Answer to Question #184470 in Calculus for Njabulo

Question #184470

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.


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Expert's answer
2021-05-11T09:22:54-0400

r(t)=(t,t2) if t[2,0]r(t)=(t,t^2)\text{ if } t\in[-2,0]

    =(t,t) if t(0,2)=(t,t2) if t[2,3]=(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]I=[-2,3]


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


limt01I(t)=limt01(t,t2)=(0,0)limt0+1I(t)=limt0+1(t,t2)=(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)=limt0I(t)I(0)=lim_{t\to 0} I(t) , I is continuous at t=0.


(c)limt21I(t)=limt21(t,t)=(2,2)limt2+1I(t)=limt2+1(t,t2)=(2,4)(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 limt2r(t)lim_{t\to 2}r(t) does npt exist. Hence r is not continuousat t=2.


(d)




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Comments

Assignment Expert
10.05.21, 12:06

Dear devy, please use the panel for submitting a new question.

devy
06.05.21, 16:23

Let f be the function defined by f (x) = xe−x (a) Determine the y–intercept. (b) Determine the horizontal and vertical asymptotes. (c) Use the sign pattern for f'(x) to determine (i) the interval(s) over which f rises and where it falls; (ii) the local extrema. (d) Use the sign pattern for f ''(x) to determine (i) where the graph of f is concave up and where it is concave down (ii) the inflection point(s) (if any).

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