Single Variable Calculus: Limits, Derivatives, and Applications

Limits: The Backbone of Calculus

What is a Limit?

The concept of a limit is foundational in calculus, describing the behavior of a function as its input approaches a particular value. Limits allow us to rigorously define continuity, derivatives, and integrals.

• Definition: The limit of a function $f(x)$ as $x$ approaches $a$ is the value that $f(x)$ gets closer to as $x$ gets closer to $a$.

• Notation: $\lim_{x \to a} f(x)$

• Example: For $f(x) = x^2$, as $x$ approaches $2$,$f(x)$approaches $4$.

x	f(x)
1.9	3.61
1.99	3.9601
1.999	3.996001
2.001	4.004001

As $x$ gets closer to $2$,$f(x)$gets closer to $4$. Thus, $\lim_{x \to 2} x^2 = 4$.

Graphical Representation of Limits

Limits can be visualized by observing how the function values approach a specific point on the graph as the input variable approaches a target value.

• Key Point: The function values (on the curve) get closer to the value at the target $x$-value, even if the function is not defined at that point.

• Example: The graph of $f(x) = x^2$ shows the function values approaching $4$as$x$approaches $2$.

Properties of Limits

Limits obey several important algebraic properties that allow us to compute them efficiently.

• Sum Rule: $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$

• Product Rule: $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

• Quotient Rule: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\lim_{x \to a} g(x) \neq 0$

Evaluating Limits Algebraically

Limits can often be evaluated by direct substitution, factoring, rationalizing, or applying limit laws.

• Direct Substitution: If $f(x)$ is continuous at $x=a$, then $\lim_{x \to a} f(x) = f(a)$.

• Factoring: Used when direct substitution yields an indeterminate form like $\frac{0}{0}$.

• Rationalizing: Multiply numerator and denominator by a conjugate to simplify expressions.

Indeterminate Forms and Infinity

Some limits result in indeterminate forms, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$, requiring special techniques to resolve.

• Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$

• Infinity: Limits can approach infinity, indicating unbounded growth as $x$ approaches a value.

Derivatives: Everybody's Favourite Linear Operator

Motivating Derivatives

The derivative of a function measures the rate at which the function value changes as its input changes. It is a fundamental concept for understanding instantaneous rates of change and slopes of tangent lines.

• Definition: The derivative of $f(x)$ at $x=a$ is $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Interpretation: The derivative represents the slope of the tangent line to the graph of $f(x)$ at $x=a$.

• Example: For $f(x) = x^2$, $f'(x) = 2x$

Derivative Rules

Several rules simplify the computation of derivatives for common functions.

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Product Rule: $\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

• Quotient Rule: $\frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Applications of Derivatives

Derivatives are used to analyze the behavior of functions, including finding local maxima and minima, determining concavity, and solving optimization problems.

• Critical Points: Points where $f'(x) = 0$ or $f'(x)$ is undefined.

• Inflection Points: Points where the concavity of $f(x)$ changes, i.e., where $f''(x) = 0$.

• Optimization: Using derivatives to find maximum or minimum values of functions in applied contexts.

Graphical Interpretation of Derivatives

The derivative at a point corresponds to the slope of the tangent line to the function's graph at that point. This provides a geometric understanding of instantaneous rate of change.

• Example: For $f(x) = x^2$, the slope at $x=2$ is $f'(2) = 4$.

Additional info:

• The notes cover further topics such as applications of derivatives, introduction to integration, techniques of integration, infinite series, and power series, as indicated by the table of contents.

• Each chapter includes definitions, properties, examples, and proofs, making the notes suitable for comprehensive exam preparation in a college-level single variable calculus course.