Continuity and Tangent Lines for functions of two variables

Transcription

1 Continuity and Tangent Lines for functions of two variables James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 4, 2014

2 Outline 1 Continuity 2

3 Continuity Let s recall the ideas of continuity for a function of one variable.consider these three versions of a function f defined on [0, 2]. f (x) = x 2, if 0 x < 1 10, if x = (x 1) 2 if 1 < x 2.

4 Continuity Let s recall the ideas of continuity for a function of one variable.consider these three versions of a function f defined on [0, 2]. f (x) = x 2, if 0 x < 1 10, if x = (x 1) 2 if 1 < x 2. This function is not continuous at x = 1 because although the lim x 1 f (x) exists and equals 1 ( lim x 1 f (x) = 1 and lim x 1 + f (x) = 1), the value of f (1) is 10 which does not match the limit. Hence, we know f here has a removeable discontinuity at x = 1. Note continuity failed because the limit existed but the value of the function did not match it.

5 Continuity The second version of f is given below. f (x) = { x 2, if 0 x 1 (x 1) 2 if 1 < x 2.

6 Continuity The second version of f is given below. f (x) = { x 2, if 0 x 1 (x 1) 2 if 1 < x 2. In this case, the lim x 1 = 1 and f (1) = 1, so f is continuous from the left. However, lim x 1 + = 0 which does not match f (1) and so f is not continuous from the right. Also, since the right and left hand limits do not match at x = 1, we know lim x 1 does not exist. Here, the function fails to be continuous because the limit does not exist.

7 Continuity The final example is below: f (x) = { x 2, if 0 x < 1 x + (x 1) 2 if 1 x 2.

8 Continuity The final example is below: f (x) = { x 2, if 0 x < 1 x + (x 1) 2 if 1 x 2. Here, the limit and the function value at 1 both match and so f is continuous at x = 1.

9 Continuity The final example is below: f (x) = { x 2, if 0 x < 1 x + (x 1) 2 if 1 x 2. Here, the limit and the function value at 1 both match and so f is continuous at x = 1. To extend these ideas to two dimensions, the first thing we need to do is to look at the meaning of the limiting process. What does lim (x,y) (x0,y 0 ) mean?

10 Continuity The final example is below: f (x) = { x 2, if 0 x < 1 x + (x 1) 2 if 1 x 2. Here, the limit and the function value at 1 both match and so f is continuous at x = 1. To extend these ideas to two dimensions, the first thing we need to do is to look at the meaning of the limiting process. What does lim (x,y) (x0,y 0 ) mean? Clearly, in one dimension we can approach a point x 0 from x in two ways: from the left or from the right or jump around between left and right. Now, it is apparent that we can approach a given point (x 0, y 0 ) in an infinite number of ways. Draw a point on a piece of paper and convince yourself that there are many ways you can draw a curve from another point (x, y) so that the curve ends up at (x 0, y 0 )!

11 Continuity We still want to define continuity in the same way; i.e. f is continuous at the point (x 0, y 0 ) if lim (x,y) (x0,y 0 ) f (x, y) = f (x 0, y 0 ). If you look at the graphs of the surface z = x 2 + y 2 we have done previously, we clearly see that we have this kind of behavior. There are no jumps, tears or gaps in the surface we have drawn. Let s make this formal. Definition Continuity: Let z = f (x, y) be a function of the two independent variables x and y defined on some domain. At each pair (x, y) where f is defined in a circle of some finite radius r, B r (x 0, y 0 ) = {(x, y) (x x 0 ) 2 + (y y 0 ) 2 < r}, If lim (x,y) (x0,y 0 ) f (x, y) exists and matches f (x 0, y 0 ), we say f is continuous at (x 0, y 0 ).

12 Continuity Here is an example of a function which is not continuous at the point (0, 0). Let { 2 x, if (x, y) (0, 0) f (x, y) = x 2 +y 2 0, if (x, y) = (0, 0).

13 Continuity Here is an example of a function which is not continuous at the point (0, 0). Let { 2 x, if (x, y) (0, 0) f (x, y) = x 2 +y 2 0, if (x, y) = (0, 0). If we show the limit as we approach (0, 0) does not exist, then we will know f is not continuous at (0, 0). If this limit exists, we should get the same value for the limit no matter what path we take to reach (0, 0).

