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Engineering Calculus Notes 290

Engineering Calculus Notes 290 - has a vertical tangent...

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278 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION 2. For each equation below, investigate several slices and use them to sketch the locus of the equation. For quadric surfaces, decide which kind it is ( e.g. , hyperbolic paraboloid, ellipsoid, hyperboloid of one sheet, etc.) (a) z = 9 x 2 + 4 y 2 (b) z = 1 x 2 y 2 (c) z = x 2 2 x + y 2 (d) x 2 + y 2 + z = 1 (e) 9 x 2 = y 2 + z (f) x 2 y 2 z 2 = 1 (g) x 2 y 2 + z 2 = 1 (h) z 2 = x 2 + y 2 (i) x 2 + 4 y 2 + 9 z 2 = 36 Theory problems: 3. Show that the gradient vector −→ f is perpendicular to the level curves of the function f ( x,y ), using the Chain Rule instead of implicit differentiation. 4. Suppose f ( x,y ) has a nonvanishing gradient at ( x 0 ,y 0 ) and f ( x 0 ,y 0 ) = c . (a) Show that if L ( f,c ) is not expressible near ( x 0 ,y 0 ) as the graph of y as a function of x , then L ( f,c ) has a vertical tangent line at
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Unformatted text preview: ) has a vertical tangent line at ( x ,y ). (b) Give an example of a function with a vertical tangent at some regular point such that L ( f,c ) near this point can be expressed as the graph of y as a function of x . (c) Show that in this situation, y cannot be di±erentiable (as a function of x ) at this regular point. Challenge problem: 5. The following example (based on [ 32 , pp. 58-9] or [ 49 , p. 201]) shows that the hypothesis that f be continuously di±erentiable cannot be ignored in Theorem 3.4.2 . De²ne f ( x,y ) by f ( x,y ) = b xy + y 2 sin 1 y if y n = 0 , if y = 0 ....
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