Functions and Rates of Change

Average and Instantaneous Rate of Change

The average rate of change of a function over an interval measures how much the function changes per unit interval. The instantaneous rate of change is the derivative at a point, representing the slope of the tangent line.

• Average Rate of Change: For over ,

• Instantaneous Rate of Change:

• Example: If , average rate from to is ; instantaneous at is .

Limits and Continuity

Evaluating Limits and One-Sided Limits

Limits describe the behavior of functions as inputs approach a value. One-sided limits consider approach from only one direction.

• Limit:

• One-Sided Limits: (from left), (from right)

• Example:

Limits Involving Infinity and Asymptotes

Limits as approaches infinity or negative infinity help identify horizontal and vertical asymptotes.

• Horizontal Asymptote:

• Vertical Asymptote:

• Example:

Continuity and Types of Discontinuities

A function is continuous at if . Discontinuities can be classified as removable, jump, or infinite.

• Removable: Limit exists, but function value is missing or different.

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

• Infinite: Limit approaches infinity.

• Example: is removable at .

Derivatives and Differentiation

Limit Definition of Derivative

The derivative at a point is defined as the limit of the average rate of change as the interval shrinks to zero.

• Example: For ,

Higher Order Derivatives and Notation

Higher order derivatives represent rates of change of derivatives. Common notations include , , etc.

• Second Derivative: or

• Units: If is position (meters), is velocity (m/s), is acceleration (m/s2).

Differentiation Rules

Several rules simplify finding derivatives:

• Power Rule:

• Exponential Rule:

• Trigonometric Rules: ,

• Sum Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Implicit Differentiation

Used when is defined implicitly by an equation. Differentiate both sides and solve for .

• Example: If ,

• Second Derivative: Differentiate with respect to .

Derivatives of Inverse, Logarithmic, and Inverse Trig Functions

• Inverse Function:

• Logarithmic:

• Logarithmic Differentiation: Useful for products, quotients, or powers: , then differentiate.

• Inverse Trig: ,

Tangent Lines

The tangent line to at has slope and equation .

Related Rates

Problems where two or more quantities change with respect to time. Use implicit differentiation with respect to .

• Example: If ,

Linearization and Differentials

Linearization approximates a function near a point using its tangent line. Differentials estimate small changes.

• Linear Approximation:

• Differential:

Applications of Derivatives

Extrema and Critical Points

Absolute and local maxima/minima are found using derivatives. Critical points occur where or is undefined.

• First Derivative Test: Determines if critical points are maxima or minima.

• Second Derivative Test: If , minimum; if , maximum.

Increasing/Decreasing and Concavity

• Increasing:

• Decreasing:

• Concave Up:

• Concave Down:

• Inflection Point: Where concavity changes.

Graph Sketching

Use critical points, intervals of increase/decrease, and concavity to sketch graphs.

Optimization

Set up equations, find critical points, and justify extrema for real-world problems.

• Example: Maximizing area, minimizing cost, etc.

L'Hôpital's Rule

Used for indeterminate forms or :

Newton's Method

An iterative method to approximate roots of equations.

• Formula:

Mean Value Theorem for Derivatives

If is continuous on and differentiable on , then such that

Integrals and Applications

Antiderivatives

Antiderivatives reverse differentiation. All antiderivatives differ by a constant .

• (for )

Riemann Sums and Definite Integrals

Riemann sums approximate area under curves. Definite integrals compute exact area.

• Riemann Sum:

• Definite Integral:

Properties of Definite Integrals

• Linearity:

• Reversal:

• Breaking Interval:

Average Value of a Function

The average value of over is

Mean Value Theorem for Integrals

There exists such that

Fundamental Theorem of Calculus

• Part 1: If , then

• Part 2: , where is any antiderivative of

Substitution in Integration

Used to simplify integrals by changing variables.

• Let , then

Area Under and Between Curves

• Net Area:

• Area Between Curves:

Applications: Acceleration, Velocity, Position

Integration relates position, velocity, and acceleration.

• Velocity from acceleration:

• Position from velocity:

Differential Equations

Separable Differential Equations

Equations that can be written as and solved by separating variables.

• General Solution:

• Apply initial conditions to find particular solutions.

Summary Table: Key Calculus Concepts