or $y = \sqrt{x^2 + 4}$, since $y(0) = 2 > 0$)

2. Linear First-Order Equations

A linear DE has the form:

$\frac{dy}{dx} + P(x)y = Q(x)$

Solve using an integrating factor $\mu(x) = e^{\int P(x) dx}$. Multiply both sides by $\mu(x)$, then the left side becomes the derivative of $\mu(x)y$:

$\frac{d}{dx}(\mu(x)y) = \mu(x)Q(x)$

Integrate both sides and solve for $y$.

Example: Solve $\frac{dy}{dx} + \frac{1}{x}y = x$ (for $x > 0$)

Step 1: Identify $P(x) = \frac{1}{x}$, $Q(x) = x$

Step 2: Compute integrating factor: $\mu(x) = e^{\int \frac{1}{x} dx} = e^{\ln x} = x$

Step 3: Multiply DE by $\mu(x) = x$: $x \cdot \frac{dy}{dx} + x \cdot \frac{1}{x}y = x \cdot x \implies x\frac{dy}{dx} + y = x^2$

Left side is $\frac{d}{dx}(xy)$ (by product rule), so: $\frac{d}{dx}(xy) = x^2$

Step 4: Integrate both sides: $xy = \int x^2 dx = \frac{x^3}{3} + C$

Step 5: Solve for $y$: $y = \frac{x^2}{3} + \frac{C}{x}$

4.2.3 Second-Order Linear DEs with Constant Coefficients

These have the form $ay'' + by' + cy = G(x)$, where $a,b,c$ are constants. They model oscillating systems (e.g., springs, pendulums).

Homogeneous case ($G(x) = 0$): Solve using the characteristic equation $ar^2 + br + c = 0$

• If roots $r_1 \neq r_2$ (real): $y = C_1e^{r_1x} + C_2e^{r_2x}$
• If roots $r_1 = r_2 = r$ (real, repeated): $y = (C_1 + C_2x)e^{rx}$
• If roots $r = \alpha \pm i\beta$ (complex): $y = e^{\alpha x}(C_1\cos\beta x + C_2\sin\beta x)$

Example: Solve $y'' + 4y' + 4y = 0$

Step 1: Characteristic equation: $r^2 + 4r + 4 = 0 \implies (r + 2)^2 = 0 \implies r = -2$ (repeated root)

Step 2: General solution: $y = (C_1 + C_2x)e^{-2x}$

4.3 Series

Series are infinite sums of terms, used to approximate functions, analyze convergence, and solve DEs.

4.3.1 Sequences and Series Basics

• A sequence is an infinite list $\{a_n\} = a_1, a_2, a_3, \ldots$. It converges if $\lim_{n \to \infty} a_n = L$ (a finite number); otherwise, it diverges
• A series is the sum of a sequence: $\sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \cdots$. The n-th partial sum is $S_n = a_1 + \cdots + a_n$. The series converges if $\lim_{n \to \infty} S_n = S$ (a finite sum); otherwise, it diverges

Example: Geometric Series

A geometric series has the form $\sum_{n=0}^\infty ar^n = a + ar + ar^2 + \cdots$, where $a \neq 0$ and $r$ is the common ratio.

• It converges if $|r| < 1$, with sum $\frac{a}{1 - r}$
• It diverges if $|r| \geq 1$

Example: Evaluate $\sum_{n=0}^\infty \left(\frac{1}{2}\right)^n$

This is a geometric series with $a = 1$, $r = \frac{1}{2}$ (since $|r| = \frac{1}{2} < 1$, it converges).

Sum: $\frac{1}{1 - \frac{1}{2}} = 2$

4.3.2 Convergence Tests

To determine if a series converges (without computing partial sums), use these tests:

• Divergence Test: If $\lim_{n \to \infty} a_n \neq 0$, the series $\sum a_n$ diverges
• Comparison Test: For positive terms, if $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges. If $\sum b_n$ diverges, $\sum a_n$ diverges
• Ratio Test: For positive terms, compute $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$
  • If $L < 1$, converges
  • If $L > 1$, diverges
  • If $L = 1$, test is inconclusive

4.3.3 Power Series

A power series is a series of the form $\sum_{n=0}^\infty c_n (x - a)^n$, where $a$ is the center and $c_n$ are coefficients. It converges for $x$ in an interval around $a$ (the interval of convergence) with radius $R$ (radius of convergence).

Example: Find the interval of convergence for $\sum_{n=0}^\infty \frac{x^n}{n!}$

Step 1: Apply the ratio test:

$$L = \lim_{n \to \infty} \left|\frac{x^{n+1}/(n+1)!}{x^n/n!}\right| = \lim_{n \to \infty} \left|\frac{x}{n+1}\right| = 0 \quad \text{(for any } x\text{)}$$

Step 2: Since $L = 0 < 1$ for all $x$, the radius of convergence $R = \infty$, and the interval of convergence is $(-\infty, \infty)$

4.3.4 Taylor and Maclaurin Series

A Taylor series approximates a function $f(x)$ around a point $x = a$ using its derivatives:

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

where $f^{(n)}(a)$ is the $n$-th derivative of $f$ at $a$. A Maclaurin series is a Taylor series with $a = 0$:

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

Common Maclaurin Series

• $e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ (converges for all $x$)
• $\sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ (converges for all $x$)