MATH 149LABORATORY ASSIGNMENT 4 IMPLICIT DIFFERENTIATION The function f ( x ) is said to be implicitly defined by the equation (*) E( x , y ) = 0provided that E( x , f ( x ) ) = 0 for all x in some interval I . For example, if E( x , y ) = + − x2y24, then = ()f1x − 4x2 and = ()f2x− − 4x2 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: = + − + x4y44( ) + + y3x2x2y 2 x2y20 at the point P = ( + 2 32 ,+ 2 332 ). 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 = + − − − + x4y44 y34 x24 x2y 2 x2y20 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()fx44()fx34x24x2()fx2x2()fx20 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);deq 4 x34()f x3⎛⎝⎜⎜⎞⎠⎟⎟ddx()f x 12 ( )f x2⎛⎝⎜⎜⎞⎠⎟⎟ddx()f x 8 x 8 x ()f x 4 x2⎛⎝⎜⎜⎞⎠⎟⎟ddx()f x 4 x ()f x2 + − − − − + := 4 x2()f x⎛⎝⎜⎜⎞⎠⎟⎟ddx()f x + 0 = Here Maple uses ddx()f x to denote f '( x ), the derivative of f( x ). We now solve for ddx()fx : with a deferred command> Diff(f(x),x)=solve(deq,diff(f(x),x)); = ddx()f x −x () − − + x222()f x ()f x2 − − + ()f x33()f x2x2x2()f x The single Maple command "implicitdiff" combines the above computations but uses "y" rather than "f(x)". > y:='y'; := yy (Previously,we had replaced y by f(x); the command "y:='y' "undoes this.) > eq; = + − − − + x4y44 y34 x24 x2y 2 x2y20> dydx:=implicitdiff(eq,y,x); := dydx−x () − − + x222yy2 − − + y33 y2x2x2y The point P has coordinates x = + 2 32 and y = + 2 332 , so in order to find m, the slope of the tangent at P, we must replace y by + 2 332 and x by + 2 32in the formula for dy/dx.> subs(y=(2*sqrt(3)+3)/2,x=(2+sqrt(3))/2,dydx);−⎛⎝⎜⎜⎞⎠⎟⎟ + 132⎛⎝⎜⎜⎜⎞⎠⎟⎟⎟ − − + ⎛⎝⎜⎜⎞⎠⎟⎟ + 1322523⎛⎝⎜⎜⎞⎠⎟⎟ + 3322 − − + ⎛⎝⎜⎜⎞⎠⎟⎟ + 33233⎛⎝⎜⎜⎞⎠⎟⎟ + 3322⎛⎝⎜⎜⎞⎠⎟⎟ + 1322⎛⎝⎜⎜⎞⎠⎟⎟ + 1322⎛⎝⎜⎜⎞⎠⎟⎟ + 332 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 32 ) + + 2 332 . We can plot the cardioid and its tangent on the same set of axes as follows:> 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. >
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