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 \(a\). It is denoted as:

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

A function \(f(x)\) is continuous at a point \(x = a\) if the following three conditions are met:

• \(f(a)\) is defined.
• \(\lim_{x \to a} f(x)\) exists.
• \(\lim_{x \to a} f(x) = f(a)\).

The derivative of a function \(f(x)\) at a point \(x = a\) is the slope of the tangent line to the function at that point. It is defined as:

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

Key differentiation rules include the power rule, product rule, quotient rule, and chain rule:

• Power Rule: \( \frac{d}{dx} [x^n] = nx^{n-1} \)
• Product Rule: \( \frac{d}{dx} [uv] = u'v + uv' \)
• Quotient Rule: \( \frac{d}{dx} \left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2} \)
• Chain Rule: \( \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \)

The equation of the tangent line to the function \(f(x)\) at the point \(x = a\) is given by:

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

Related rates problems involve finding the rate at which one quantity changes with respect to another. Typically, you differentiate an equation with respect to time \(t\) to relate the rates.

Optimization involves finding the maximum or minimum value of a function subject to constraints. Typically, you set the derivative equal to zero and solve for critical points, then use the second derivative test or endpoints to determine maxima or minima.

The Mean Value Theorem states that if a function \(f(x)\) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \(c\) in \((a, b)\) such that:

\[ f'(c) = \frac{f(b) - f(a)}{b - a} \]

The definite integral of a function \(f(x)\) from \(a\) to \(b\) is the signed area under the curve \(f(x)\) between \(x = a\) and \(x = b\). It is defined as:

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

where \( \Delta x = \frac{b - a}{n} \) and \( x_i^* \) is a sample point in the ith subinterval.

The Fundamental Theorem of Calculus connects differentiation and integration. It states:

• Part 1: If \(f(x)\) is continuous on \([a, b]\), then the function \(F(x) = \int_{a}^{x} f(t) \, dt\) is continuous on \([a, b]\), differentiable on \((a, b)\), and \(F'(x) = f(x)\).
• Part 2: If \(F(x)\) is an antiderivative of \(f(x)\), then \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).

An indefinite integral represents a family of functions whose derivative is the integrand. It is expressed as:

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

where \(F(x)\) is an antiderivative of \(f(x)\) and \(C\) is the constant of integration.

Basic Integration Rules:

• Power Rule: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), for \(n \neq -1\)
• Exponential Rule: \( \int e^x \, dx = e^x + C \)
• Trigonometric Rule: \( \int \sin(x) \, dx = -\cos(x) + C \), \( \int \cos(x) \, dx = \sin(x) + C \)

Integration by substitution is used to simplify integrals by making a substitution that transforms the integral into a simpler form. It is often used when the integrand is a composite function.

General Formula:

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

where \(u = g(x)\) and \(du = g'(x) \, dx\).

Integration by parts is based on the product rule for differentiation. It is used when the integrand is a product of two functions. The formula is:

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

The area under a curve \(y = f(x)\) from \(x = a\) to \(x = b\) is given by the definite integral:

\[ \text{Area} = \int_{a}^{b} f(x) \, dx \]

The volume of a solid of revolution generated by rotating a curve \(y = f(x)\) around the x-axis from \(x = a\) to \(x = b\) is given by:

\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]

The average value of a continuous function \(f(x)\) on the interval \([a, b]\) is given by:

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