Chapter 2: Limits and Continuity

2.2 Definition of Limits

The concept of a limit is foundational in calculus, describing the behavior of a function as its input approaches a particular value. The formal definition uses the language of approaching a value arbitrarily closely.

• Limit Notation: means that as approaches , approaches .

• Existence of a Limit: For a limit to exist at , the left-hand and right-hand limits must both exist and be equal.

• Formal (Epsilon-Delta) Definition: For every , there exists a such that if , then .

2.3 Techniques for Evaluating Limits

Several algebraic techniques are used to evaluate limits, especially when direct substitution leads to indeterminate forms such as .

• Factoring: Factor numerator and denominator to cancel common terms.

• Simplification: Simplify the expression to remove the indeterminate form.

• Example: Evaluate .

After canceling , the limit simplifies to .

2.4 Infinite Limits and Limits at Infinity

Infinite limits describe the behavior of functions as they grow without bound near a certain point, while limits at infinity describe the behavior as approaches infinity or negative infinity.

• Infinite Limit: means increases without bound as approaches .

• Limit at Infinity: describes the end behavior of as becomes very large.

2.6 Continuity

A function is continuous at a point if its limit exists at that point, the function is defined there, and the value of the function equals the limit.

• Three Conditions for Continuity at :

  1. exists

  2. is defined

• Intermediate Value Theorem (IVT): If is continuous on and is between and , then there exists in such that .

Chapter 4: Applications of the Derivative

4.1 Maxima and Minima

Maxima and minima are the highest and lowest points on a function, respectively. These points are critical for optimization problems and are found using derivatives.

• Critical Points: Points where or does not exist.

• Local Maximum/Minimum: Determined by the sign of the first and second derivatives.

4.2 Mean Value Theorem (MVT)

The Mean Value Theorem states that for a function continuous on and differentiable on , there exists in $(a, b)$ such that:

4.3 What Derivatives Tell Us & 4.4 Graphing Functions

Derivatives provide information about the increasing/decreasing behavior of functions and their concavity. This is essential for sketching accurate graphs.

• First Derivative Test: Determines intervals of increase/decrease.

• Second Derivative Test: Determines concavity and inflection points.

• Graphing: Use and to identify local maxima, minima, and inflection points.

• Example: Draw a graph so that on , etc.

4.7 L'Hôpital's Rule

L'Hôpital's Rule is used to evaluate limits that result in indeterminate forms such as or by differentiating the numerator and denominator.

• Rule: If yields or , then (if the latter limit exists).

4.9 Antiderivatives

An antiderivative of a function is a function such that . Finding antiderivatives is the reverse process of differentiation and is fundamental to integration.

Chapter 5: Integration

5.2 Area Under a Curve

The area under a curve between and is given by the definite integral:

• This represents the net area between the function and the -axis.

5.3 Fundamental Theorem of Calculus (FTC)

The Fundamental Theorem of Calculus links differentiation and integration, providing a way to evaluate definite integrals using antiderivatives.

• Part 1: If is an antiderivative of , then .

• Part 2: .

5.5 Substitution (U-Substitution)

U-substitution is a technique for evaluating integrals by substituting part of the integrand with a new variable to simplify the integral.

• Steps:

  1. Let , compute .

  2. Rewrite the integral in terms of and .

  3. Integrate with respect to , then substitute back .

Chapter 6: Applications of Integration

6.2 Area Between Curves

The area between two curves and from to is given by:

• This computes the vertical distance between the curves, integrated over the interval.

6.3 Washer and 6.4 Shell Methods

These methods are used to find the volume of solids of revolution.

• Washer Method:

• Shell Method:

6.5 Arc Length

The arc length of a curve from to is given by:

Appendix: Derivatives of Trigonometric Functions

The derivatives of the six basic trigonometric functions are essential for solving calculus problems involving trigonometric expressions.