# 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. $$x=3\ \ln(t),\ y=4t^{\frac{1}{2}},\ z=t^{3},\ (0,\ 4,\ 1)$$

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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