The directional derivative of f x, y in the direction of at x, y u. f x sa y sb f x y (, ) (, ) 0 0 y 0 0

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1 Review: 0, lim D f u u The directional derivative of f, in the direction of at, is denoted b D f, : u a, b must a unit vector u f sa sb f s 0 (, ) (, ) s f (, ) a f (, ) b D f, f u where f f, f u Fastest increase is f, in the direction of 0 0 Fastest decrease is f, in the direction of 0 0 f u f f u f f (, ) is orthogonal to the level (contour) curve f (, ) f (, ) If w f (,, z), then the directional derivative is: D f,, z f u where f f, f, f u z

2 Eample: Find the gradient vector of f,, z z z f ( z ) ( k i) (z ) (k + j) ( z ) ( z ) i + ( z) j+ ( ) k ( z )

3 Let S be a surface with equation F,, z k S is a level surface of a function F of three variables Let P,, z be a point on S. Let C be an curve that lies on the surface S and passes through the point P. Let C defined b r t t, t, z t An point t, t, z t on C is also on S.,, F t t z t k If,, and z are differentiable functions of t and F is also differentiable, then F d F d F dz 0 dt dt z dt Another wa to write this is: F F F d d dz,,,, 0 z dt dt dt t 0 Fr F r t Let t be the parameter value corresponding to P. 0 r The tangent vector r t,, z 0 0 lies in the tangent plane to the surface S at the point P Gradient vector in 3-space t F r t 0 F,, z is the normal 0 vector to the tangent plane to S at P

4 The equation of the plane with normal a, b, c containing the point,, z : a b c z z 0,, b F,, z c F,, z a F z F,, z is the normal vector to the tangent plane to S at P z When the plane is the tangent plane to the surface F,, z k at the point,, z : F (,, z ) F (,, z ) F (,, z ) z z z 0 Eample: 3 Compute the tangent plane to the surface 4z z 0 at the point (1,, 1) F 4z F 1,, F z 1,,1 F 1z F 1 1 z 1,,1 1 z F z z 6 0 5z 3 0 z

5 14.5 Tangent Planes and Differentials

6 Recall: If a surface in 3-space is described as a level surface, i.e. F,, z k, then the tangent plane at,, z is given b F (,, z ) F (,, z ) F (,, z ) z z z 0 Often, a surface is given as the graph of a function z f (, ). What is a good formula for the tangent plane? Let F f (, ) z Then the surface is also describe b F 0, i.e., it is a level surface of F Then F f, F f and F 1 The z So f f z z 0 tangent plane to the surface z f, at the point P,, z is z z f (, ) f (, ) 0

7 Eample: 4 Find the equation of the tangent plane to z e at 3,0,. 4, f e f 1 e 4 f 3,0 1 3 e f 4e 4 e 4 4 e e 4 f 3,0 e 40 3 e ,, z z f f 0 1 z z 3 4 4z z 5 0

8 The tangent plane approimates the surface: z The elliptic paraboloid appears to coincide with its tangent plane as we zoom in toward (1, 1, 3). The tangent plane and the elliptic paraboloid become virtuall Indistinguishable the closer we get to (1, 1, 3). locall the surface looks linear The tangent plane is the linearization of the function at the point of tangenc.

9 The linearization of f at a, b is L, f a, b f a, b a f a, b b The approimation f, f a, b f a, b a f a, b b This can be used to approimate the function at "nearb" points. L, Recall, the tangent plane is z f ( a, b) f ( a, b) a f ( a, b) b is called the linear approimation of f at a, b or the tangent plane approimation of f at a, b.

10 Eample: L, f a, b f a, b a f, a b b Find the linear approimation of the function f, 0 7 at,1 and use it to approimate f 1.95,1.08. f, 0 7,,1 f a b f 1,1 f 14,1 f, 3 1 f f f f, Linear approimation: L(, ) , , , A calculator shows the difference is If (, ) is close to (,1) f 7 3

11 Recall the eample: f (, ), f (0,0) 0 f f is not continuous at (0,0). 0,0 0 f Tangent plane would be: 0,0 0 z z f (, ) f (, ) 0 0 or z z 0 The "tangent plane" would be the - plane Clearl a bad approimation near (0,0) (Recall though that for f ( ) the tangent line approimates f near a if f '( a) eists!) So we need a better definition of when f is differentiable at ( a, b) It is not sufficient to assume that f ( a, b) and f ( a, b) eists.

12 Definition : z f, is called differentiable at ( a, b) if z f (, ) f ( a, b) is approimatel f ( a, b) f ( a, b) for small, (and the approimation gets better the smaller, ) Theorem : If z f, and f and f are continuous near ( a, b), then f is differentiable at ( a, b) Corollar : If f is differentiable at ( a, b), then it is also continuous at ( a, b). Indeed, if z f (, ) f ( a, b) is approimatel f ( a, b) f ( a, b) for small,, then as (, ) ( a, b), 0 and 0 and hence z 0. Thus lim f (, ) f ( a, b ) (, ) ( a, b)

13 Recall the concept of a differential for a function of one variable d a f a h f a f '( a) lim lim d h0 h h0 hence f '( a) Differentials : d and d f '( a) d Recall tangent line is f ( a) f '( a)( a) hence along the tangent line the change in d is f ( a) f '( a)( a) is the change along the tangent line f '( a) f '( a) d

14 The linear approimation formula f, f a, b f a, b a f a, b b the increment of the increment of the d differential of the d differential of for independent variables, the increment is the same as the differential d What about the dependent variable z?,, z f f a b dz? d

15 The differential dz or the total differential is dz f, d f, d The linear approimation formula f, f a, b f a, b a f a, b b f, f a, b f a, b a f a, b b z dz but dz is much easier to calculate

16 Geometric interpretation of the differential dz and the incerement z dz goes along the tangent plane from ( a, b) to ( a, b )

17 Problem: 1 a A Watch out for units! A 1 da Ad Ad d d 1 da da m Relative error b z error actual 1 1 A 1 the maimum error in the measurement 5 1 d 0.00 m. d 0.00 similarl d 0.00 the maimum error in the calculated value of the area % 30 dz d d m Relative error % 13 z

18 Problem: z 4 The ellipsoid 1 bounds a region of volume V abc. a b c 3 At a moment when a 1, b, and c 3, a is changing at a rate of and b is changing at a rate of 3. At what rate must c change for the volume to be constant?