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Ch12Notes

# Of moles of gas present rconstant ttemperature

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Unformatted text preview: x(no. of items sold) A function in TWO variables: Assign a unique number to each pair (x,y) The graph is a surface in A function of THREE variables: In General: Gives the temperature at some point in space A function of n independent variables Let one of the variables equal a constant c. Ie: x=c or y=c or z=c The result is a function of a single variable. Graph the curve in the appropriate plane (ie: x = c). This curve is called the trace of the graph of in the plane x = c. A Level Curve is a 2-D sketch of in the xy-plane. in the xy-plane. Each constant value c represents a unique level curve Examples: Topographical Maps: Each curve represents a different altitude Weather Maps: Each curve represents an isotherm (equal temperature) or isobar (equal pressure) Equipotential Lines: The electric potential or Voltage (these lines are always perpendicular to the electric field) We cannot sketch graphs of three variables Example: BUT we can sketch their Level Surfaces Isothermal Surfaces An important observation: Any function of two variables can be considered as a Level Surface for a function of three variables . We will use this observation to find tangent planes to functions of two variables. A Contour Plot is where many level curves are drawn and ALL are put in the xy-plane. So a contour plot is a twodimensional picture. A Density Plot is a spectrum of colors where each color represents a constant value. 12.3: Limits and Continuity We can rewrite this with vector notation: Show a Limit does not exist (using 2 different paths) Show a Limit exists (using Theorem 2.1) Determine whether a piecewise function of two variables is continuous Limits: Formal Definition of a Limit Calc 101A Version (2D) Notice the condition is equivalent to an open interval centered at c . Calc 101C Version (3D) Notice the condition is equivalent to an open disk centered at (for the 3D version) If we were using four variables then we would have an open ball instead. We lose graphical meaning beyond that dimension. In 2D we were restricted to approaching c along the x-axis. But in 3...
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