Crash Course: Calculus

Calculus is a branch of mathematics focused on change (differentiation) and accumulation (integration). It is divided into differential calculus and integral calculus.

1. Differential Calculus

Goal:

Understand how things change.

Key Concept:

Derivative - A derivative measures the rate of change of a function with respect to one of its variables.

Notation:

If \(y = f(x)\), the derivative of \(y\) with respect to \(x\) is written as:

\[\frac{dy}{dx} \quad \text{or} \quad f'(x)\]

This gives the slope of the function at any point \(x\).

Basic Rules:

• Constant Rule: The derivative of a constant is zero.

\[\frac{d}{dx}(c) = 0\]

• Power Rule: The derivative of \(x^n\) is \(nx^{n-1}\).

\[\frac{d}{dx}(x^n) = nx^{n-1}\]

• Sum Rule: The derivative of a sum is the sum of the derivatives.

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

• Product Rule: The derivative of a product is:

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

• Quotient Rule: The derivative of a quotient is:

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

• Chain Rule: The derivative of a composite function is:

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

Geometrically:

The derivative gives the slope of the tangent line to the curve of \(f(x)\) at a specific point.

2. Integral Calculus

Goal:

Understand how to accumulate or total quantities over a range.

Key Concept:

Integral - An integral represents the area under a curve. It can be thought of as the reverse operation of differentiation.

Notation:

The indefinite integral (anti-derivative) of \(f(x)\) with respect to \(x\) is written as:

\[\int f(x) \, dx\]

The definite integral of \(f(x)\) from \(a\) to \(b\) is:

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

This calculates the area under the curve from \(x = a\) to \(x = b\).

Basic Rules:

• Constant Rule: The integral of a constant is:

  \[\int c \, dx = cx + C\]

  where \(C\) is the constant of integration.

• Power Rule: The integral of \(x^n\) is:

  \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for} \, n \neq -1\]

• Sum Rule: The integral of a sum is the sum of the integrals.

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

Fundamental Theorem of Calculus:

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

It connects differentiation and integration, showing that differentiation reverses integration and vice versa.

3. Applications of Calculus

• Differential Calculus is used to:
  • Find instantaneous rates of change (e.g., velocity, acceleration).
  • Maximize or minimize functions (e.g., optimizing profits or efficiency).
• Integral Calculus is used to:
  • Calculate areas under curves, volumes of solids.
  • Accumulate quantities (e.g., total distance, mass, or charge over time or space).

Summary:

• Derivative: Measures how a function changes.
• Integral: Measures how much a function accumulates.
• Key Rules: Power rule, product rule, chain rule (for derivatives); power rule, sum rule (for integrals).