Murphy: Math 19a (Calculus) So Far

Murphy: Math 19a (Calculus) So Far

1. Functions

   • Denoted as f(x) most commonly
     • Strictly one output for every one input, can be checked with the vertical line test
       • Can be combined through specific processes, but limitations of the host functions carry over
         • Inverse Functions
           • Inverse functions are denoted as f^-1(x) and are the inverses of functions, meaning they swap (x, f(x)) to (f(x), x)
             • One to one functions have an inverse without any domain restriction necessary

2. Limits

   • Limits are denoted as Limx->c f(c)
     • Limts are a great descriptor of behavior and show how a function acts approaching a specific point c, they can also describe the behavior of a function at points where the function does not exist
       • Limits can be combined using algebraic methods, similarly using algebra, indeterminate limits can be evaluated
         • Continuity
           • Continuity is a devlopment of limits and a further description of behavior
             • A function is continuous if: f(a) is defined; limx->a f(x) exists; limx->a f(x)= f(a)
               • Squeeze Theorem: a function's limit at a can be evaluated by squeezing it between two other functions that have the same limit at a, thus the original function must also have the same limit at a
                 • Intermediate Value Theorem: if a function is continuous on an interval [a,b] then there is a number c that has a position between [f(a), f(b)] that exists f(c)=z
   • Derivatives
     • Derivatives are a vital part of calculus and display the rate of change of a function over time in the form of another function
       • Denoted as f'(x) for a derivative of f(x)
         • Can be found through the Limit definition of derivatives, but also other processes such as the chain rule
           • Derivates are the inverse operation of integrating