Question

5. (Sections 6.1, 6.3) Consider the R − R 2 function r defined by r (t) = ￾ t, t2
; t ∈ [−3, 3] . (a) Determine the vector derivative r 0 (1) by using Definition 6.1.1(b) Sketch the curve r together with the vector r 0 (1), in order to illustrate the geometric meaning of the vector derivative. Note: The curve r is the image of r, so it consists of all points (x, y) = (t, t2 ); t ∈ [−3, 3]

Answer 1

$r(t)=(t,t^2), t\in [-3,3]$

(a) $r'(t)=(\dfrac{dt}{dt},2\dfrac{dt}{dt}) =(1,2t)$

 

   $r'(1)=(1,2)$

(b) $r=(t,t^2)=t\hat{i}+t^2\hat{j}, (x,y)=(t,t^2), y=x^2$

  $r'(1)=(1,2)=\hat{i}+2\hat{j}$

    The curve is-