How do you find parametric equations for the tangent line to the curve with the given parametric equations at the specified point x = 1 + 10 ⋅ t , y = t 5 - t , and z = t 5 + t ; (11, 0, 2)?

Question

How do you find parametric equations for the tangent line to the curve with the given parametric equations at the specified point       x    =    1    +    10    ⋅                  t            ,       y    =          t      5        -    t  , and       z    =          t      5        +    t  ; (11, 0, 2)?

nuvolor8 · Accepted Answer

Step 1Find the value of t that creates the point (11,0,2)Substitute 11 for x:      11    =    1    +    10                  t                  10                  t              =    10        t    =    1  check that       t    =    1   works for the y and z values:            1      5        -    1    =    0              1      5        +    1    =    2  Step 2These check,       t    =    1  Find the tangent vector:                    d        x                    d        t              =          5                        t                                        d        y                    d        t              =    5          t      4        -    1                      d        z                    d        t              =    5          t      4        +    1  The tangent vector for all points,             v      ¯              (      t      )      , is:            v      ¯              (      t      )        =          {              5                              t                              }              i      ^        +          {      5              t        4            -      1      }              j      ^        +          {      5              t        4            +      1      }              k      ^      We are interested in the tangent vector at the given point:            v      ¯              (      1      )        =    5          i      ^        +    4          j      ^        +    6          k      ^      The vector equation of the tangent line is            (      x      ,      y      ,      z      )        =          (      11      ,      0      ,      2      )        +    s          (      5              i        ^            +      4              j        ^            +      6              k        ^            )      The parametric equations for this line are:      x    =    11    +    5    s        y    =    4    s        z    =    2    +    6    s

How do you find parametric equations for the tangent line to the curve with the given parametric equ

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