1. Exponential and Logarithmic Functions

1.1 Exponential Functions

The exponential function is defined as , where and . It is a fundamental function in calculus, modeling growth and decay.

• Key Properties: , ,

• Applications: Population growth, radioactive decay, compound interest

• Example: Solve

1.2 Logarithmic Functions

The logarithmic function is the inverse of the exponential function, defined as if and only if .

• Key Properties: ,

• Applications: pH scale, Richter scale, information theory

• Example: Find if

1.3 Solving Exponential and Logarithmic Equations

• Isolate the exponential or logarithmic term

• Apply logarithms or exponentiate both sides as needed

• Example: Solve

2. Inverse Trigonometric Functions

2.1 Definitions and Domains

Inverse trigonometric functions include , , and . They return the angle whose trigonometric value is .

• Domain and Range:

  • : , range

  • : , range

  • : , range

• Example:

2.2 Trigonometric Identities and Inverses

• Know basic identities: ,

• Use identities to simplify expressions involving inverse functions

• Example:

3. Solving Trigonometric Equations

3.1 Equations on Specified Intervals

To solve equations like on , use identities and consider all solutions within the interval.

• Apply double angle and Pythagorean identities

• Check all possible solutions in the given interval

• Example: Solve for

4. Expressing Trigonometric Functions Algebraically

4.1 Algebraic Formulas

Express trigonometric functions in terms of using inverse functions and identities.

• Example:

5. Inverse Functions and Their Domains

5.1 One-to-One Functions

A function is one-to-one if each output is produced by exactly one input. Only one-to-one functions have inverses.

• To find the inverse, solve for in terms of

• Example: ; inverse is

5.2 Domain and Range of Inverse Functions

• The domain of is the range of

• Example: Find the domain and range of

6. Limits and Their Computation

6.1 Definition and Techniques

The limit describes the value approaches as approaches .

• Use algebraic simplification, factoring, rationalization, and squeeze theorem

• Understand right/left-hand limits: and

• Example:

6.2 Indeterminate Forms

• Forms like , require special techniques

• Apply algebraic manipulation or squeeze theorem

• Example:

7. Intermediate Value Theorem (IVT)

7.1 Statement and Application

The Intermediate Value Theorem states that if is continuous on and is between and , then there exists such that .

• Used to show existence of roots or solutions

• Example: Show has a root in

8. Derivatives and Their Meaning

8.1 Definition of the Derivative

The derivative of at is .

• Represents the instantaneous rate of change

• Used to find slopes of tangent lines

• Example: If , then

8.2 Related Limits

• Limits involving derivatives:

• Example: Compute

9. Piecewise Functions and Continuity

9.1 Continuity

A function is continuous at if .

• Check continuity at points where the formula changes

• Example:

9.2 Differentiability

A function is differentiable at if the derivative exists at .

• Differentiability implies continuity, but not vice versa

• Example: is continuous everywhere but not differentiable at

10. Asymptotes of Functions

10.1 Horizontal and Vertical Asymptotes

Vertical asymptotes occur where the function grows without bound as approaches a certain value. Horizontal asymptotes describe the behavior as or .

• Find vertical asymptotes by setting denominator to zero

• Find horizontal asymptotes by evaluating

• Example: has a vertical asymptote at

11. Tangent Lines and Applications

11.1 Equation of the Tangent Line

The tangent line to at has equation .

• Find using the derivative

• Example: Find the tangent to at

12. Continuity and Differentiability of Piecewise Functions

12.1 Checking Continuity and Differentiability

• Check limits from both sides at the point where the formula changes

• Check if derivatives from both sides match

• Example:

13. Derivatives of Sums, Products, Quotients, and Inverse Functions

13.1 Rules for Derivatives

• Sum Rule:

• Product Rule:

• Quotient Rule:

• Inverse Function Rule: If and are inverses,

• Example: If , then and

Additional info:

• This study guide covers all major topics for a first college Calculus exam, including functions, limits, derivatives, continuity, and applications such as tangent lines and asymptotes.

• Example problems are provided for each topic to illustrate typical exam questions.