Find a parametrization of the tangent line to $$r(t) = (\ln t)\mathbf{i} + t^{-10}\mathbf{j} + 11t\mathbf{k}$$ at the point $$t = 1$$ (Use symbolic notation and fractions where needed.) $$L(t) = < \boxed{\phantom{X}} >$$ help (fractions)
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The formula for the tangent line $$L(t)$$ to a vector function $$r(t)$$ at a point $$t_0$$ is given by: $$L(t) = r(t_0) + (t - t_0)r'(t_0)$$ In this case, $$t_0 = 1$$. Step 2: First, we need to find $$r(t_0) = r(1)$$. Given $$r(t) = (\ln t)\mathbf{i} + Show more…
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Sri K.
As mentioned in the text, the tangent line to a smooth curve $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ at $t=t_{0}$ is the line that passes through the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), ~ h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0} .$ In Exercises $19-22,$ find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ . $$\mathbf{r}(t)=\ln t \mathbf{i}+\frac{t-1}{t+2} \mathbf{j}+t \ln t \mathbf{k}, \quad t_{0}=1$$
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As mentioned in the text, the tangent line to a smooth curve $\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}$ at $t=t_{0}$ is the line that passes through the point $\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)$ parallel to $\mathbf{v}\left(t_{0}\right),$ the curve's velocity vector at $t_{0}$. Find parametric equations for the line that is tangent to the given curve at the given parameter value $t=t_{0}$ $$\mathbf{r}(t)=\ln t \mathbf{i}+\frac{t-1}{t+2} \mathbf{j}+t \ln t \mathbf{k}, \quad t_{0}=1$$
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