The domain of the vector function r–(t)=t2i–+ln(2−t)j–+3k––𝑟_(𝑡)=𝑡2𝑖_+𝑙𝑛(2−𝑡)𝑗_+3𝑘_ is given by the interval

The domain of the vector function F––(t)=t−3−−−−√i–+ln(10−2t−−−−−−√)–j+1t−3√k––𝐹_(𝑡)=𝑡−3𝑖_+𝑙𝑛(10−2𝑡)_𝑗+1𝑡−3𝑘_ is given by the interval

Now let H––(t)=t3i–+t2j–+tk––𝐻_(𝑡)=𝑡3𝑖_+𝑡2𝑗_+𝑡𝑘_ be another vector function defined for each t∈R𝑡∈𝑅. The derivative of (F––(t)×G––(t)).H––(t)(𝐹_(𝑡)×𝐺_(𝑡)).𝐻_(𝑡) with respect to t𝑡, at t=1𝑡=1 is given by Answer

Let F––(t)=t2i–+tj–+k––𝐹_(𝑡)=𝑡2𝑖_+𝑡𝑗_+𝑘_ and G––(t)=i–+tj–+t2k––𝐺_(𝑡)=𝑖_+𝑡𝑗_+𝑡2𝑘_ be two vector functions defined for each t∈R𝑡∈𝑅.

Let F––(t)=t2i–+tj–+k––𝐹_(𝑡)=𝑡2𝑖_+𝑡𝑗_+𝑘_ and G––(t)=i–+tj–+t2k––𝐺_(𝑡)=𝑖_+𝑡𝑗_+𝑡2𝑘_ be two vector functions defined for each t∈R𝑡∈𝑅. (a) The derivative of F––(t)×G––(t)𝐹_(𝑡)×𝐺_(𝑡) with respect to t𝑡, at t=1𝑡=1 is given by

For the given position vectors r(t) compute the unit tangent vector T(t) for the given value of t A) let r(t)=<cos2t,sin2t> Then T(pi/4) <__,__>B) let r(t)=<t^2,t^3> Then T(1)=<_,_>c) let r(t)= e^2t i+ e^-t j+ t K Then T(2)= _i+_j+_k