1. Exponential and Logarithmic Functions

1.1 Exponential Functions

Exponential functions are functions of the form , where and . They model growth and decay in many real-world contexts.

• Key Properties:

  • Domain:

  • Range:

  • Always positive, never zero

• Example:

1.2 Logarithmic Functions

The logarithmic function is the inverse of the exponential function . It is defined for and , .

• Key Properties:

  • Domain:

  • Range:

  • and

• Example: because

1.3 Solving Exponential and Logarithmic Equations

• To solve , take logarithms:

• To solve , rewrite as

1.4 Example Problems

• Solve

• Find such that

2. Inverse Trigonometric Functions

2.1 Definitions and Properties

Inverse trigonometric functions (arcsin, arccos, arctan) return the angle whose trigonometric function equals a given value.

• Domain and Range:

  • : , range

  • : , range

  • : , range

• Key Identities:

  • for

  • for

  • for all

2.2 Example Problems

• Find

• Find

3. Trigonometric Equations and Identities

3.1 Solving Trigonometric Equations

To solve equations like , use the inverse function and consider the periodicity of trigonometric functions.

• Example: Solve for in

3.2 Trigonometric Identities

• Pythagorean:

• Double Angle:

• Sum-to-Product, Product-to-Sum, etc.

4. Limits and Continuity

4.1 Definition of a Limit

The limit of as approaches is if gets arbitrarily close to as approaches .

• Notation:

• One-sided limits: and

4.2 Techniques for Computing Limits

• Direct substitution

• Factoring and canceling

• Rationalization

• L'Hospital's Rule for indeterminate forms

4.3 Continuity

• A function is continuous at if

• Piecewise functions: check continuity at the boundaries

4.4 Example Problems

• Compute

• Determine where is continuous for a piecewise function

5. The Intermediate Value Theorem (IVT)

5.1 Statement and Application

If is continuous on and is between and , then there exists such that .

• Used to show existence of roots

5.2 Example Problems

• Show that has a root in

6. The Derivative

6.1 Definition and Interpretation

The derivative of at is the limit:

• Represents the instantaneous rate of change or slope of the tangent line

6.2 Rules of Differentiation

• Sum Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

6.3 Derivatives of Inverse Functions

• If and are inverses, then where

6.4 Example Problems

• Find the derivative of

• Find the equation of the tangent line to at

7. Applications of Derivatives

7.1 Tangent Lines

• The slope of the tangent line at is

• Equation:

7.2 Continuity and Differentiability

• If a function is differentiable at , it is continuous at

• The converse is not always true

7.3 Piecewise Functions

• Check continuity and differentiability at the points where the formula changes

8. Asymptotes

8.1 Horizontal and Vertical Asymptotes

• Vertical asymptotes: where the function grows without bound as approaches a value

• Horizontal asymptotes: the value the function approaches as or

8.2 Example Problems

• Find the vertical and horizontal asymptotes of

9. Summary Table: Key Concepts

10. Practice and Application

• Practice solving equations involving exponentials, logarithms, and trigonometric functions

• Be able to compute limits, including indeterminate forms using L'Hospital's Rule

• Understand and apply the definition of the derivative

• Find equations of tangent lines and analyze continuity/differentiability of piecewise functions

• Find asymptotes and apply the Intermediate Value Theorem

Additional info: This study guide is based on a sample exam syllabus and covers foundational topics in Calculus I, including functions, limits, derivatives, and their applications. The examples and problems are representative of typical first-semester calculus content.