Second-Order Polynomial Approximation

Transcription

1 > with(plots): Warning, the name changecoords has been redefined M:8 Spring 7 J. Simon Second-Order Polynomial Approimation In Calc I or II, you studied Taylor polynomial approimations. If y=f() is some function, and the function f is smooth, we can write polynomials P() that approimate f near a given point =a with high accuracy. The idea is to find a polynomial P() such that P(a) = f(a) There is a -degree polynomial, namely P() = f(a) for all, that does this. P'(a) = f'(a) There is a st degree polynomial that has P(a)=f(a) and P'(a)=f'(a), namely P() = f(a) + f'(a)(-a) P''(a) = f''(a) There is a nd degree polynomial that has P(a)=f(a) AND P'(a)=f'(a) AND P''(a) = f''(a), namely etc. P() = f(a) + f'(a)(-a) + (/) f''(a)(-a)^. For any given n (assuming f is smooth enough around =a that all the needed derivatives eist), we can write a polynomial P_n() = f(a) + f'(a)(-a) + (/) f''(a)(-a)^ + (/3!) f'''(a) (-a)^ (/n!) f^(n)(a) (-a)^n where f^(n)(a) denotes the nth derivative of f at point =a. There are (at least) two important uses for Taylor polynomial approimations:. In computers and calculators, we need to have algorithms for computing values of functions such as ep(), sin(), log() --- how can a machine that only knows how to add, subtract, multiply, and divide numbers ever know how to calculate sin()?? The actual algorithms may be fancier, but you could write a 3rd degree polynomial approimation of the sine function, using your knowledge of sine and cosine at =, and get an estimate for sin(). > p8:=series(sin(), =, 8); p8 := O 8 ( ) > P8:=convert(p8,polynom); > subs(=,p8);evalf(%); P8 := > sin();evalf(%);

2 sin( ) In theoretical analysis of ma/min problems, you developed the "second derivative test" for deciding (in many situations) whether a critical point of a function f represents a local maimum vs. local minimum vs. saddle point. The key here is that the second-order Taylor polynomial approimation of f() near =critical point "a" will capture BOTH phenomena of having "a" be a critical point and will have the same ma vs min behavior as f. Eamples: > f:=cos(); > f:=subs(=,f); > df:=diff(f,); f := cos( ) f := cos( ) df := -sin( ) > df:=subs(=,df);df:=value(df); df := -sin( ) df := > df:=diff(df,); df := -cos( ) > df:=subs(=,df);df:=value(df); df := -cos( ) df := - > p:=f+df* + df*^/; p := - > plot({f, p}, =-4..4, thickness=, view=[-4..4, ], scaling=constrained);

3 (Remark: In the case of a saddle-point, the second derivative will be, and the quadratic Taylor polynomial will be the same as the st degree polynomial, just a line tangent to the graph of f. ########################################## There is a similar theory of Taylor polynomials for functions of several variables. Consider the following function, near the point (,y)=(,) > f:=cos()*ln(+y); f := cos( ) ln( + y) > plot3d(f, =-..3, y=..3, aes=boed, orientation=[54,75]);

4 y > :=;y:=; > f:=subs(=, y=y,f); > df:=diff(f,); > df:=subs(=,y=y,df); > dfy:=diff(f,y); := y := f := cos( ) ln( 3) df := -sin( ) ln( + y) df := -sin( ) ln( 3) dfy := cos( ) + y > dfy:=subs(=,y=y,dfy);

5 dfy := 3 cos( ) > df:=diff(df,); > df:=subs(=,y=y,df); > dfy:=diff(dfy,y); > dfy:=subs(=,y=y,dfy); df := -cos( ) ln( + y) df := -cos( ) ln( 3) dfy := - cos( ) ( + y) dfy := - 9 cos( ) > dfy:=diff(df,y); > dfy:=diff(dfy,); dfy := - sin( ) + y dfy := - sin( ) + y Notice these last two are equal, which we epect; but it is nice to have the reassuring calculation. > dfy:=subs(=,y=y,dfy); dfy := - 3 sin( ) > P:=f+df*(-)+dfy*(y-)+(/)*df*(-)^ + dfy*(-)*(y-)+(/)*dfy*(y-)^;p:=evalf(p);p:=evalf(epand(p)) ; P := cos( ) ln( 3) - sin( ) ln( 3) ( - ) + 3 cos( ) ( y - ) - cos( ) ln( 3) ( - ) - 3 sin( ) ( - ) ( y - ) - 8 cos( ) ( y - ) P := y ( -. ) ( -. ) ( y -. ) ( y -. ) P := y y y > plot3d(p, =-..3, y=..3, aes=boed, orientation=[54,75]);

