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I. Differentiation – 3. Tangentplan 56 til koordinatkurverne på grafen Γf . x ↦→ (x,y0,f(x,y0)), y ↦→ (x0,y,f(x0,y)) 3.7. Ligning for tangentplan ☞ [S] 11.4 Tangent planes and linear approx. 2 Sætning Antag at f har kontinuerte partielle afledede fx,fy i en lille cirkelskive om (x0,y0). Tangentplanen for grafen i et punkt (x0,y0,z0), z0 = f(x0,y0) har ligning Bevis Indsættes z − z0 = fx(x0,y0)(x − x0) + fy(x0,y0)(y − y0) (x,y,z) = (x0,y0,z0) + (1,0,fx(x0,y0)) = (x0 + 1,y0,z0 + fx(x0,y0)) er ligningen opfyldt. Ligeså for den anden tangentvektor. 3.8. Find tangentplan ☞ [S] 11.4 Tangent planes and linear . . . Eksempel 1 Find ligningen for tangentplanen til z = 2x 2 + y 2 i punktet (1,1,3). Løsning
I. Differentiation – 3. Tangentplan 57 De partielle afledede er I punktet (1,1,3) er tangentplanen givet ved zx = 4x,zy = 2y z(1,1) = 3, zx(1,1) = 4, zy(1,1) = 2 z − 3 = 4(x − 1) + 2(y − 1) 3.9. Tangentplan ☞ [S] 11.4 Tangent planes and linear approximations Figur - Eksempel 1 x z Tangentplan i (1,1,3) y
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CALCULUS "SLIDES" TIL CALCULUS 1 +
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3. Generelle områder 201 4. Koordi
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Litteratur • Arnold, Ordinary dif
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