Lesson: ____ Section: 3.7 Implicit Implicit Functions Functions Review: If y is a function of x and k is a constant, find: = ( ) y is an explicitly defined function of x. = The output is + = y is an implicit function of x This relation cant be written as an explicit function. Wed need to write it as two functions, one for the top half of the circle and another for the bottom half.
Consider the circle as a whole. It is still a curve with an equation and this curve has a tangent line at each point. so the idea of a derivative still makes sense The slope of the tan line can be found by Differentiating both sides of the equation with respect to x. + = ( + )= ()Chain
Rule! ( )+ ( )= 2x 2y 2y Note that the derivative depends on both x and y rather than x. points on the This worksjust for all circle excepts where a vertical
tangent line exists such as (2.0) or (2,0) Discuss the polarity of the slope of this circle in the second quadrant. Does this agree with the derivative? How about QIII? Ex.1 Make a table of x and approximate y values for the equation near Since we cant solve for y in (7,2). terms of x, lets find an = equation for the tangent line ( )= ( ) and use that tangent line to
approximate other points near (7,2). Chain Rule ( ) ( )= & Product =0 Rule =0 When x=7 and y=2
=y =y = ( ) = = ( () ) ( ) =
= . . Tangent line at (7,2) Notice that in a small neighborhood around (7,2), the tangent line gives a good approximation of the curve. Find the coordinates of some other points near (7,2) x y 6.8
1.92 6.9 1.96 7 2 7.1 2.04 7.2 2.08
en g n Ta ne i l t Finding other points on the curve by plugging into the original equation is a long and cumbersome process. 4y' 2x 8 cos(7y) 7y' As this cant be solved for y, this is, once again, an implicit differentiation
question. implicit differentiation: 1 Differentiate both sides wrt x. 2 Solve for . Check out the graph! We can now use dy/dx to find the slope of the curve at any point on this crazy thing. Find the equations of the lines tangent and normal to the curve at .
Note product rule. Find the equations of the lines tangent and normal to the curve at tangent: 4 =2+ ( +1) 5 . normal: 5 =2 ( +1) 4
Higher Order Derivatives Find if . Substitute back into the equation. Another crazy looking graph just for fun
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