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Engineering Calculus Notes 186

Engineering Calculus Notes 186 - −→ p t speciFes a...

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174 CHAPTER 2. CURVES 8. Prove Proposition 2.3.12 . 9. Prove the following parts of Theorem 2.3.15 : (a) d dt bracketleftBig α −→ f ( t ) + β −→ g ( t ) bracketrightBig = α vector f ( t ) + βvectorg ( t ) (b) d dt bracketleftBig −→ f ( t ) × −→ g ( t ) bracketrightBig = vector f ( t ) × −→ g ( t ) + −→ f ( t ) × vectorg ( t ) 10. Prove that the moment of velocity about the origin is constant if and only if the acceleration is radial ( i.e. , parallel to the position vector). 11. Prove Lemma 2.3.19 . ( Hint: Look at each component separately.) 12. Use the fact that each component of the linearization T t 0 −→ p ( t ) of −→ p ( t ) has first-order contact with the corresponding component of −→ p ( t ) to prove Remark 2.3.18 . Challenge problem: 13. (David Bressoud) A missile travelling at constant speed is homing in on a target at the origin. Due to an error in its circuitry, it is consistently misdirected by a constant angle α . Find its path. Show that if | α | < π 2 then it will eventually hit its target, taking 1 cos α times as long as if it were correctly aimed.
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Unformatted text preview: −→ p ( t ) speciFes a curve by “tracing it out”. This approach is particularly useful for specifying curves in space, but as we shall see it is also a natural setting (even in the plane) for applying calculus to curves. However, it has the intrinsic complication that a given curve can be traced out in many di²erent ways. In this section, we study how di²erent vector-valued functions specify the same curve, and which properties of a function encode geometric properties of the curve it traces out, independent of which particular function is used to describe it. Along the way, we will formulate more carefully which kinds of vector-valued functions are appropriate parametrizations of a curve, and hopefully end up with a better understanding of what, exactly, constitutes a “curve”....
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