Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=3 ln(t), y=4t^{frac{1}{2}}, z=t^{3}, (0, 4, 1)

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=3 ln(t), y=4t^{frac{1}{2}}, z=t^{3}, (0, 4, 1)

Question
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. \(x=3\ \ln(t),\ y=4t^{\frac{1}{2}},\ z=t^{3},\ (0,\ 4,\ 1)\)

Answers (1)

2021-02-13
Step 1 Given: \(x = 3\ \ln (t)\) ......(1) \(y = 4t^{1/2}\) ......(2) \(z = t^{3}\) ......(3) \((x_{0},\ y_{0},\ z_{0}) = (0,\ 4,\ 1)\) For the specified points, Substitute 1 for z in equation 3. \(z = t^{3}\)
\(1 = t^{3}\)
\(t = 1\) The function r(t) of these equations is, \)r(t) =\ <\ x,\ y,\ z\ >\) Substitute the values in the above equation. \(r(t)=\ <\ 3\ \ln(t), 4t^{\frac{1}{2}},\ t^{3}\ >\) Step 2 Derivative of the vector r(t) is, \(r'(t)=\ <\ \frac{3}{t},\ \frac{2}{\sqrt{t}},\ 3t^{2}\ >\) Substitute 1 for t in the above vector. \(r'(1)=\ <\ 3,\ 2,\ 3\ >\) Step 3 The parametric equations for the tangent line are, \(X = x_{0}\ +\ (r'(1))_{x} t\)
\(Y = y_{0}\ +\ (r'(1))_{y} t\)
\(Z = z_{0}\ +\ (r'(1))_{z} t\) Substitute the values in the above equation. \(X=3t\)
\(Y=4\ +\ 2t\)
\(Z=1\ +\ 3t\) Thus, the parametric equations for the tangent line are \(X = 3t,\ Y = 4\ +\ 2t,\ and\ z = 1\ +\ 3t.\)
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