Limit Notation:

\[ \lim_{x \to a} f(x) = L \]

This is read as: "The limit of \( f(x) \) as \( x \) approaches \( a \) is \( L \)."

Meaning: As \( x \) gets arbitrarily close to \( a \) (but not necessarily equal to \( a \)), the function values \( f(x) \) get arbitrarily close to \( L \).

Epsilon-Delta Definition (Formal):

For every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that:

\[ 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon \]

If \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), then:

Sum Rule:

\[ \lim_{x \to a} [f(x) + g(x)] = L + M \]

Product Rule:

\[ \lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M \]

Quotient Rule:

\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M} \quad (M \neq 0) \]

Power Rule:

\[ \lim_{x \to a} [f(x)]^n = L^n \]

Special Trigonometric Limits:

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]

\[ \lim_{x \to 0} \frac{1 - \cos x}{x} = 0 \]

Exponential Limit:

\[ \lim_{x \to 0} \frac{e^x - 1}{x} = 1 \]

Definition of e:

\[ e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \]

1. \( f(a) \) exists (the function is defined at \( a \))
2. \( \lim_{x \to a} f(x) \) exists
3. \( \lim_{x \to a} f(x) = f(a) \)

Types of Discontinuities:

• Removable Discontinuity: A hole in the graph that can be "filled in"
• Jump Discontinuity: Left and right limits exist but are not equal
• Infinite Discontinuity: The function approaches infinity (vertical asymptote)

Definition of the Derivative (First Principles):

\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

Alternative Form:

\[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]

Leibniz Notation:

\[ \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} \]

Product Rule:

\[ \frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) \]

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:

\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]

Or in Leibniz notation: \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)

The derivative of a derivative is called a second derivative:

\[ f''(x) = \frac{d^2y}{dx^2} = \frac{d}{dx}\left[\frac{dy}{dx}\right] \]

Notation for higher derivatives:

• First derivative: \( f'(x) \), \( \frac{dy}{dx} \), \( y' \)
• Second derivative: \( f''(x) \), \( \frac{d^2y}{dx^2} \), \( y'' \)
• Third derivative: \( f'''(x) \), \( \frac{d^3y}{dx^3} \), \( y''' \)
• nth derivative: \( f^{(n)}(x) \), \( \frac{d^ny}{dx^n} \), \( y^{(n)} \)

The equation of the tangent line to \( y = f(x) \) at point \( (a, f(a)) \) is:

\[ y - f(a) = f'(a)(x - a) \]

Strategy:

1. Identify all variables and their rates of change
2. Write an equation relating the variables
3. Differentiate both sides with respect to time \( t \)
4. Substitute known values and solve

Definition: If \( F'(x) = f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \).

Notation:

\[ \int f(x) \, dx = F(x) + C \]

Where \( C \) is the constant of integration

Key Point: The indefinite integral represents a family of functions that differ by a constant.

Definition (as a Riemann Sum):

\[ \int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x \]

Geometric Interpretation: The definite integral represents the signed area between the curve \( y = f(x) \) and the x-axis from \( x = a \) to \( x = b \).

Properties of Definite Integrals:

• \( \int_a^a f(x) \, dx = 0 \)
• \( \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx \)
• \( \int_a^b [f(x) + g(x)] \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx \)
• \( \int_a^b cf(x) \, dx = c\int_a^b f(x) \, dx \)
• \( \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx \)

If \( f \) is continuous on \([a, b]\) and \( F \) is defined by:

\[ F(x) = \int_a^x f(t) \, dt \]

Then \( F'(x) = f(x) \) for all \( x \) in \((a, b)\).

In other words: Differentiation undoes integration.

\[ \frac{d}{dx}\left[\int_a^x f(t) \, dt\right] = f(x) \]

If \( f \) is continuous on \([a, b]\) and \( F \) is any antiderivative of \( f \), then:

\[ \int_a^b f(x) \, dx = F(b) - F(a) \]

Notation: Often written as:

\[ \int_a^b f(x) \, dx = \left[F(x)\right]_a^b = F(b) - F(a) \]

In other words: To evaluate a definite integral, find any antiderivative and evaluate it at the bounds.

Method: Let \( u = g(x) \), then \( du = g'(x) \, dx \)

\[ \int f(g(x))g'(x) \, dx = \int f(u) \, du \]

For definite integrals: Change the limits of integration

\[ \int_a^b f(g(x))g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du \]

Formula:

\[ \int u \, dv = uv - \int v \, du \]

Strategy: Choose \( u \) using LIATE priority:

• L - Logarithmic functions
• I - Inverse trigonometric functions
• A - Algebraic functions
• T - Trigonometric functions
• E - Exponential functions

The area between \( y = f(x) \) and \( y = g(x) \) from \( x = a \) to \( x = b \):

\[ A = \int_a^b |f(x) - g(x)| \, dx \]

Disk Method:

\[ V = \pi \int_a^b [f(x)]^2 \, dx \]

Washer Method:

\[ V = \pi \int_a^b \left([R(x)]^2 - [r(x)]^2\right) \, dx \]

Shell Method:

\[ V = 2\pi \int_a^b x \cdot f(x) \, dx \]

The length of the curve \( y = f(x) \) from \( x = a \) to \( x = b \):

\[ L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx \]

The average value of \( f(x) \) on \([a, b]\):

\[ f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x) \, dx \]

For a function \( f(x, y) \), the partial derivative with respect to \( x \) is:

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

Higher-Order Partial Derivatives:

• \( f_{xx} = \frac{\partial^2 f}{\partial x^2} \)
• \( f_{xy} = \frac{\partial^2 f}{\partial y \partial x} \)
• \( f_{yx} = \frac{\partial^2 f}{\partial x \partial y} \)
• If \( f \) is continuous, then \( f_{xy} = f_{yx} \) (Clairaut's Theorem)

The gradient of \( f(x, y, z) \) is:

\[ \nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle \]

Properties:

• The gradient points in the direction of greatest increase
• The magnitude is the rate of maximum increase
• The gradient is perpendicular to level curves/surfaces