Engineering Calculus Notes 355

Engineering Calculus Notes 355 - about the way the graph...

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3.7. HIGHER DERIVATIVES 343 for every x S , as follows: given x S , by part (a), there is a sequence x k x with x k S k . Thus, f ( m k ) f ( x k ) f ( M k ) and so by properties of limits (which?) the desired conclusion follows. 14. Suppose −→ a satisFes f ( −→ a ) = b and g ( −→ a ) = c and is not a critical point of either function; suppose furthermore that −→ g n = −→ 0 everywhere on the level set L ( g,c ) (that is, c is a regular value of g ), and max L ( g,c ) f ( x ) = b. (a) Show that L ( f,b ) and L ( g,c ) are tangent at −→ a . (b) As a corollary, show that the restriction of g ( −→ x ) to L ( f,b ) has a local extremum at −→ x = −→ a . 3.7 Higher Derivatives ±or a function of one variable, the higher-order derivatives give more subtle information about the function near a point: while the Frst derivative speciFes the “tilt” of the graph, the second derivative tells us
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Unformatted text preview: about the way the graph curves, and so on. SpeciFcally, the second derivative can help us decide whether a given critical point is a local maximum, local minimum, or neither. In this section we develop the basic theory of higher-order derivatives for functions of several variables, which can be a bit more complicated than the single-variable version. Most of our energy will be devoted to second-order derivatives. Higher-order Partial Derivatives The partial derivatives of a function of several variables are themselves functions of several variables, and we can try to Fnd their partial derivatives. Thus, if f ( x,y ) is di²erentiable, it has two Frst-order partials ∂f ∂x , ∂f ∂y...
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