นิพจน์ 21: vector left parenthesis, "P" Subscript, 0 , Baseline , "P" Subscript, 0 , Baseline plus left parenthesis, "a" , "b" , 0 , right parenthesis , right parenthesisvectorP0,P0+a,b,0
21
นิพจน์ 22:
22
How fast is f changing in this direction at P_0, ie the directional derivative?
23
นิพจน์ 24: "a" "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "b" "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesisafxx0,y0+bfyx0,y0
equals=
negative 3.8 6 4 6−3.8646
24
We can understand this as a change up or down.
25
นิพจน์ 26: vector left parenthesis, "P" Subscript, 0 , Baseline plus left parenthesis, "a" , "b" , 0 , right parenthesis , "P" Subscript, 0 , Baseline plus left parenthesis, "a" , "b" , "a" "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "b" "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesisvectorP0+a,b,0,P0+a,b,afxx0,y0+bfyx0,y0
26
นิพจน์ 27: vector left parenthesis, "P" Subscript, 0 , Baseline , "P" Subscript, 0 , Baseline plus left parenthesis, "a" , "b" , "a" "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "b" "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis , right parenthesis , right parenthesisvectorP0,P0+a,b,afxx0,y0+bfyx0,y0
27
Notice that the direction and the directional derivative together give the tangent line
28
Cross section in some other direction
29
นิพจน์ 30: left parenthesis, "x" Subscript, 0 , Baseline plus "a" "t" , "y" Subscript, 0 , Baseline plus "b" "t" , "f" left parenthesis, "x" Subscript, 0 , Baseline plus "a" "t" , "y" Subscript, 0 , Baseline plus "b" "t" , right parenthesis , right parenthesisx0+at,y0+bt,fx0+at,y0+bt
โดเมน t ค่าต่ำสุด: negative 3−3
less than or equal to "t" less than or equal to≤t≤
โดเมน t ค่าสูงสุด: 33
30
Equation of tangent line in other direction
31
นิพจน์ 32: "P" Subscript, 0 , Baseline plus "t" left parenthesis, "a" , "b" , "a" "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "b" "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis , right parenthesisP0+ta,b,afxx0,y0+bfyx0,y0
โดเมน t ค่าต่ำสุด: negative 3−3
less than or equal to "t" less than or equal to≤t≤
โดเมน t ค่าสูงสุด: 33
32
Tangent plane
33
นิพจน์ 34: "z" equals "f" left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis plus "f" Subscript, "x" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis left parenthesis, "x" minus "x" Subscript, 0 , Baseline , right parenthesis plus "f" Subscript, "y" , Baseline left parenthesis, "x" Subscript, 0 , Baseline , "y" Subscript, 0 , Baseline , right parenthesis left parenthesis, "y" minus "y" Subscript, 0 , Baseline , right parenthesisz=fx0,y0+fxx0,y0x−x0+fyx0,y0y−y0
34
Niveaulinien
ซ่อนโฟลเดอร์นี้จากนักเรียน
35
นิพจน์ 36: "z" equals "f" left parenthesis, "x" , "y" , right parenthesis left brace, "z" equals "z" Subscript, 0 , Baseline , right bracez=fx,yz=z0