Calculus-Sec-12-2-Solution


   12.2      Calculus of Vector Functions                  by SGLee - HSKim - JHLee

                                                                                                                              http://youtu.be/VS5rPyOjP2I 

 

 1-5. Find  and draw the position vector  and the tangent vector  for the given value of .

 

1.  

    http://matrix.skku.ac.kr/cal-lab/cal-13-2-1.html 

 

 




 So, we can see that .




2. 

    http://matrix.skku.ac.kr/cal-lab/cal-13-2-2.html 

 




 So, we can see that 




3. 

    http://matrix.skku.ac.kr/cal-lab/cal-13-2-3.html 

 

 




 So,  we can see that .




4. 

    http://matrix.skku.ac.kr/cal-lab/cal-13-2-4.html 

 




 So, we can see that .




5. 

    http://matrix.skku.ac.kr/cal-lab/cal-13-2-5.html 

 

 




 So, we can see that .




6-10. Find the derivatives of the vector functions.

 

6. 

 

 

      .




7. 

 

      .




8. 

 

      .

 




9. 

 

      .

 




10. 

 

      .

 




11. 

 

 

       ,

      ,

      Hence .   <-- r '(0)/abs(r'(0))







12. 

 

 

      .

      .

      Therefore, .

 







13. 

 

 

      ,

      ,

      .







14. If , find , and .

 

       .

      Therefore, ,

         .

         .

 




15. If , find  and .

 

      .

      .

      .

     .

     .

     .




16-19. Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.

 

16.        <-- z=t^4

 

 

      We can see that .

      For . So, .

      Hence .

 







17. 

 

      We can see that .

      For .

      Therefore, .

      Hence .    <-- <1, 1, 0> + s <-1, -1, 1> = < 1-s, 1-s, s> 

 







18. 

 

      We can see that .

      For . So, .

      Hence .

 




19. 

 

 

      We can see that .

      For . So, .

      Hence .

 20-21. Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.

                 Illustrate by graphing both the curve and the tangent line together.

 

 




20. 

      http://matrix.skku.ac.kr/cal-lab/cal-13-2-20.html 

 

     We can see that .




 So, parametric equation is .




21. 

     http://matrix.skku.ac.kr/cal-lab/cal-13-2-21.html 

 

      We can see that .




 So, parametric equation is .




22-24. Determine whether the curve are smooth.

 

22. 

 

      . Since , this curve is not smooth.

 




23. 

 

      .

      First coordinate of  cannot be zero for any real number , so this curve is smooth.

 




24. 

 

      .

      Second coordinate of  cannot be zero for any real number , so this curve is smooth.

 




25-30. Evaluate the integrals.

 

25. 

 

 

    .

 




26. 

 

     .




27. 

 

 

     .

 




28. 

 

 

     where  is a constant vector.

 




29. 

 

      where  is a constant vector.    <-- e^(2t) j

 




30. 

 

 

    .




31. Find  if  and .

 

 

      , where  is a constant vector.

       Since , we finally have .

 




32. Find  if  and .

 

 

     , where  is a constant vector.

      Since .

      So, we finally have .

 




33. If  and ,

     use Theorem 4(Rules of differentiation) to find 

 

      .

      .

      So, .




34. If  and ,

     use Theorem 2(Rules of differentiation) to find .

 

      ,

      ,

      ,

      .




35. Show that if  is a vector function such that  exists, then .

 

 Proof

       Using Theorem 4(Rules of differentiation),

       .

36. If , show that .

 

 Proof

      We already know that .

       Differentiate both sides, .

       So, we finally have .

 37. Find unit tangent vector to the curve  at any time .

             http://matrix.skku.ac.kr/cal-lab/cal-14-1-19.html 

 

  










 And we can draw a curve with given unit tangent vector.




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