Calculus I Final Exam Review: Limits, Derivatives, and Integrals

Limits

Understanding Limits

Limits are fundamental to calculus, describing the behavior of a function as its input approaches a particular value. They are essential for defining continuity, derivatives, and integrals.

• Limit of a Function: The value that f(x) approaches as x approaches a specific point a.

• Notation: $\lim_{x \to a} f(x)$

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

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

• Limits at Infinity: $\lim_{x \to \infty} f(x)$ describes the end behavior of a function.

Example: $\lim_{x \to 2} (3x + 1) = 7$

Continuity

A function is continuous at a point a if the following three conditions are met:

• 1. f(a) is defined

• 2. $\lim_{x \to a} f(x)$ exists

• 3. $\lim_{x \to a} f(x) = f(a)$

Types of Discontinuities:

• Removable Discontinuity: The limit exists, but the function is not defined at that point or is defined differently.

• Jump Discontinuity: The left and right limits exist but are not equal.

• Infinite Discontinuity: The 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).

Evaluating Limits Algebraically

• Direct Substitution: Substitute the value of x directly if the function is continuous at that point.

• Factoring: Factor numerator and denominator to cancel common terms.

• Rationalization: Multiply by the conjugate to simplify expressions with square roots.

• Special Limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$, $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

Table: Types of Discontinuities

Type	Description	Example
Removable	Limit exists, function not defined or mismatched at point	$f(x) = \frac{x^2-1}{x-1}$ at $x=1$
Jump	Left and right limits not equal	Piecewise function with different values at $x=a$
Infinite	Function approaches $\infty$ or $-\infty$	$f(x) = \frac{1}{x}$ at $x=0$

Derivatives

Definition and Interpretation

The derivative of a function measures the rate at which the function value changes as its input changes. It is the foundation of differential calculus.

• 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.

• Physical Meaning: Instantaneous rate of change (e.g., velocity is the derivative of position).

Basic Differentiation Rules

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

• Constant Rule: $\frac{d}{dx} c = 0$

• Constant Multiple Rule: $\frac{d}{dx} [c f(x)] = c f'(x)$

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

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

Derivatives of Common Functions

• $\frac{d}{dx} \sin x = \cos x$

• $\frac{d}{dx} \cos x = -\sin x$

• $\frac{d}{dx} e^x = e^x$

• $\frac{d}{dx} \ln x = \frac{1}{x}$

Example: $f(x) = x^3 - 2x^2 + 5x - 7$ $f'(x) = 3x^2 - 4x + 5$

Integrals

Definition and Interpretation

The integral of a function represents the accumulation of quantities, such as areas under curves. It is the central concept of integral calculus.

• Indefinite Integral: $\int f(x) dx$ gives the family of antiderivatives of $f(x)$.

• Definite Integral: $\int_a^b f(x) dx$ gives the net area under $f(x)$ from $x=a$ to $x=b$.

• Fundamental Theorem of Calculus: If $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$.

Basic Integration Rules

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)

• Constant Multiple Rule: $\int c f(x) dx = c \int f(x) dx$

• Sum Rule: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

Integrals of Common Functions

• $\int e^x dx = e^x + C$

• $\int \sin x dx = -\cos x + C$

• $\int \cos x dx = \sin x + C$

• $\int \frac{1}{x} dx = \ln|x| + C$

Example: $\int (3x^2 - 4x + 5) dx = x^3 - 2x^2 + 5x + C$

Table: Fundamental Theorem of Calculus

Part	Description
First	If $F(x)$ is an antiderivative of $f(x)$, then $\frac{d}{dx} \int_a^x f(t) dt = f(x)$
Second	$\int_a^b f(x) dx = F(b) - F(a)$

Summary Table: Key Calculus Concepts

Concept	Definition	Key Formula
Limit	Value function approaches as input nears a point	$\lim_{x \to a} f(x)$
Derivative	Instantaneous rate of change	$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
Integral	Accumulated area under a curve	$\int_a^b f(x) dx$

Additional info: The original file is a comprehensive set of exam review questions covering limits, continuity, derivatives, and integrals, with a focus on problem-solving and application of core calculus concepts. The above notes provide the necessary theoretical background and examples to support the types of problems found in the review.