14 Continuity Here is an example of a function which is not continuous at the point (0, 0). Let { 2 x, if (x, y) (0, 0) f (x, y) = x 2 +y 2 0, if (x, y) = (0, 0). If we show the limit as we approach (0, 0) does not exist, then we will know f is not continuous at (0, 0). If this limit exists, we should get the same value for the limit no matter what path we take to reach (0, 0). Let the first path be given by x(t) = t and y(t) = 2t. Then, as t 0, (x(t), y(t)) (0, 0) as desired. Plugging in to f, we find for t 0, f (t, 2t) = 2 t / t 2 + 4t 2 = 2/ 5 and hence the limit along this path is this constant value 2/ 5. On the other hand, along the path x(t) = t and y(t) = 3t, for t 0, we have f (t, 3t) = 2/3 which is not the same. Since the limiting value differs on two paths, the limit can t exist. Hence, f is not continuous at (0, 0).

15 Let s go back to our simple surface example and look at the traces again. In this figure, we show the traces for the base point x 0 = 0.5 and y 0 = 0.5. We have also drawn vertical lines down from the traces to the x y plane to further emphasize the placement of the traces on the surface. The surface itself is not shown as it is somewhat distracting and makes the illustration too busy.

16 You can generate this type of graph yourself with the function DrawFullTraces as follows: D r a w F u l l T r a c e s ( f, 0. 5, 2, 0. 5, 2, 0. 5, 0. 5 ) ;

17 You can generate this type of graph yourself with the function DrawFullTraces as follows: D r a w F u l l T r a c e s ( f, 0. 5, 2, 0. 5, 2, 0. 5, 0. 5 ) ; Note, that each trace has a well-defined tangent line and derivative at the points x 0 and y 0. We have d dx f (x, y 0) = d dx (x 2 + y 2 0 ) = 2x as the value y 0 in this expression is a constant and hence its derivative with respect to x is zero. We denote this new derivative as f x which we read as the partial derivative of f with respect to x. It s value as the point (x 0, y 0 ) is 2x 0 here. For any value of (x, y), we would have f x = 2x.

18 We also have d dy f (x 0, y) = d dy (x y 2 ) = 2y We then denote this new derivative as f y which we read as the partial derivative of f with respect to y. It s value as the point (x 0, y 0 ) is then 2y 0 here. For any value of (x, y), we would have f y = 2y.

19 We also have d dy f (x 0, y) = d dy (x y 2 ) = 2y We then denote this new derivative as f y which we read as the partial derivative of f with respect to y. It s value as the point (x 0, y 0 ) is then 2y 0 here. For any value of (x, y), we would have f y = 2y. The tangent lines for these two traces are then T (x, y 0 ) = f (x 0, y 0 ) + d dx f (x, y 0) (x x 0 ) x0 = (x0 2 + y0 2 ) + 2x 0 (x x 0 ) T (x 0, y) = f (x 0, y 0 ) + d dy f (x 0, y) (y y 0 ) y0 = (x y 2 0 ) + 2y 0 (y y 0 ).

20 We also have d dy f (x 0, y) = d dy (x y 2 ) = 2y We then denote this new derivative as f y which we read as the partial derivative of f with respect to y. It s value as the point (x 0, y 0 ) is then 2y 0 here. For any value of (x, y), we would have f y = 2y. The tangent lines for these two traces are then T (x, y 0 ) = f (x 0, y 0 ) + d dx f (x, y 0) (x x 0 ) x0 = (x0 2 + y0 2 ) + 2x 0 (x x 0 ) T (x 0, y) = f (x 0, y 0 ) + d dy f (x 0, y) (y y 0 ) y0 = (x y 2 0 ) + 2y 0 (y y 0 ).

