Question

Find the unit tangent vector to r(t)=e^(2t)i+e^-tj+(t^2+4)k at (1,1,4)

Answer 1

\[r(t)=e ^{2t} i+e^{-t}j+(t^2+4)k \]

reformatted to be clearer.

Answer 2

take the derivative

Answer 3

the <...> looks better to me than the ijks but thats immaterial

Answer 4

2 things we need; r'(t) and that value of "t" to get that point; then we can create a unti tangent

Answer 5

and the other is ||r'(t)||, so T=r'(t)/||r'(t)||... could you check my derivative? I got \[2e^{2t}i+e^{-t}j+2tk\]

Answer 6

middle is -e^-t i beleive

Answer 7

|r'| is after we establish the "t" value

Answer 8

i spose you could determine a general for |r'(t)| but it tends to be a bit easier after you apply a t value