Definition of a Derivative

The derivative is defined as the limit of the difference quotient as h approaches zero.

Geometric Interpretation

Geometrically, the derivative at a point gives the slope of the tangent line to the function's graph at that point.

Applications

• Finding rates of change
• Optimization problems (minima/maxima)
• Approximating functions (linearization)
• Analyzing motion (velocity/acceleration)

Common Derivative Rules

• Power Rule: If f(x) = x^n, then f'(x) = n·x^(n-1)
• Product Rule: (f·g)' = f'·g + f·g'
• Chain Rule: (f(g(x)))' = f'(g(x))·g'(x)
• Derivative of sin(x) is cos(x)
• Derivative of e^x is e^x

Definite Integral

The definite integral represents the signed area between the function and the x-axis from a to b.

Fundamental Theorem of Calculus

If F(x) is an antiderivative of f(x), then:

This connects differentiation and integration as inverse operations.

Numerical Integration

• Rectangle Method: Approximates area using rectangles
• Trapezoidal Rule: Uses trapezoids for better approximation
• Simpson's Rule: Approximates using parabolas for even higher accuracy

Applications

• Computing areas and volumes
• Finding average values
• Work and energy calculations
• Probability distributions
• Solving differential equations

Definition of a Limit

The limit of f(x) as x approaches a is L if f(x) can be made arbitrarily close to L by taking x sufficiently close to a.

Epsilon-Delta Definition

Formally, for every ε > 0, there exists δ > 0 such that:

One-Sided Limits

• Left-hand limit: \(\lim_{x \to a^-} f(x)\)
• Right-hand limit: \(\lim_{x \to a^+} f(x)\)
• For the limit to exist, both one-sided limits must exist and be equal

Special Limits

• \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\)
• \(\lim_{x \to \infty} (1 + \frac{1}{x})^x = e\)
• \(\lim_{x \to 0} \frac{e^x - 1}{x} = 1\)

Convergence of Series

A series converges if the sequence of partial sums converges to a finite value.

Common Series

• Geometric Series: \(\sum_{n=0}^{\infty} ar^n\) (converges when |r| < 1)
• p-Series: \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) (converges when p > 1)
• Harmonic Series: \(\sum_{n=1}^{\infty} \frac{1}{n}\) (diverges)
• Alternating Harmonic: \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\) (converges)

Convergence Tests

• Comparison Test
• Ratio Test
• Root Test
• Integral Test
• Alternating Series Test

Power Series

Power series are expansions of functions as infinite sums of power terms:

Examples include Taylor and Maclaurin series.

Types of Differential Equations

• Ordinary Differential Equations (ODEs): Involve derivatives with respect to a single variable
• Partial Differential Equations (PDEs): Involve derivatives with respect to multiple variables
• Linear vs. Nonlinear
• First-order vs. Higher-order

First-Order Linear ODEs

Can be solved using an integrating factor.

Numerical Methods

• Euler's Method: Simple approximation using tangent lines
• Runge-Kutta Methods: Higher-order approximations
• Finite Difference Methods: Discretization techniques

Applications

• Physics: Mechanics, waves, fields
• Engineering: Circuits, vibrations
• Biology: Population dynamics, epidemiology
• Economics: Growth models, finance
• Chemistry: Reaction rates