Explanation Given info: r ( s ) = r ( t o ) + 1 k N + 1 k [ ( sin k s ) T − ( cos k s ) N ] Concept used: Are length ( s ) = ∫ a b ( d x d t ) 2 + ( d y d t ) 2 + ( d z d t ) 2 dt Calculation: r ( t ) = r ( t o ) + 1 k N T ( sin k s − cos k s ) = r ( t o ) + R N + R t < cos t , sin t > An are length parameterization of the osculating circle r ' ( t ) = 0 + R 0 + R < − sin t , cos t > ‖ r ' t ‖ = R 2 [ ( − sin t ) 2 + ( cos t ) 2 ] = R sin 2 t + cos 2 t = R . Now, are length at point t can be calculated as s = ∫ 0 t ‖ r ' ( t ) ‖ d t = ∫ 0 t R d t = R | t | 0 t = R t . Now, the inverse of the are length function can be calculated as s = g ( t ) is t = g − 1 ( s ) = S R The are length parameterization of the osculating circle is r ( s ) = r ( g 1 ( x ) ) = r × ( S R ) Now, r ( t ) = r ( t o ) + 1 k N + 1 k T ( cos k s , sin k s )

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Chapter 14.4, Problem 80E

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An are length of the osculating circle at $r (t o)$ the function $r (s) = r (t o) + \frac{1}{k} N + \frac{1}{k} [(\sin k s) T - (\cos k s) N]$

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Chapter 14.4, Problem 79E

Chapter 14.4, Problem 81E

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الاسم يمنع استخدام الآلة الحاسبة ظلل في ورقة الإجابة الدائرة التي تحتوي على الحرف (ص) للإجابة الصحيحة والحرف (خ) للإجابة الخطأ بحسب رقم الفقرة لكل مما يأتي: درجة لكل فقرة. ( ) نها جا 元 جتا = صفر س ۱ س س -۱ ( ) يمكن إعادة تعريف الدالة د(س) = س قاس لكي تكون متصلة عند س = 7 ( ) إذا كانت د(س) = (٢) س - س ) ؛ فإن د(١) = ٦ ٢ س ص ( ) إذا كانت س + 0= ؛ فإن عند ) - ١ ، - ٦ ) تساوي (٦) ( ) إذا كانت د(س) = س ه ، و (س) = ٣ س ٢ + ٢ س ؛ فإن ( د ) (۱) = ۸ ) ( معادلة ناظم الدالة ص = د(س) عند النقطة ) ( ، د (۲)) هي ص - (د (م) - - د (۲) ( س - م ) ( ) إذا كانت ص = ظتا٢ س ؛ فإن ص = ٢ ص قتا ٢س ) ( إذا كانت د(س) = س ؛ فإن د (T) = جتاس 1- T ( ) إذا كانت د(س) = 1 - جناس جاس ؛ فإن د () = - 1 ( ) إذا كانت الدالة د (س) تحقق شروط مبرهنة القيمة المتوسطة على [ ، ب ] ، فإنه يوجد جـ ] ، ب [ بحيث (جـ) = (P) + (~)- - ب + P 1 2 3 4 5 6 7 8 9 10 11 ( ) للدالة د(س) = لو ( س ) + (٣) نقطة حرجة عند س = . ( ) إذا كان س = - ٢ مقارباً رأسياً للدالة د(س) 12 10 13 14 15 16 17 س = لو|س | + ث - = ۲ س + ٣ ب س + ٤ ، فإن معادلة…

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An are length of the osculating circle at r ( t o ) the function r ( s ) = r ( t o ) + 1 k N + 1 k [ ( sin k s ) T − ( cos k s ) N ]

Solution Summary: The author calculates the length of the osculating circle for the function r ( t ) = a b.

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An are length of the osculating circle at $r (t o)$ the function $r (s) = r (t o) + \frac{1}{k} N + \frac{1}{k} [(\sin k s) T - (\cos k s) N]$

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An are length of the osculating circle at r ( t o ) the function r ( s ) = r ( t o ) + 1 k N + 1 k [ ( sin k s ) T − ( cos k s ) N ]

Solution Summary: The author calculates the length of the osculating circle for the function r ( t ) = a b.

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An are length of the osculating circle at r(to) the function r(s)=r(to)+1kN+1k[(sinks)T−(cosks)N]

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An are length of the osculating circle at $r (t o)$ the function $r (s) = r (t o) + \frac{1}{k} N + \frac{1}{k} [(\sin k s) T - (\cos k s) N]$