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.


1
Expert's answer
2021-05-11T09:22:54-0400

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

    "=(t,t) \\text{ if } t\\in (0,2)\\\\\n\n = (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]\n\n\n\n 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)\n\n \\\\[9pt]\n\n 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)




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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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