21 We can also write these tangent line equations like this using our new notation for partial derivatives.

22 We can also write these tangent line equations like this using our new notation for partial derivatives. T (x, y 0 ) = f (x 0, y 0 ) + f x (x 0, y 0 ) (x x 0 ) = (x0 2 + y0 2 ) + 2x 0 (x x 0 ) T (x 0, y) = f (x 0, y 0 ) + f y (x 0, y 0 ) (y y 0 ) = (x y 2 0 ) + 2y 0 (y y 0 ).

23 We can draw these tangent lines in 3D. To draw T (x, y 0 ), we fix the y value to be y 0 and then we draw the usual tangent line in the x z plane. This is a copy of the x z plane translated over to the value y 0 ; i.e. it is parallel to the x z plane we see at the value y = 0.

24 We can draw these tangent lines in 3D. To draw T (x, y 0 ), we fix the y value to be y 0 and then we draw the usual tangent line in the x z plane. This is a copy of the x z plane translated over to the value y 0 ; i.e. it is parallel to the x z plane we see at the value y = 0. We can do the same thing for the tangent line T (x, y 0 ); we fix the x value to be x 0 and then draw the tangent line in the copy of the y z plane translated to the value x 0.

25 We can draw these tangent lines in 3D. To draw T (x, y 0 ), we fix the y value to be y 0 and then we draw the usual tangent line in the x z plane. This is a copy of the x z plane translated over to the value y 0 ; i.e. it is parallel to the x z plane we see at the value y = 0. We can do the same thing for the tangent line T (x, y 0 ); we fix the x value to be x 0 and then draw the tangent line in the copy of the y z plane translated to the value x 0. We show this in the next figure. Note the T (x, y 0 ) and the T (x 0, y) lines are determined by vectors as shown below. A = 1 0 d dx f (x, y 0) x0 = 1 0 2x and B = d dy f (x 0, y) y0 = 0 1 2y 0

26 Note that if we connect the lines determined by the vectors A and B, we determine a flat sheet which you can interpret as a piece of paper laid on top of these two lines. Of course, we can only envision a small finite subset of this sheet of paper as you can see in the figure below.

27 Note that if we connect the lines determined by the vectors A and B, we determine a flat sheet which you can interpret as a piece of paper laid on top of these two lines. Of course, we can only envision a small finite subset of this sheet of paper as you can see in the figure below. Imagine that the sheet extends infinitely in all directions! The sheet of paper we are plotting is called the tangent plane to our surface at the point (x 0, y 0 ). We will talk about this more formally later.

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29 To draw this picture with the tangent lines, the traces and the tangent plane, we use the function DrawTangentLines which has arguments (f,fx,fy,delx,nx,dely,ny,r,x0,y0).

30 To draw this picture with the tangent lines, the traces and the tangent plane, we use the function DrawTangentLines which has arguments (f,fx,fy,delx,nx,dely,ny,r,x0,y0). There are three new arguments: fx which is f / x, fy which is f / y and r which is the size of the tangent plane that is plotted.

31 To draw this picture with the tangent lines, the traces and the tangent plane, we use the function DrawTangentLines which has arguments (f,fx,fy,delx,nx,dely,ny,r,x0,y0). There are three new arguments: fx which is f / x, fy which is f / y and r which is the size of the tangent plane that is plotted. For the picture shown next, we ve removed the tangent plane because the plot was getting pretty busy. We did this by commenting out the line that plots the tangent plane. It is easy for you to go into the code and add it back in if you want to play around. The MatLab command line is f x x, y ) 2 x ; f y x, y ) 2 y ; % DrawTangentLines ( f, fx, fy, 0. 5, 2, 0. 5, 2,. 3, 0. 5, 0. 5 ) ;

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33 f you want to see the tangent plane as well as the tangent lines, all you have to do is look at the following lines in DrawTangentLines.m. % s e t up a new l o c a l mesh g r i d n e a r ( x0, y0 ) [U, V] = m e s h g r i d ( u, v ) % s e t up t h e t a n g e n t p l a n e a t ( x0, y0 ) W = f ( x0, y0 ) + f x ( x0, y0 ) (U x0 ) + f y ( x0, y0 ) (V y0 ) % p l o t the t a n g e n t p l a n e s u r f (U, V,W, EdgeColor, b l u e ) ;

