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's output changes per unit change in input. The instantaneous rate of change at a point is the derivative at that point, representing the slope of the tangent line.

• Average Rate of Change:

• Instantaneous Rate of Change:

• Example: For , the average rate of change from to is .

Limits and Continuity

Understanding and Evaluating Limits

Limits describe the behavior of a function as the input approaches a certain value. They are foundational for defining derivatives and continuity.

• Limit Notation:

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

• Special Trigonometric Limit:

• Limits at Infinity: Used to determine end behavior and asymptotes.

• Example:

Continuity and Types of Discontinuities

A function is continuous at a point if the limit exists and equals the function's value there. Discontinuities can be classified as removable, jump, or infinite.

• Definition of Continuity:

• Types of Discontinuities:

  • Removable: Limit exists, but function is not defined or not equal to the limit.

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

  • Infinite: Function approaches infinity near the point.

Derivatives and Differentiation

Definition and Calculation of Derivatives

The derivative measures the instantaneous rate of change of a function. It can be defined at a point or as a function.

• Limit Definition at a Point:

• Limit Definition as a Function:

• Units: If is in meters and in seconds, is in meters per second.

Higher Order Derivatives and Notation

• Second Derivative:

• Notation: , , ,

Rules of Differentiation

• Power Rule:

• Exponential Rule:

• Trigonometric Rules: ,

• Sum Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Implicit Differentiation

• Used when is defined implicitly by an equation involving and .

• Example: For , differentiate both sides with respect to to find .

• Can also find by differentiating again.

Derivatives of Inverse, Logarithmic, and Inverse Trig Functions

• Inverse Function Rule:

• Logarithmic Differentiation: Useful for functions of the form .

• Derivative of :

• Inverse Trig Derivatives: ,

Applications of Derivatives

• Tangent Lines: The equation of the tangent line at is

• Rates of Change: Position, velocity, and acceleration are related by derivatives.

• Related Rates: Problems involving rates at which related variables change.

• Linearization: Approximating a function near a point using its tangent line.

Analysis of Functions Using Derivatives

Critical Points, Extrema, and the First/Second Derivative Tests

• Critical Points: Where or does not exist.

• Absolute and Local Extrema: Highest or lowest values on a domain or interval.

• First Derivative Test: Determines if a critical point is a local max or min based on sign changes of .

• Second Derivative Test: If , local min at ; if , local max at .

Increasing/Decreasing Intervals and Concavity

• Increasing/Decreasing: means increasing; means decreasing.

• Concavity: means concave up; means concave down.

• Inflection Points: Where concavity changes (where and sign changes).

Graph Sketching

• Use information about critical points, inflection points, and intervals of increase/decrease and concavity to sketch graphs.

Optimization Problems

• Set up a function to optimize (maximize or minimize), find critical points, and justify the solution using derivative tests.

• Include correct units in answers.

L'Hôpital's Rule and Indeterminate Forms

• Used to evaluate limits of the form or .

• Rule: if the original limit is indeterminate.

• Other indeterminate forms (e.g., , ) can often be manipulated into or .

Newton's Method

• An iterative method to approximate roots of a function.

• Formula:

Mean Value Theorem (MVT) for Derivatives

• If is continuous on and differentiable on , then there exists in such that

Integration and Its Applications

Antiderivatives and Indefinite Integrals

• Antiderivative: A function such that

• General Form:

• Common Antiderivatives:

  • (for )

Riemann Sums and Definite Integrals

• Riemann Sum: Approximates area under a curve using rectangles.

• Summation Notation:

• Definite Integral: gives the net area under from to .

• Properties: Linearity, additivity, reversing limits, etc.

Average Value and Mean Value Theorem for Integrals

• Average Value:

• Mean Value Theorem for Integrals: There exists in such that

Fundamental Theorem of Calculus (FTC)

• Part 1: If , then

• Part 2: , where is any antiderivative of

Integration Techniques

• Substitution: Used to simplify integrals by changing variables.

• Definite and Indefinite Integrals: Apply substitution to both types.

Applications of Integration

• Net Area: gives net area; area between curves is

• Motion: Integrate acceleration to get velocity, and velocity to get position.

Differential Equations

Separable Differential Equations

• Can be written as and solved by separating variables.

• General Solution:

• Apply initial conditions to find particular solutions.

Additional info: This guide covers all major Calculus I topics, including limits, derivatives, applications, integrals, and basic differential equations, as outlined in the provided syllabus.