Tangent Planes and Linear Approximations

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1 February 21, 2007

2 Tangent Planes

3 Tangent Planes Let S be a surface with equation z = f (x, y).

4 Tangent Planes Let S be a surface with equation z = f (x, y). Let P(x 0, y 0, z 0 ) be a point on S.

5 Tangent Planes Let S be a surface with equation z = f (x, y). Let P(x 0, y 0, z 0 ) be a point on S. Let C 1 and C 2 be the curves obtained by intersecting the vertical planes y = y 0 and x = x 0 with the surface S.

6 Tangent Planes Let S be a surface with equation z = f (x, y). Let P(x 0, y 0, z 0 ) be a point on S. Let C 1 and C 2 be the curves obtained by intersecting the vertical planes y = y 0 and x = x 0 with the surface S. Let T 1 and T 2 be the tangent lines to the curves C 1 and C 2.

7 Tangent Planes Let S be a surface with equation z = f (x, y). Let P(x 0, y 0, z 0 ) be a point on S. Let C 1 and C 2 be the curves obtained by intersecting the vertical planes y = y 0 and x = x 0 with the surface S. Let T 1 and T 2 be the tangent lines to the curves C 1 and C 2. The tangent plane to the surface S at the point P is dened to be the plane that contains both tangent lines T 1 and T 2.

8 Equations of the tangent plane

9 Equations of the tangent plane Suppose f has a continuous partial derivatives.

10 Equations of the tangent plane Suppose f has a continuous partial derivatives. An equation of the tangent plane to the surface z = f (x, y) at the point P(x 0, y 0, z 0 ) is z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 )

11 Examples Example Find the tangent plane to the elliptic paraboloid z = 2x 2 + y 2 at the point (1, 1, 3) y

12 Linear Approximations y x

13 Linear Approximations The linear function whose graph is this tangent plane L(x, y) = f (a, b) + f x (a, b)(x a) + f y (a, b)(y b) is called the linearization of f at (a, b) and the approximation f (x, y) f (a, b) + f x (a, b)(x a) + f y (a, b)(y b) is called the linear approximation or the tangent plane approximation of f at (a, b).

14 Examples Examples

15 Examples Examples Find the linearization of the function f (x, y) = xy at the point (4, 16).

16 Examples Examples Find the linearization of the function f (x, y) = xy at the point (4, 16). Find the linearization of the function f (x, y) = 1 + y + x cos y at P 0 (0, 0).

17 The increment of z

18 The increment of z Recall that for a function of one variable, y = f (x), if x changes from a to a + x, we dened the increment of y as y = f (a + x) f (a).

19 The increment of z Recall that for a function of one variable, y = f (x), if x changes from a to a + x, we dened the increment of y as y = f (a + x) f (a). If f is dierentiable at a, then where ɛ 0 as x 0. y = f (a) x + ɛ x,

20 The increment of z

21 The increment of z If z = f (x, y) and x changes from (a, b) to (a + x, b + y), then the increment of z is z = f (a + x, b + y) f (a, b)

22 The increment of z If z = f (x, y) and x changes from (a, b) to (a + x, b + y), then the increment of z is z = f (a + x, b + y) f (a, b) If z = f (x, y), then f expressed in the form is dierentiable at (a, b) if z can be z = f x (a, b) x + f y (a, b) y + ɛ 1 x + ɛ 2 y, where ɛ 1 and ɛ 2 0 as ( x, y) (0, 0).

23 Fact If the partial derivatives f x and f y exist near (a, b) and are continuous at (a, b), then f is dierentiable at (a, b).

24 Example Example Show that f (x, y) = xe xy linearization there. is dierentiable at (1, 0) and nd its

25 Dierentials

26 Dierentials For a dierentiable function z = f (x, y) we dene the dierential dz, also called the total dierential, is dened by dz = f x (x, y)dx + f y (x, y)dy = z x dz + z y dy, where the dierentials dx and dy are independent variables.

27 Dierentials For a dierentiable function z = f (x, y) we dene the dierential dz, also called the total dierential, is dened by dz = f x (x, y)dx + f y (x, y)dy = z x dz + z y dy, where the dierentials dx and dy are independent variables. If dx = x = x a and dy = y = y b the the dierential of z is dz = f x (a, b)(x a) + f y (a, b)(y b)

28 Examples Examples

29 Examples Examples If f (x, y) = x 2 + 3xy y 2, nd the dierential dz.

30 Examples Examples If f (x, y) = x 2 + 3xy y 2, nd the dierential dz. If x changes from 2 to 2.05 and y changes 3 to 2.96, compare the values of z and dz.

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