Question

Consider the R −R2 function r r defined by   (t) = (a) Write down the domain of r (b) Is r (c) Is r continuous at t = 0? continuous at t = 2? (d) Sketch the curve r 26 . .  (t, t2) if t ∈ [−2,0] (t, t) t, t2 if if t ∈ (0,2) t ∈ [2,3]

Answer 1

$$r(t)= \begin{cases} (t, t^2) &\text{if } t\in [-2,0] \\ (t, t) &\text{if } t\in (0,2) \\ (t, t^2) &\text{if } t\in [2,3] \\ \end{cases}$$

  

(a) Domain: $[-2, 3]$

(b)

$$r(0)=(0, 0)$$

$$\lim\limits_{t\to 0^-}r(t)=(0,0)$$

$$\lim\limits_{t\to 0^+}r(t)=(0,0)$$

Then

$$\lim\limits_{t\to 0}r(t)=(0,0)=r(0)$$

The function $r(t)$ is continuous at $t=0.$

(c)

$$r(2)=(2, 4)$$

$$\lim\limits_{t\to 2^-}r(t)=(2,2)$$

$$\lim\limits_{t\to 2^+}r(t)=(2,4)$$

$$\lim\limits_{t\to 2^-}r(t)\not=\lim\limits_{t\to 2^+}r(t)$$

Then

$$\lim\limits_{t\to 2}r(t)=\text{does not exist}$$

The function $r(t)$ is not continuous at $t=2.$

(d)