Question

2. Consider the R − R 2 function r defined by r (t) =    (t, t2 ) if t ∈ [−2, 0] (t, t) if t ∈ (0, 2) t, t2 if t ∈ [2, 3] (a) Write down the domain of r. (1) (b) Is r continuous at t = 0? (2) (c) Is r continuous at t = 2? (2) (d) Sketch the curve r.

Answer 1

$r(t)=(t,t^2)\text{ if } t\in[-2,0]$

    $=(t,t) \text{ if } t\in (0,2)\\ = (t,t^2)\text{ if } t\in [2,3]$

(a) Domain of I is the set of all parts t Where I is defined. So, Domain of $I=[-2,3]$

$(b) I(o)=(0,0)$

$lim_{t\to 0^{-1}} I(t)=lim_{t\to 0^{-1}}(t,t^2)=(0,0)\\[9pt] lim_{t\to 0^{+1}} I(t)=lim_{t\to 0^{+1}}(t,t^2)=(0,0)$

 Since $I(0)=lim_{t\to 0} I(t)$ , I is continuous at t=0.

$(c) lim_{t\to 2^{-1}}I(t)=lim_{t\to 2^{-1}}(t,t)=(2,2) \\[9pt] lim_{t\to 2^{+1}}I(t)=lim_{t\to 2^{+1}}(t,t^2)=(2,4)$

As Above both limits are not equal, So $lim_{t\to 2}r(t)$ does npt exist. Hence r is not continuousat t=2.

(d)