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

$r(t)= (t,t^2)\\$

a)

$r'(t)=(1,2t)\\r'(1)=(1,2)$

b)

$(x,y)=(t,t^2)\\y=x^2\ \ \ for\ \ \ x\in\ (-3,3)\\$

$r'(1)=(1,2)\\$

This is tangent line at x=1, y=1 with slope m=2