Find the derivatives of r(t) = (1 + t2)i – te-tj + sin 2tk and find the unit tangent vector at t=0.

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Solution:

Given:

r(t) = (1 + t2)i – te-tj + sin 2tk

To find the derivative, we will differentiate with respect to t. So derivative is

r(t) = 2ti – [e-t + te-t]j + 2cos 2tk

= 2ti – e-t(1 + t) + 2cos 2tk

 

The unit tangent vector is obtained using the formula r'(0)/|r(0)|. r(0) is the value of r'(t) at t = 0, and |r(0)| is the modulus of r(0).

now,

r'(0) = 2 . 0 . i – e0 (1 + 0)j + 2 cos 2.0.t

= -j + 2k

 

r(0) = (1 + 0)i – 0 . e0 . j + sin0 k

= i

|r(0)| = √(1)2 = 1

 

So, Unit vector is

\(\frac{r'(0)}{|r(0)|} = \frac{-j + 2k}{1}\)

= -i + 2k

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