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Engineering Calculus Notes 252

Engineering Calculus Notes 252 - 240 CHAPTER 3 REAL-VALUED...

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240 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION This result ensures that functions defined by algebraic or analytic expressions such as polynomials (in two or three variables) or combinations of trigonometric, exponential, logarithmic functions and roots are generally differentiable, since by the formal rules of differentiation the partials are again of this type, and hence are continuous wherever they are defined; the only difficulties arise in cases where differentiation introduces a denominator which becomes zero at the point in question. The Gradient and Directional Derivatives Recall from § 3.2 that a linear function can be viewed in three different ways: as a homogeneous polynomial of degree one, as multiplication of the coordinate matrix by its matrix representative , and as the dot product of the input with a fixed vector. We have seen that when f ( −→ x ) is differentiable at −→ x = −→ x 0 , then the coefficients of the differential d −→ x 0 f ( −→ v ),
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