Engineering Calculus Notes 200

Engineering Calculus Notes 200 - p ( t ) and q ( s ) are...

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188 CHAPTER 2. CURVES (c) −→ p ( t ) = (cos t, sin t ) − ∞ < t < −→ q ( t ) = (sin t, cos t ) − ∞ < t < (d) −→ p ( t ) = (cos t, sin t, sin 2 t ) π 2 t π 2 −→ q ( t ) = ( r 1 t 2 ,t,t r 4 4 t 2 ) 1 t 1 Theory problems: 2. Prove Lemma 2.4.2 as follows: (a) Suppose Frst that −→ p ( t ) = ( t,f ( t )), and write −→ q ( s ) = ( q 1 ( s ) ,q 2 ( s )) . Then show that t ( s ) = q 1 ( s ) satisFes −→ p ( t ( s )) = −→ q ( s ). (b) Show that the derivatives of the components of −→ q ( s ) are related by q 2 ( s ) = f ( q 1 ( s )) · q 1 ( s ) so that if v q ( s 0 ) n = −→ 0 , then also q 1 ( s 0 ) n = 0. (c) Conclude that, when −→ p ( t ) is the standard graph parametrization of C and −→ q ( s ) is another regular parametrization, then t ( s ) = q 1 ( s ) satisFes all the conditions of the lemma. (d) In general, if −→
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Unformatted text preview: p ( t ) and q ( s ) are arbitrary regular parametrizations of C , let r ( x ) = ( x,f ( x )) be the standard graph parametrization of gr ( f ) and let t ( x ) ( resp . s ( s )) be the calibration function for p in terms of r ( resp . r in terms of q ). Then show that their composition is the required recalibration between p and q . 3. Let C be given by the polar equation r = f ( ) and set p ( ) = ( f ( )cos ,f ( )sin ) . Assume that the function f is C 1 ....
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