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Calculus: Continuity problem.

Posted 10 years ago

Find the values of a and b that make the given function continuous.

f(x)={a(tan^-1 x +2), if x<0, 
        2e^bx +1, if 0<(or equal to) x <(or equal to) 3, 
        ln(x-2) +x^2, if x>3

I cannot get WolframAlpha to understand this question and everyone I have asked for help also has not been able to figure it out, does anyone here know how to solve this or know how to make Wolfram understand it? Hopefully I worded this appropriately to actually get help this time. Thanks!

POSTED BY: Janet Gordon
4 Replies
POSTED BY: Daniel Lichtblau

Thank you for the help!

POSTED BY: Janet Gordon

I'm attempting to answer for the "Mathematica" listed above.

f[a_, x_] := a (ArcTan[x] + 2);(*x<0*)
g[b_, x_] := 2 Exp[b x] + 1;       (*0<=x<=3*)
h[x_]     := Log[x - 2] + x^2;  (*x>3*)

solb = b /. Solve[ Limit[g[b, x], x -> 3] == h[3], b][[1]] 
sola = a /. Solve[ Limit[ f[a, x], x -> 0] == g[b, 0], a][[1]]

Unfortunately the values I get namely a=Log[4]/3=0.46 and b=3/2 do not result in the three pieces having the same slope at the "joins". Enforcing the same slope at the two joins gave a=1.18 and b=0.59. Therefore I don't think there is a solution. Maybe someone else will post a better solution.

POSTED BY: Isaac Abraham

Thank you for the input.

POSTED BY: Janet Gordon
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