Calculus I: Core Concepts and Study Guide

Chapter 1: Functions

Definition and Properties of Functions

Functions are foundational objects in calculus, describing relationships between sets of inputs and outputs. Understanding their properties is essential for further study in calculus.

• Function: A rule that assigns to each element in a set (domain) exactly one element in another set (range).

• Domain: The set of all possible input values for the function.

• Range: The set of all possible output values.

• Function Notation: If f is a function, then f(x) denotes the output when the input is x.

• Evaluating Functions: Substitute the input value into the function rule to find the output.

Example: For $f(x) = x^2 + 3$, the domain is all real numbers, and $f(2) = 2^2 + 3 = 7$.

Types of Functions

• Polynomial Functions: Functions of the form $f(x) = a_n x^n + \ldots + a_1 x + a_0$.

• Rational Functions: Ratios of polynomials, $f(x) = \frac{p(x)}{q(x)}$, where $q(x) \neq 0$.

• Radical Functions: Functions involving roots, such as $f(x) = \sqrt{x}$.

• Absolute Value Functions: $f(x) = |x|$, which outputs the non-negative value of $x$.

Graphs and Transformations

Understanding how basic graphs change under various transformations is crucial for visualizing functions.

• Shifts: Moving the graph horizontally or vertically.

• Stretches: Expanding or compressing the graph vertically or horizontally.

• Reflections: Flipping the graph over a line, such as the x-axis or y-axis.

Example: The graph of $f(x) = (x - 2)^2 + 3$ is the graph of $y = x^2$ shifted right by 2 units and up by 3 units.

Function Operations and Composition

• Operations: Functions can be added, subtracted, multiplied, or divided (where defined).

• Composition: The composition $f(g(x))$ means applying $g$ first, then $f$ to the result.

Example: If $f(x) = 2x$ and $g(x) = x + 1$, then $f(g(x)) = 2(x + 1) = 2x + 2$.

Inverse Functions

• Inverse Function: If $f$ is one-to-one, its inverse $f^{-1}$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Example: If $f(x) = 3x + 2$, then $f^{-1}(x) = \frac{x - 2}{3}$.

Chapter 2: Limits & Continuity

Limits and One-Sided Limits

Limits describe the behavior of a function as the input approaches a particular value.

• Limit: $\lim_{x \to a} f(x) = L$ means as $x$ approaches $a$, $f(x)$ approaches $L$.

• One-Sided Limits: $\lim_{x \to a^-} f(x)$ (from the left), $\lim_{x \to a^+} f(x)$ (from the right).

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

Limit Laws

• Limits can be combined using algebraic rules, such as:

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

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

• $\lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, if $\lim_{x \to a} g(x) \neq 0$

Infinite Limits and Limits at Infinity

• Infinite Limits: When $f(x)$ increases or decreases without bound as $x$ approaches a value.

• Limits at Infinity: Describes the behavior as $x$ approaches $\infty$ or $-\infty$.

Example: $\lim_{x \to 0^+} \frac{1}{x} = +\infty$; $\lim_{x \to \infty} \frac{1}{x} = 0$.

Continuity and Types of Discontinuity

• Continuity at a Point: $f$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$.

• Types of Discontinuity:

  • Removable: Limit exists, but function is not defined or not equal to the limit at that point.

  • Jump: Left and right limits exist but are not equal.

  • Infinite: Function approaches infinity at the point.

Example: $f(x) = \frac{x^2 - 1}{x - 1}$ is not defined at $x = 1$, but $\lim_{x \to 1} f(x) = 2$ (removable discontinuity).

Chapter 3: Derivatives

Definition of the Derivative

The derivative measures the instantaneous rate of change of a function with respect to its variable.

• Definition: $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

• Geometric Meaning: The slope of the tangent line to the curve at a point.

• Normal Line: The line perpendicular to the tangent at a point.

• Rate of Change: The derivative represents how a quantity changes with respect to another.

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

Rules of Differentiation

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

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

• Constant Rule: $\frac{d}{dx} [c] = 0$

• 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 and Implicit Differentiation

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

• Implicit Differentiation: Used when $y$ is defined implicitly by an equation involving $x$ and $y$.

Example: If $y^2 + x^2 = 1$, then $\frac{d}{dx} (y^2) + \frac{d}{dx} (x^2) = 0$ leads to $2y \frac{dy}{dx} + 2x = 0$.

Trigonometric Derivatives

• $\frac{d}{dx} \sin x = \cos x$

• $\frac{d}{dx} \cos x = -\sin x$

• $\frac{d}{dx} \tan x = \sec^2 x$

Applications of Derivatives

• Related Rates: Problems involving rates at which related variables change.

• Linearization: Approximating a function near a point using its tangent line.

• Differentials: Small changes in variables, $dy = f'(x) dx$.

Example: If a balloon's radius increases at 2 cm/s, how fast is the volume increasing when $r = 5$ cm? Use related rates with $V = \frac{4}{3}\pi r^3$.

Study Plan

• Practice daily and solve 10–15 problems per topic to reinforce understanding.

• Focus on mastering limits, derivatives, the chain rule, and their applications.

• Review mistakes carefully and simulate exam conditions for effective preparation.