6 y The function P is "just" a quadratic in and y, so the graph should be recognizable as a paraboloid or a saddle surface. To see the overall shape better, let's re-plot the graph with a larger range of values of and y. > plot3d(p, =-5..5, y=-5..5, aes=boed, orientation=[54,75], color=proc(,y) abs() end proc);

7 y So we see the graph is a saddle surface. Now let's see how the graph of P fits with the graph of f. > plot3d([f, P], =-..3, y=..3, aes=boed, orientation=[54,75], color=[blue, pink]);

8 y It is hard to see how the two surfaces relate near the point (,). So let's redraw the graphs, with ranges of and y limited to near (,). > plot3d([f, P], =.5...5, y=.5...5, aes=boed, orientation=[54,75], color=[blue, pink]);

9 y Now we can see that near the point (,), the graph z=f(,y) and the graph z=p(,y) are nearly identical. ########################################## ########################################## Here is a second eample. Consider the following function, near the point (,y)=(,) > f:=sin()+cos(+y); f := sin( ) + cos( + y)

10 > plot3d(f, =..5, y=..7, aes=boed, orientation=[8,83]); y We will select the point (, y) to be where we think the graph has a local maimum (more on this later in the chapter...) > df:=diff(f,); > dfy:=diff(f,y); df := cos( ) - sin( + y) dfy := -sin( + y) We want to find where both partial derivatives are zero; from the picture, we think there is such a "critical point" near =, y=5. The command "fsolve" tells Maple to numerically solve one or more equations, and we can tell it roughly were to look (in case there are other solutions far away from where we want to look). > fsolve({df=,dfy=}, {=,y=5}); { = , y = } Actually, we can solve these equations eactly. The second equation says sin(+y)=. But then the first equation says cos()=. For near, this says =Pi/. So sin(+y)= says "Pi/ + y = some whole multiple of Pi, i.e. y=some odd multiple of Pi/. For y near 5, this says y=3*pi/.

11 > :=Pi/;evalf(); := π > y:=3*pi/; evalf(y); y := 3 π > f:=subs(=, y=y,f); æ f := sinç è π ö ø + cos( π) > df:=diff(f,); df := cos( ) - sin( + y) > df:=subs(=,y=y,df);df:=value(df); æ df := cosç π ö - sin( π) è ø df := > dfy:=diff(f,y); dfy := -sin( + y) > dfy:=subs(=,y=y,dfy);dfy:=value(dfy); dfy := -sin( π) dfy := > df:=diff(df,); df := -sin( ) - cos( + y) > df:=subs(=,y=y,df);df:=evalf(df); æ df := -sinç π ö - cos( π) è ø df := -. > dfy:=diff(dfy,y); dfy := -cos( + y) > dfy:=subs(=,y=y,dfy);dfy:=value(dfy); dfy := -cos( π) > dfy:=diff(df,y); dfy := -

12 dfy := -cos( + y) > dfy:=diff(dfy,); dfy := -cos( + y) Notice these last two are equal, which we epect; but it is nice to have the reassuring calculation. > dfy:=subs(=,y=y,dfy);dfy:=value(dfy); dfy := -cos( π) dfy := - > P:=f+df*(-)+dfy*(y-y)+(/)*df*(-)^ + dfy*(-)*(y-y)+(/)*dfy*(y-y)^;p:=value(epand(p));p:=evalf( P); P := -. æ ç è - π ö æ - ç - ø π ö è ø æ ç è y - 3 π ö ø - æ ç è y - 3 π ö ø P := π -.5 π - y + π y - y P := y y -.5 y > plot3d(p, =..5, y=..7, aes=boed, orientation=[54,75]);

13 y The function P is "just" a quadratic in and y, so the graph should be recognizable as a paraboloid or a saddle surface. To see the overall shape better, let's re-plot the graph with a larger range of values of and y. > plot3d(p, =-5..9, y=-3.., aes=boed, orientation=[63,75], color=pink);

14 y - -4 The graph is a paraboloid, concave down. Now let's see how the two graphs "snuggle" together near the point (, y). > plot3d([f,p], =-3..+3, y=y-3..y+3, color=[blue, pink], view=[..5,..7, ], aes=boed, orientation=[,68]);

15 y > plot3d([f,p], =-..+, y=y-..y+, color=[blue, pink],orientation=[68,79] );

16 ######## End of handout #########