34 f you want to see the tangent plane as well as the tangent lines, all you have to do is look at the following lines in DrawTangentLines.m. % s e t up a new l o c a l mesh g r i d n e a r ( x0, y0 ) [U, V] = m e s h g r i d ( u, v ) % s e t up t h e t a n g e n t p l a n e a t ( x0, y0 ) W = f ( x0, y0 ) + f x ( x0, y0 ) (U x0 ) + f y ( x0, y0 ) (V y0 ) % p l o t the t a n g e n t p l a n e s u r f (U, V,W, EdgeColor, b l u e ) ; These lines setup the tangent plane and the tangent plane is turned off is there is a percent % if front of surf(u,v,w, EdgeColor, blue );. We edited the file to take the % out so we can see the tangent plane. We then see the plane in the next figure.

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36 The ideas we have been discussing can be made more general. When we take the derivative with respect to one variable while holding the other variable constant (as we do when we find the normal derivative along a trace ), we say we are taking a partial derivative of f.

37 The ideas we have been discussing can be made more general. When we take the derivative with respect to one variable while holding the other variable constant (as we do when we find the normal derivative along a trace ), we say we are taking a partial derivative of f. Here there are two flavors: the partial derivative with respect to x and the partial derivative with respect to y.

38 The ideas we have been discussing can be made more general. When we take the derivative with respect to one variable while holding the other variable constant (as we do when we find the normal derivative along a trace ), we say we are taking a partial derivative of f. Here there are two flavors: the partial derivative with respect to x and the partial derivative with respect to y. We can now state some formal definitions and introduce the notations and symbols we use for these things. We define the process of partial differentiation carefully below.

39 Definition Let z = f (x, y) be a function of the two independent variables x and y defined on some domain. At each pair (x, y) where f is defined in a circle of some finite radius r, B r (x 0, y 0 ) = {(x, y) (x x 0 ) 2 + (y y 0 ) 2 < r}, it makes sense to try to find the limits f (x, y 0 ) f (x 0, y 0 ) lim x x 0,y=y 0 x x 0 f (x 0, y) f (x 0, y 0 ) lim x=x 0,y y 0 y y 0 If these limits exists, they are called the partial derivatives of f with respect to x and y at (x 0, y 0 ), respectively.

40 Comment For these partial derivatives, we use the symbols and f x (x 0, y 0 ), f y (x 0, y 0 ), f x (x 0, y 0 ), z x (x 0, y 0 ), f y (x 0, y 0 ), z y (x 0, y 0 ), z x (x 0, y 0 ) z y (x 0, y 0 )

41 Comment We often use another notation for partial derivatives. The function f of two variables x and y can be thought of as having two arguments or slots into which we place values. So another useful notation is to let the symbol D 1 f be f x and D 2 f be f y. We will be using this notation later when we talk about the chain rule. Comment It is easy to take partial derivatives. Just imagine the one variable held constant and take the derivative of the resulting function just like you did in your earlier calculus courses.

42 Example Let z = f (x, y) = x 2 + 4y 2 be a function of two variables. Find z z x and y. Solution Thinking of y as a constant, we take the derivative in the usual way with respect to x. This gives z x = 2x as the derivative of 4y 2 with respect to x is 0. So, we know f x = 2x. In a similar way, we find z y. We see z y = 8y as the derivative of x 2 with respect to y is 0. So f y = 8y.

43 Example Let z = f (x, y) = 4x 2 y 3. Find z x and z y. Solution Thinking of y as a constant, take the derivative in the usual way with respect to x: This gives z x = 8xy 3 as the term 4y 3 is considered a constant here. So f x = 8xy 3. Similarly, z y = 12x 2 y 2 as the term 4x 2 is considered a constant here. So f y = 12x 2 y 2.

44 Homework 67 These are for you: for each of these functions, find f x and f y. First, functions with no cross terms f (x, y) = x 2 + 3y f (x, y) = 4x 2 + 5y f (x, y) = 3x + 2y 8. Next, functions with cross terms f (x, y) = x 2 y f (x, y) = 2x 3 y 2 + 5x f (x, y) = 3x y f (x, y) = x 2 y 5.