Calculus with Vector Functions — Question 1

Question 1

For $\vec r(t)=\left\langle t^2-1,e^t,\sin(2t)\right\rangle$, find $\vec r'(t)$ and $\vec r''(t)$. Find the tangent line at $t=0$ and a second-order approximation to $\vec r(0.1)$.

Question 1 – Solution

Strategy Differentiate componentwise; use the vector Taylor polynomial through second order.

Derivatives $\vec r'(t)=\left\langle 2t,e^t,2\cos(2t)\right\rangle$ and $\vec r''(t)=\left\langle 2,e^t,-4\sin(2t)\right\rangle$. At $t=0$, $\vec r(0)=\left\langle-1,1,0\right\rangle$ and $\vec r'(0)=\left\langle 0,1,2\right\rangle$.

Tangent line $\boxed{\vec L(s)=\left\langle-1,1,0\right\rangle+s\left\langle 0,1,2\right\rangle}$.

Approximation Since $\vec r''(0)=\left\langle 2,1,0\right\rangle$, $$\vec r(0.1)\approx\vec r(0)+0.1\vec r'(0)+\frac{0.1^2}{2}\vec r''(0)=\boxed{\left\langle-0.99,1.105,0.2\right\rangle}.$$