Unformatted text preview: n that of using the linear approximation. A nautral question to ask is, how small is the error? T HEOREM 16.1 [TAYLOR ’ S F ORMULA ]. Suppose is such that . Then for all , there exists a point on the line segment joining in some neighborhood of , say such that and where Recall the linear approximation to at is given by Then this theorem is saying that there exists a such that differs from the linear approximation by . This gives us a useful tool in bounding the error on the linear approximation. 27 WATERLOO SOS E XAM -AID: MATH237 E XAMPLE 31. If , show that with error S OLUTION. Let . Differentiating, Then Calculating the second partial derivatives, For , has continuous second partial derivatives, so we can apply Taylor’s theorem to get that there exists a point on the line segment joining and such that where Since the theorem does not give us the value of , we will simply ﬁnd an upper bound for this error. Since , triangle inequality gives so that By symmetry, 28 WATERLOO SOS E XAM -AID: MATH237 Finally, so that Then by triangle inequality, since as required. C OROLLARY 16.2. Suppose is such that in some closed neighborhood of , say Then there exists a positive constant such that for all , we have E XERCISE 32. If at most and . , show that the error in the linear approximation is 29 MATH 237 PRACTICE MIDTERM 1
NIALL W. MACGILLIVRAY 1. Consider the function f , where f (x, y ) = ln (2xy − y 2 ). (a) Write down the domain and range of f . [2marks] (b) Sketch the level curve of f on the x − y plane. [3marks] (d) Write down the equation of the tangent plane at (2,1). [2marks] 2. Let F (x, y, z ) = f ( y−z , z −x , x−y ). x y z Show that x ∂F ∂x + y ∂F ∂y + z ∂F ∂z =0 [4marks] 3. Suppose g : R → R and f (x, y ) = g ( x ). What are the mixed ordered partials of f ? y [4marks] 4. Consider the following questions on directional derivatives. (a) Suppose that Susan knows that the directional derivative is deﬁned by d Du f (x, y ) = ds f (a + su) where a is a position vector and u is a direction vector and s approaches 0. Prove to Susan that for diﬀerentiable f at a, Du f (a, b) = f (a, b) · u. [4marks] (b) Calculate the directional derivative of f (x, y, z ) = sin(xyz ) at (1, 1, π ) in the 4 √ direction v = 1, − 2, 1 . [2marks] (c) Suppose f in part (b) is the path a car follows. Determine in what direction the car must travel to experience no rate of change if we already know the car does not travel in the z -direction. [4marks] 5. Let f : R2 → R where f (x, y ) =
Date : August 29, 2010.
1 1−e−|xy| √ x2 +y 2 : (x, y ) = (0, 0) : (x, y ) = (0, 0) 0 2 NIALL W. MACGILLIVRAY Prove f is continuous ∀(x, y ) ∈ R2 [6marks] 6. Consider f (x, y ) = |x| |y |
1 2 2 3 (a) Determine whether f is diﬀerentiable at (0,0). [7marks] (b) On the basis of part (a), can we draw a conclusion about the continuity of f ? [1mark ] (c) On the basis of part (a), can we draw a conclusion about the continuity of the partials of f ? [1mark ] 7. Consider the approximation ln(x + 2y ) = (x − 3) + 2(y + 1) for (x, y ) suﬃciently close to (3, -1). Prove that if x ≥ 3 ∩ y ≥ −1, the error of the approximation is satisﬁed by:
7 |R1,(3,−1) (x, y )| ≤ 2 ((x − 3)2 + (y + 1)2 ) [7marks] MAXIMUM MARKS: 47...
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