Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=3 ln⁡(t), y=4t12, z=t3, (0, 4, 1)

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Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x=3 ln⁡(t), y=4t12, z=t3, (0, 4, 1)

timbalemX · Accepted Answer

Step 1 Given: x=3 ln⁡(t) ......(1) y=4t1/2 ......(2) z=t3 ......(3) (x0, y0, z0)=(0, 4, 1) For the specified points, Substitute 1 for z in equation 3. z=t31=t3t=1 The function r(t) of these equations is, r(t)= &amp;lt; x, y, z &amp;gt;Substitute the values in the above equation. r(t)= &amp;lt; 3 ln⁡(t),4t12, t3 &amp;gt; Step 2 Derivative of the vector r(t) is, r′(t)= &amp;lt; 3t, 2t, 3t2 &amp;gt; Substitute 1 for t in the above vector. r′(1)= &amp;lt; 3, 2, 3 &amp;gt; Step 3 The parametric equations for the tangent line are, X=x0 + (r′(1))xtY=y0 + (r′(1))ytZ=z0 + (r′(1))zt Substitute the values in the above equation. X=3tY=4 + 2tZ=1 + 3t Thus, the parametric equations for the tangent line are X=3t, Y=4 + 2t, and z=1 + 3t.

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)

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