CALCULUS

CALCULUS

1. FUNCTIONS

   • Functions and their graphs
     • Functions are a tool for describing the real world in mathematical terms
       • Takes in inputs (x) and provides outputs (y)
     • Domain
       • The domain of a function is a list of ALL POSSIBLE input values.
       • A domain which equals all real numbers is called the NATURAL DOMAIN
       • Interval Notation
         • Simply an array of values within parenthesis (excluding) and brackets (including)
         • When infinity is present, always use parenthesis
         • Can combine two intervals using U or show where they intersect using an upside down U.
     • Range
       • The range of a function is a list of ALL POSSIBLE output values
     • Definition of a function
       • A function must have only one output per input
         • Functions must pass the vertical line test
     • Piecewise Functions
       • A piecewise function means that a function changes its behavior over time
     • Vertical Asymptotes
       • Vertical asymptotes are vertical lines on the graph where f(x) cannot equal.
       • Can easily be solved by seeing where the function is undefined (denominator equals zero)
     • Symmetry
       • Even Functions
         • Even functions are symmetrical with respect to the y-axis
           • f(-x) = f(x)
       • Odd Functions
         • Odd functions are symmetrical with respect to the origin
         • f(-x) = -f(x)
       • No symmetry
         • It is possible for graphs to have no symmetry
       • Symmetry helps us understand the behavior of graphs without analyzing
   • Combining Functions: Shifting and Scaling
     • Combining f(x) and g(x)
       • (f+g)(x) = f(x)+g(x)
       • (f-g)(x) = f(x)-g(x)
       • (f/g)(x) = f(x)/g(x)
       • (f*g)(x) = f(x)g(x)
       • Composite Functions
         • (fog)(x) = f(g(x))
     • Shifting Functions
       • Shifting Lines
         • Horizontal Shifts
           • EXAMPLE: y = f(x-h) + k
             • Shifts the graph right or left h units
               • EXAMPLE: if h=2: The graph shifts left 2 units
         • Vertical Shifts
           • EXAMPLE: y= f(x-h) + k
             • Shifts the graph up or down k units
               • EXAMPLE if k=2: The graph shifts up 2 units
   • Trigonometric Functions
   • Graphing with software
   • Exponential Functions
   • Inverse Functions and Logarithms

2. LIMITS AND CONTINUITY

   • Rates of change and tangent lines to curves
   • Limits
     • Limit of a function
     • Limit Laws
     • Precise Definition of a limit
     • Limits Involving Infinity
       • Horizontal Asymptotes
   • Continuity
     • Discontinuity