(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The graph shows the vector-valued function r(t). It is given that r(1) = <-1,1> and r'(1) = <1,2>. Find the unit tangent vector T(1) and unit normal vector N(1).

(the graph given is that of y = x^2 shifted 2 units left.)

2. Relevant equations

(1) T(t) = r'(t)/||r'(t)||

(2) N(t) = T'(t)/||T'(t)||

3. The attempt at a solution

After playing around for a bit I found r(t) = <t-2,t^2> so r'(t) = <1,2t>

Using this in eq. (1) I get T(t) = <1,2t>/sqrt(1+4t^2).

T(t) = <1/sqrt(1+4t^2),2t/sqrt(1+4t^2)>

T(1) = <1/sqrt5,2sqrt5> (this vector im fairly confident is correct)

T'(t) = <-4t/(4t^2+1)^(3/2),2/(4t^2+1)^(3/2)>

||T'(t)|| = sqrt((-4t/(4t^2+1)^(3/2))^2+(2/(4t^2+1)^(3/2))^2)

which reduces to sqrt(16t^2/(4t^2+1)^3+(4/(4t^2+1)^3))

Then using T'(t) and ||T'(t)|| in eq. (2) I get

N(t) = <-4t/(4t^2+1)^(3/2),2/(4t^2+1)^(3/2)>/sqrt(16t^2/(4t^2+1)^3+(4/(4t^2+1)^3)

N(1) = <-4/5^(3/2),2/5^(3/2)>/(4/25) or approx. <-2.236,1.118>

Drawing this vector on the graph it seems too big to be a unit vector. Can anyone spot any errors in my calculations? Am I even close? I feel like I may be making this harder then neccesary. Thanks in advance for any help!

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Finding unit tangent and unit normal vectors

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