Limits and Continuity

Understanding Limits

Limits are foundational to calculus, describing the behavior of a function as its input approaches a particular value. They are essential for defining derivatives and integrals.

• Definition: The limit of a function f(x) as x approaches a value a is the value that f(x) approaches as x gets arbitrarily close to a.

• Notation:

• One-sided limits: (from the left), (from the right)

• Existence: The limit exists only if both one-sided limits exist and are equal.

Example: For a piecewise function, check the left and right limits at the point of interest to determine continuity.

Continuity

A function is continuous at a point if the limit exists and equals the function's value at that point.

• Definition: f(x) is continuous at x = a if

• Types of discontinuities: Removable, jump, and infinite discontinuities.

Example: A graph with a hole at x = 2 is discontinuous at x = 2 (removable discontinuity).

Intro to Derivatives

Definition of the Derivative

The derivative measures the instantaneous rate of change of a function, or the slope of the tangent line at a point.

• Definition:

• Interpretation: The derivative at x = a gives the slope of the tangent to the curve at that point.

Example: For , .

Differentiability

A function is differentiable at a point if its derivative exists there. Differentiability implies continuity, but not vice versa.

• Sharp corners or cusps: Not differentiable at those points.

• Vertical tangents: Not differentiable where the tangent is vertical.

Example: is not differentiable at x = 0.

Techniques of Differentiation

Basic Rules

• Power Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Example:

Applications of Derivatives

Critical Points and Extrema

Critical points occur where the derivative is zero or undefined. These points are candidates for local maxima or minima.

• Critical Point: Where or does not exist.

• First Derivative Test: Determines if a critical point is a local maximum, minimum, or neither.

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

Example: For , find , set to zero, and solve for critical points.

Increasing and Decreasing Functions

• Increasing: on an interval.

• Decreasing: on an interval.

Example: Analyze the sign of to determine intervals of increase or decrease.

Concavity and Inflection Points

• Concave Up:

• Concave Down:

• Inflection Point: Where concavity changes (where and changes sign)

Example: For , , so inflection at x = 0.

Graphical Applications of Derivatives

Sketching Graphs Using Derivatives

Derivatives help identify key features of graphs, such as maxima, minima, and inflection points.

• Use to find critical points and determine intervals of increase/decrease.

• Use to determine concavity and locate inflection points.

Example: Given a graph of , determine where is increasing or decreasing.

Definite and Indefinite Integrals

Antiderivatives and Indefinite Integrals

An antiderivative of a function f(x) is a function F(x) such that . The indefinite integral represents the family of all antiderivatives.

• Notation:

• Basic Rules: (for )

Example:

Definite Integrals and Area

The definite integral computes the net area under a curve between two points.

• Notation:

• Fundamental Theorem of Calculus: , where F is any antiderivative of f.

Example:

Techniques of Integration

Substitution Method

Substitution simplifies integration by changing variables.

• Let , then

• Rewrite the integral in terms of u and du, integrate, then substitute back.

Example: ; let , .

Applications of Integrals

Area Between Curves

The area between two curves and from to is:

Example: Find the area between and from to .

Table: Summary of Key Calculus Concepts

Practice Problems and Applications

• Identify intervals where a function is increasing or decreasing using the sign of .

• Find and classify critical points using the first and second derivative tests.

• Determine points of inflection by analyzing .

• Evaluate definite and indefinite integrals using basic rules and substitution.

• Sketch graphs based on information about , , and .

Additional info: The original file consists of exam review questions covering graphical analysis, critical points, intervals of increase/decrease, concavity, asymptotes, and basic integration, all central to Calculus I topics.