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solution_exam1 - Math 251:H1 Exam #1 Name : Problem Number...

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Unformatted text preview: Math 251:H1 Exam #1 Name : Problem Number Points Possible Points 1 15 2 10 3 10 4 15 5 10 6 10 7 10 8 10 9 10 10 (Bonus) 10 Total 1. [15 pts] Let C be the curve given by the vector function r ( t ) = h cos t, sin t,t i . Find the equation of the line that has the direction of the binormal vector B to the curve C at the the point (- 1 , , ). Solution: Note that the point (- 1 , , ) corresponds to r ( ). Hence, lets first compute B ( ). T ( t ) = r ( t ) | r ( t ) | = 1 2 h- sin t, cos t, 1 i , N ( t ) = T ( t ) | T ( t ) | = h- cos t,- sin t, i B ( t ) = T ( t ) N ( t ) = i j k- 1 2 sin t 1 2 cos t 1 2- cos t- sin t = 1 2 h sin t,- cos t, 1 i . Hence B ( ) = h , 1 2 , 1 2 i . This is then the direction of the line of interest. We know that the point P = (- 1 , , ) belongs to the line. Hence, the equation is- OP + t B ( ) = h- 1 , , i + t h , 1 2 , 1 2 i . 2. [10 pts] Reparametrize the curve of equation r ( t ) = h 4sin t, 3 t, 4cos t i with respect to arc length measured from the point where t = 1 5 in the direction of increasing t . Solution: We have that s ( t ) = Z t 1 5 | r ( u ) | du = Z t 1 5 p (4cos u ) 2 + (3) 2 + (- 4sin u ) 2 du = Z t 1 5 5 du = 5 t- 1 . Hence by letting t = s +1 5 , we reparametrized the curve with respect to arc length: r ( s ) = h 4sin s + 1 5 , 3 s + 1 5 , 4cos s + 1 5 i . 3. [10 pts] Imagine youre having a nice walk in the x- y plane. The temperature at a general point ( x,y ) is given by T ( x,y ) = x 2- 3 y 2 . You stop at the point P = (2 , 3) and realize youre getting pretty cold. From that point, what is the immediate direction u = ( a,b ) you should take to get the warmest? Justify your answer. Solution: At the point (2 , 3), the direction where the temperature increases the most is given by the direction of the gradient vector at (2 , 3) which is given by T (2 , 3) = h T x (2 , 3) ,T y (2 , 3) i . Since T x ( x,y ) = 2 x and T y ( x,y ) =- 6 y , then T (2 , 3) = h 4 ,- 18 i . Therefore, the direction to be taken is u = h 4 ,- 18 i |h 4 ,- 18 i| = 1 340 h 4 ,- 18 i ....
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solution_exam1 - Math 251:H1 Exam #1 Name : Problem Number...

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