Level Curves and Contour Maps (sec 14.1) & Partial Derivatives (sec 14.3) 11 February 2015 Today, We Will: Wrap-up our discussion of level curves and contour maps. Define partial derivatives of a function of two or more variables, and see how partial derivatives relate to derivatives of functions of a single variable (as studied in Calc I). Practice computing partial derivatives. Level Curves A level curve is a path in the domain of a function on which the function values remain constant. For example: If the function measures height of a mountain above sea level, the height of the mountain above a level curve is constant. If the function measures the temperature on a plate, the temperature is constant along a level curve. More technically: Let c be a constant. A level curve of a function f (x, y) is a curve in the xy-plane such that, for all points (x, y) on the curve, f (x, y) = c.
Equations of Level Curves of a Function z = f (x, y) Set z equal to a constant c. The level curves of z = f (x, y) are the curves in the xy-plane satisfying the equation c = f (x, y). Contour Maps A contour map is a collection of level curves f (x, y) = c, where the difference between c-values on adjacent curves is constant. The difference between c-values on adjacent curves of a contour map is called the contour interval. The closer together contour lines (level curves) are, the steeper the graph. An example of a contour map is a topographic map. The distance between topo-lines (contour lines) represents a (constant) change in elevation. Example/Discussion Question: Level Curves/Contour Maps Use a graphing app (or online applet) to compare the contour maps of the following functions: f (x, y) = x 2 + y 2 (paraboloid) g(x, y) = x 2 + y 2 (cone) What do the contour maps have in common? How are they different? Two online apps: WolframAlpha: http://www.wolframalpha.com. Surface/Level Curve Grapher (D. Ensley, B. Kaskosz): http://www.flashandmath.com/mathlets/multicalc/ contours/combo.html.
Example: Contour Maps and Rates of Change Use the contour map find: The change in the function value from the point A to the point C. The average rate of change of the function from the point A to the point C. The average rate of change of the function from the point A to the point B. Summary of Properties of Level Curves and Contour Maps Level curves of a function lie in the domain of the function for example, the level curves of a function f (x, y) lie in the xy-plane. Level curves represent the curves in the domain over which the function value remains constant. The rate of change of the function is zero along level curves. A contour map is a collection of level curves representing evenly-spaced function values (the spacing is called the contour interval). The closer together the curves, the steeper the graph of the function. The Story So Far The derivative of a scalar-valued function of a single variable (the type of function studied in Calc I) is: The limit of difference quotients. The slope of the tangent line to the curve. The instantaneous rate of change of the function. We are now studying scalar-valued functions of two and three variables z = f (x, y) and w = g(x, y, z). What is a derivative for such functions?
Partial Derivatives of a Function f (x, y) at a Point The partial derivative of f with respect to x at the point (x 0, y 0 ) is: x (y is held constant) (x0,y 0) = lim h 0 f (x 0 + h, y 0 ) f (x 0, y 0 ) h The partial derivative of f with respect to y at the point (x 0, y 0 ) is: y (x is held constant) (x0,y 0) = lim h 0 provided these limits exist. f (x 0, y 0 + h) f (x 0, y 0 ) h These are Calc I -type limits, since one variable is treated as a constant. Geometry of Partial Derivatives Partial derivatives have the same interpretations as the derivatives of single-variable functions: they represent instantaneous rates of change, and slopes of tangent lines. For example: the partial derivative y (x0,y 0) is the limit of the slopes of the secant lines through the point (x 0, y 0, f (x 0, y 0 )) and nearby points (x 0, y 0 + g, f (x 0, y 0 + g)). If this limit (the derivative) exists, it is the slope of the tangent line to the curve (x 0, y, f (x 0, y)) at the point (x 0, y 0, f (x 0, y 0 )). Partial Derivatives as Functions If a partial derivative of a function f is defined on a region R, then the partial derivative can be considered a function on the region R. Technically, this region must be open. This means every point P in the region is contained in an open disk D, which lies entirely inside the region. An open neighborhood in R 3 is defined analogously, using an open ball instead of an open disk.
Partial Derivatives of a Function of Three or More Variables Partial derivatives of functions of three or more variables are defined in an analogous manner. If f (x, y, z) is a function of three variables, a partial derivative is the limit of a difference quotient with two variables held constant. If f (x 1, x 2,..., x n ) is a function of n variables, a partial derivative is the limit of a difference quotient with n 1 variables held constant. Partial Derivative Notation The partial derivative as a function: x = f x(x, y) y = f y(x, y) x = f x(x, y, z) y = f y(x, y, z) z = f z(x, y, z) The partial derivative evaluated at the point (x 0, y 0, z 0 ): = f x (x 0, y 0 ) = f y (x 0, y 0 ) x (x0,y0) y (x0,y0) x = f x(x 0, y 0, z 0 ) y = f y(x 0, y 0, z 0 ) z = f z(x 0, y 0, z 0 ) Computing Partial Derivatives (Jedi Mind Tricks) When computing the partial derivative with respect to one variable, treat the other variables as though they were constant. Then use the differentiation rules from Calc I.
Example: Computing Partial Derivatives Compute the partial derivatives for the following functions: f (x, y) = 3x 2 y + sin(x) g(x, y) = y ln(xy) w(x, y, z) = exyz z P(V, n, T ) = nrt, R is a constant. V