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

Limits

Understanding Limits

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

• Definition: The limit of a function f(x) as x approaches a value a is the value that f(x) gets closer to as x gets closer to 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: If f(x) increases or decreases without bound as x approaches a, the limit is infinite.

Example:

Given a graph of f(x), you may be asked to find $f(5)$, $\lim_{x \to 5^-} f(x)$, $\lim_{x \to 5^+} f(x)$, and $\lim_{x \to 5} f(x)$. Use the graph to determine the function's value and its behavior from both sides.

Continuity

A function is continuous at a point if its limit exists at that point and equals the function's value.

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

• Types of discontinuities:

  • Removable: The limit exists, but f(a) is not defined or not equal to the limit.

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

  • Infinite: The function approaches infinity at the point.

Example:

Given a piecewise function, determine values of parameters to make it continuous everywhere.

Evaluating Limits Algebraically

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

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

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

• Rationalizing: Multiply by a conjugate to simplify square roots.

• Special limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$, $\lim_{x \to \infty} \frac{1}{x} = 0$

Example:

Find $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$ by factoring numerator as $(x-2)(x+2)$ and canceling.

Derivatives

Definition and Interpretation

The derivative of a function measures its instantaneous rate of change. It is defined as the limit of the average rate of change as the interval approaches zero.

• 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 point x.

• Physical meaning: If f(x) represents position, f'(x) is velocity.

Basic Differentiation Rules

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

• Constant Rule: $\frac{d}{dx} c = 0$

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

Example:

Find the derivative of $f(x) = x^3 + 2x^2$ using the power rule:

$f'(x) = 3x^2 + 4x$

Derivatives of Common Functions

• Exponential: $\frac{d}{dx} e^x = e^x$

• Logarithmic: $\frac{d}{dx} \ln x = \frac{1}{x}$

• Trigonometric: $\frac{d}{dx} \sin x = \cos x$, $\frac{d}{dx} \cos x = -\sin x$

Example:

Find $\frac{d}{dx} \sin(2x)$ using the chain rule:

$\frac{d}{dx} \sin(2x) = 2 \cos(2x)$

Integrals

Indefinite Integrals

Indefinite integrals, or antiderivatives, reverse the process of differentiation. They represent a family of functions whose derivative is the integrand.

• Notation: $\int f(x) dx$

• General form: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$

• Constant of integration: Always add $+C$ for indefinite integrals.

Example:

$\int 3x^2 dx = x^3 + C$

Definite Integrals

Definite integrals compute the net area under a curve between two points. They are evaluated using the Fundamental Theorem of Calculus.

• Notation: $\int_a^b f(x) dx$

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

Example:

Find $\int_0^2 x dx$:

Antiderivative is $\frac{1}{2}x^2$, so $\int_0^2 x dx = \frac{1}{2}(2)^2 - \frac{1}{2}(0)^2 = 2$

Basic Integration Rules

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

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

• Substitution: Useful for integrals involving composite functions.

Example:

Use substitution for $\int 2x \cos(x^2) dx$:

Let $u = x^2$, $du = 2x dx$, so $\int 2x \cos(x^2) dx = \int \cos(u) du = \sin(u) + C = \sin(x^2) + C$

Table: Types of Discontinuities

Type	Description	Example
Removable	Limit exists, but function value is undefined or not equal to the limit	$f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$
Jump	Left and right limits exist but are not equal	Piecewise function with different values on each side of a point
Infinite	Function approaches infinity at the point	$f(x) = \frac{1}{x}$ at $x = 0$

Summary

• Limits describe function behavior near a point and are essential for defining continuity and derivatives.

• Derivatives measure instantaneous rate of change and are computed using various rules.

• Integrals reverse differentiation and compute areas under curves, with definite integrals using limits of integration.

• Understanding types of discontinuities is crucial for analyzing function behavior.

Additional info: These study notes expand upon the exam review questions by providing definitions, rules, and examples for each major topic covered in Calculus I: limits, derivatives, and integrals.