Analysis II (MA131L)

      • Graph of Functions. To understand functions, we look at the graphs of functions. As an exercise produce a function in formula and get it graphed. But take care when an induction is made using a graph, a graph can be misleading if it is not properly zoomed.
        Could you tell whether the function is continuous? differentiable? twice differentiable?
      • Graph a function with the aid of a computer.
        • For Mac users, go to Applications, Utilities, inside Utilities you find Grapher. Open Grapher. Open a new file, choose 2D. Type a function, e.g. y=exp(-1/x^2) and hit return.
          Other graphing tools: graphic calculator, Mathematica, maple matlab.
        • Use of Mathematica. Open the program. Open a new notebook. Begin to type and when you finished typing, hold shift key and press return.
          e.g. type 1/5+2 Pi.
          Plot[x^2 + 3 x - 1, {x, -10, 10}]Plot[Sin[2 x + 1], {x, 0, Pi}]
          Plot[x^2 + 3 x - 1, {x, -10, 10}]
          The letters P in Plot and Sin in sin(x) must be in capital letter.
          {x, 0, Pi} indicates the variable and the range of the vraible.
          Other use: D[f, x], givs the partial derivative of f with respect to x variable.
          e.g. D[Sin(x^2+1), x]
          Integrate[x^7+2x, x]
          Useful names of functions: Log, Exp, ArcTan.
      • Some Graphs