Calculus with Vector Functions — Question 8

Question 8 – Solution

Strategy Use $\vec r(0)+t\vec r'(0)+\frac 12t^2\vec r''(0)$.

Data $\vec r(0)=\left\langle 1,0,1\right\rangle$, $\vec r'(0)=\left\langle 0,1,1\right\rangle$, and $\vec r''(0)=\left\langle-1,0,1\right\rangle$. Hence $$\boxed{\vec P_2(t)=\left\langle 1-t^2/2,t,1+t+t^2/2\right\rangle}.$$ At $t=0.05$, $\vec r(0.05)\approx\boxed{\left\langle 0.99875,0.05,1.05125\right\rangle}$.

Error Because the components have bounded third derivatives near zero, the remainder is $O(t^3)$, here on the scale of $0.05^3$.