Functions and Their Properties

Rational Functions

Rational functions are quotients of polynomials and are fundamental in calculus for analyzing asymptotic behavior and domain restrictions.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is nonzero. For , set to find vertical asymptotes.

• Horizontal Asymptotes: Determined by comparing degrees of numerator and denominator. If degrees are equal, the asymptote is the ratio of leading coefficients.

• Domain: All real numbers except where the denominator is zero.

Example: For , vertical asymptotes at and .

Limits and Continuity

Asymptotic Behavior

Limits are used to analyze the behavior of functions near points of discontinuity or infinity.

• Vertical Asymptote:

• Horizontal Asymptote: or

Techniques of Differentiation

Basic Differentiation

Differentiation is the process of finding the rate of change of a function.

• Power Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Example: ;

Applications of Derivatives

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to another.

• Volume of a Sphere:

• Given: , find using

Example: If and , then

Implicit Differentiation

Implicit Functions

Implicit differentiation is used when a function is not given explicitly as .

• Differentiate both sides with respect to : For ,

• Solve for :

Example: Find the tangent line at using the derivative and point-slope form.

Graphical Applications of Derivatives

Critical Points and Extrema

Critical points occur where or is undefined. These are candidates for local maxima, minima, or points of inflection.

• Increasing/Decreasing Intervals: Use sign of

• Relative Maximum/Minimum: Use first or second derivative test

• Concavity: Use sign of

• Points of Inflection: Where and concavity changes

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

Derivatives of Exponential and Logarithmic Functions

Exponential Decay Models

Exponential functions model growth and decay, common in applications such as depreciation.

• General Form:

• Derivative:

Example: For ,

Chain Rule and Composite Functions

Using the Chain Rule

The chain rule is essential for differentiating composite functions.

• Example:

• Apply Chain Rule:

Further Differentiation Practice

Implicit and Explicit Differentiation

Practice differentiating both explicit and implicit functions, including those involving logarithms and roots.

• Implicit Example: ; differentiate both sides with respect to .

• Explicit Example: ; rewrite as for .

Comprehensive Function Analysis

Full Function Investigation

Analyzing a function involves finding its domain, critical points, intervals of increase/decrease, extrema, asymptotes, concavity, and intercepts.

• Domain: Where the function is defined

• Critical Values: Where or is undefined

• Intervals of Increase/Decrease: Use sign of

• Relative Extrema: Use first or second derivative test

• Vertical/Horizontal/Slant Asymptotes: Analyze limits and degrees

• Concavity and Inflection Points: Use

• Intercepts: Set or as appropriate

Example: For , analyze all properties above.

Additional info: This guide covers all major calculus topics found in the exam, including rational functions, differentiation, related rates, implicit differentiation, and full function analysis. These are core to college-level Calculus I.