Engineering Calculus Notes 202

Engineering Calculus Notes 202 - ( x,f ( x )) equals the...

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190 CHAPTER 2. CURVES Figure 2.27: Lima¸con of Pascal: ( x 2 2 x + y 2 ) 2 = x 2 + y 2 (c) Find the equations of the two “tangent lines” at the crossing point at the origin. 7. Complete the proof of the second statement in Proposition 2.4.7 by showing how to modify the argument if either x ( t 0 ) is negative, or if x ( t 0 ) is zero but y ( t 0 ) is not. 8. (a) Show that if −→ p ( t ) is a regular vector-valued function whose image satis±es the vertical line test—so that the curve it traces out is the graph of a function f ( x )—then this function is C 1 . ( Hint: Show that the “slope” of the velocity vector at any point
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Unformatted text preview: ( x,f ( x )) equals the derivative of f ( x ) at x .) (b) Show that the vector-valued function p ( t ) = ( t | t | ,t 2 ) is C 1 and one-to-one on the whole real line, but its image is the graph of the function y = | x | , which fails to be dierentiable at x = 0. This shows the importance of the condition that the velocity is always nonzero in our denition of regularity. 9. Prove Remark 2.4.8 , as follows: Suppose C is parametrized by p ( t ) = ( x ( t ) ,y ( t ) ,z ( t ))...
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