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Skill check: Mean value theorem, L'Hopital, and local linearization

Let the function f(x)=x2f(x)=x^2 be defined on the closed interval [1,5][1,5]. For what value of cc, c(1,5)c \in (1,5), is the slope of the tangent to the graph of ff equal to the slope of the secant line connecting (1,f(1))\left(1,f(1)\right) and (5,f(5))\left(5,f(5)\right)?
  • Your answer should be
  • an integer, like 66
  • an exact decimal, like 0.750.75
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4