Calculus 1 - Full College Course

Introduction to Rational Expressions

• A rational expression is a fraction involving variables, e.g., ( \frac{x + 2}{x^2 - 3} ).

• Operations: Adding, subtracting, multiplying, dividing, and simplifying to lowest terms.

Simplifying Rational Expressions

• For numeric fractions like ( \frac{21}{45} ): reduce by finding common factors.

• Example: ( \frac{21}{45} = \frac{7}{15} ).

  • Factor: 21 = 3 x 7; 45 = 3 x 15 -> Cancel 3.

• For variable expressions, factor both numerator and denominator then cancel:

  • Example: ( \frac{3(x + 2)}{(x + 2)(2x + 2)} \rightarrow \frac{3}{2x + 2} ).

Multiplying and Dividing Rational Expressions

• Multiply: ( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} )

• For division, multiply by reciprocal:

  • Example: ( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} )

Adding and Subtracting Rational Expressions

• Find a common denominator; use least common denominator (LCD).

• For ( \frac{a}{m} + \frac{b}{n} ):

  • Example:( \frac{7}{6} - \frac{15}{15} = \frac{35 - 8}{30} = \frac{27}{30} = \frac{9}{10} )

Finding Common Denominators

• Factor denominators to find LCD. E.g., for 6 and 15:

  • LC = 2 x 3 x 5 = 30.

Basic Concepts in Functions

• Average rate of change: Difference quotient for functions, e.g., average growth of a function over an interval; slope of secant line between two points.

  • Example: ( \frac{f(b) - f(a)}{b - a} )

• Instantaneous rate of change (derived from the slope of tangent lines).

Implicit Differentiation and Rates of Change

• Higher-order derivatives: If ( f'(x) = 3x^2 ), then ( f''(x) ) computes the second derivative, which indicates concavity.

Concavity and Inflection Points

• A function is concave up if its first derivative is increasing, indicated by ( f''(x) > 0 ).

• A function is concave down if its first derivative is decreasing, indicated by ( f''(x) < 0 ).

• Inflection points occur where the second derivative changes sign.

Mean Value Theorem for Integrals

• The average value of a function on an interval is equal to the integral of the function divided by the interval's length.

  • ( f(c) = \frac{1}{b-a} \int_{a}^{b} f(x) dx )

Logarithms and Properties

• Log properties include:

  • Product, Quotient, and Power rules for simplification.

• Logarithm function's behavior: Defined for positive values only; log of zero or negative is undefined.

Introduction to Inverse Trig Functions

• Inverse trigonometric functions like arcsin, arccos, and arctan have specific ranges to maintain function properties.

Summary

• Rational expressions can be analyzed with algebraic methods and graphical interpretations. Functions exhibit characteristics in their graphs, such as maximums, minimums, critical points, and inflection points. The mean value theorem relates the average rate of change to instantaneous rates, and properties of logarithms provide behaviors for exponential relationships. Inverse functions reverse the roles of inputs and outputs, revealing practical applications in geometry and calculus.