Engineering Calculus Notes 257

Engineering Calculus Notes 257 - 245 3.3. DERIVATIVES In...

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Unformatted text preview: 245 3.3. DERIVATIVES In the first term above, the first factor is fixed and the second goes to zero as △t → 0, while in the second term, the first factor is bounded (since → → △− /△t converges to − ) and the second goes to zero. Thus, the whole x v mess goes to zero, proving that the affine function inside the absolute value in the numerator on the left above represents the linearization of the composition, as required. An important aspect of Proposition 3.3.6 (perhaps the important aspect) is that the rate of change of a function applied to a moving point depends only on the gradient of the function and the velocity of the moving point at the given moment, not on how the motion might be accelerating, etc. → For example, consider the distance from a moving point − (t) to the point p (1, 2): the distance from (x, y ) to (1, 2) is given by f (x, y ) = (x − 1)2 + (y − 2)2 with gradient → − ∇ f (x, y ) = (x − (x − 1) 1)2 + (y − 1)2 → −+ ı (x − (y − 2) 1)2 + (y − 1)2 → −. If at a given moment our point has position → − (t ) = (5, −3) p0 and velocity → − (t ) = −2− − 3− → → v0 ı then regardless of acceleration and so on, the rate at which its distance from (1, 2) is changing is given by d dt t=t0 → − → → [f (− (t))] = ∇ f (5, −3) · − (t0 ) p v 4− → − 3 − · (−2− − 3− ) → → → ı ı 5 5 89 =− + 55 1 =. 5 The other kind of chain rule that can arise is when we compose a real-valued function f of a vector variable with a real-valued function g of a real variable: = ...
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