Engineering Calculus Notes 243

Engineering Calculus Notes 243 - −→ x ): R 3 → R...

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3.3. DERIVATIVES 231 6. Suppose −→ p ( s,t ) is a parametrization of a plane in R 3 , and f : R 3 R is linear. Show that f −→ p : R 2 R is an aFne function. 7. A level set of a function is the set of points where the function takes a particular value. Show that any level set of an aFne function on R 2 is a line, and a level set of an aFne function on R 3 is a plane. When does the line/plane go through the origin? 8. Prove Remark 3.2.1 . 9. Prove Remark 3.2.2 . 10. Carry out the calculation that establishes Equation ( 3.2 ). 3.3 Derivatives In this section we carry out the program outlined at the beginning of § 3.2 , trying to formulate the derivative of a real-valued function of several variables f ( −→ x ) in terms of an aFne function making ±rst-order contact with f ( −→ x ). Defnition 3.3.1. A real-valued function of n variables f : R 3 R is diferentiable at −→ x 0 R 3 if f is deFned for all −→ x su±ciently near −→ x 0 and there exists an a±ne function T −→ x 0 f (
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Unformatted text preview: −→ x ): R 3 → R which has Frst-order contact with f ( −→ x ) at −→ x = −→ x : v v v f ( −→ x ) − T −→ x f ( −→ x ) v v v = o ( b −→ x − −→ x b ) (3.3) which is to say lim −→ x → −→ x v v v f ( −→ x ) − T −→ x f ( −→ x ) v v v b −→ x − −→ x b = 0 . (3.4) When such an a±ne function exists, we call it the linearization of f ( −→ x ) or the linear approximation to f ( −→ x ) , at −→ x = −→ x . 5 Since functions with ±rst-order contact must agree at the point of contact, we know that T −→ x f ( −→ x ) = f ( −→ x ); then Remark 3.2.2 tells us that T −→ x f ( −→ x + △ −→ x ) = f ( x ) + ℓ ( △ −→ x ) . (3.5) 5 Properly speaking, it should be called the afne approximation ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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