MATH 149 LABORATORY ASSIGNMENT 4 IMPLICIT DIFFERENTIATION The function f x is said to be implicitly defined by the equation E x y 0 provided that E x f x 0 for all x in some interval I For example if E x y x2 y2 4 then f1 x 4 x2 and f2 x 4 x2 both satisfy the equation E x y 0 and thus both are implicitly defined by this equation It is however frequently difficult or impossible to solve an equation of the form E x y 0 for y in terms of x this is the case in the example and exercises below Try it Nevertheless such an equation may implicitly define one or more functions of x It is possible to find the derivative of an implicitly defined function at a given point without first solving the equation for y in terms of x by using a technique called implicit differentiation We differentiate both sides of the equation with respect to x viewing y as a function of x applying the usual differentiation rules and ultimately solving for dy dx in terms of x and y This procedure can be carried out by using appropriate Maple commands Example Find the slope m of the tangent line to the graph of the cardioid with equation x 4 y4 4 y3 x2 x2 y 2 x2 y2 0 at the point P 2 3 2 2 3 3 2 First we enter the equation of the cardioid restart eq x 4 y 4 4 y 3 x 2 x 2 y 2 x 2 y 2 0 eq x4 y4 4 y3 4 x2 4 x2 y 2 x2 y2 0 Next we sketch a graph of the cardioid in the coordinate plane using the Maple command implicitplot which is located in the plots package This part of Maple can be accessed by first entering the command with plots Note that we ended the command with a colon rather than a semicolon if you terminate it with a semicolon Maple responds with the contents of the package Try it with plots implicitplot eq x 4 4 y 1 4 grid 100 100 color black Now we tell Maple to treat y as a function of x y f x y f x eq x4 f x 4 4 f x 3 4 x2 4 x2 f x 2 x2 f x 2 0 Observe that Maple has replaced each occurrence of y by f x To differentiate this equation with respect to x we use the diff command deq diff eq x d d d deq 4 x3 4 f x 3 f x 12 f x 2 f x 8 x 8 x f x 4 x2 f x 4 x f x 2 dx dx dx d 4 x2 f x f x 0 dx d Here Maple uses f x to denote f x the derivative of f x dx d We now solve for f x with a deferred command dx Diff f x x solve deq diff f x x d x x2 2 2 f x f x 2 f x dx f x 3 3 f x 2 x2 x2 f x The single Maple command implicitdiff combines the above computations but uses y rather than f x y y y y Previously we had replaced y by f x the command y y undoes this eq x4 y4 4 y3 4 x2 4 x2 y 2 x2 y2 0 dydx implicitdiff eq y x dydx The point P has coordinates x 2 3 2 x x2 2 2 y y2 y3 3 y2 x2 x2 y and y the slope of the tangent at P we must replace y by 2 3 3 2 2 3 3 2 so in order to find m and x by 2 3 2 in the formula for dy dx subs y 2 sqrt 3 3 2 x 2 sqrt 3 2 dydx 2 2 3 3 1 1 5 2 3 3 3 2 2 2 3 3 3 3 3 3 1 2 2 2 3 2 2 1 3 2 2 3 3 2 Simplifying this complicated fraction we find the slope m of the tangent line to the graph at P to be m simplify m 1 In order to get a decimal representation for m accurate to ten significant digits we may use the evalf command evalf m 1 The equation of the tangent is y m x 2 3 2 3 3 2 2 cardioid and its tangent on the same set of axes as follows We can plot the A implicitplot eq x 4 4 y 1 4 grid 100 100 color black B plot m x 2 sqrt 3 2 2 sqrt 3 3 2 x 4 4 y 1 5 NOTE THAT THE ABOVE COMMANDS END WITH A COLON AND NOT A SEMI COLON THIS IS DONE TO SUPPRESS THE OUTPUT OF THE COMMAND display A B
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