Question

Consider the given vector equation. R(t) = 5 sin(t)i + 2 cos(t)j (a) find r'(t). R'(t) = incorrect: your answer is incorrect. (b) sketch the plane curve together with position vector r(t) and the tangent vector r'(t) for the given value of t

Answer 1

solution:

$a) r(t)=5sint\widehat{i}+2cost\widehat{j}\\\Rightarrow r'(t)=\frac{d}{dt}(5sint\widehat{i}+2cost\widehat{j})\\\Rightarrow r'(t)=\frac{d}{dt}(5sint)\widehat{i}+\frac{d}{dt}(2cost)\widehat{j}\\\Rightarrow r'(t)=5cost\widehat{i}-2sint\widehat{j}$$b) at (t)=\frac{\pi }{4},we have\\r'(\frac{\pi }{4})=\frac{5}{\sqrt{2}}\widehat{i}-\frac{2}{\sqrt{2}}\widehat{j}\\so, the tangent vector r'(\frac{\pi }{4}) start at(\frac{5}{\sqrt{2}},\frac{-2}{\sqrt{2}})\\and moves \frac{5}{\sqrt{2}} right and \frac{2}{\sqrt{2}}down$