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

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)=2ti–−5j–+t2k––𝐹_(𝑡)=2𝑡𝑖_−5𝑗_+𝑡2𝑘_ and G––(t)=(1−t)i–+1tk––𝐺_(𝑡)=(1−𝑡)𝑖_+1𝑡𝑘_ where t∈R+𝑡∈𝑅+.

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

Suppose 𝐹(𝑥)=𝑥2𝑥2−7F(x)= x 2 −7x 2 ​ , and 𝐹(𝑥)=(ℎ∘𝑔)(𝑥)F(x)=(h∘g)(x). Which of the following are possible definitions for h and g?I. ℎ(𝑥)=𝑥,𝑔(𝑥)=𝑥2𝑥2−7II. ℎ(𝑥)=𝑥2,𝑔(𝑥)=𝑥𝑥−7III. ℎ(𝑥)=𝑥4𝑥4−7,𝑔(𝑥)=∣𝑥∣IV. ℎ(𝑥)=𝑥𝑥−7,𝑔(𝑥)=𝑥2​ I. h(x)=x,g(x)= x 2 −7x 2 ​ II. h(x)=x 2 ,g(x)= x−7x​ III. h(x)= x 4 −7x 4 ​ ,g(x)= ∣x∣​ IV. h(x)= x−7x​ ,g(x)=x 2 ​ A.II and III onlyB.I and II onlyC.I and IV onlyD.I, III and IV onlyE.I, II, III and IV