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Find all numbers c that satisfy the mean value theorem, f(x)=e^2x, [0,3]
 2 years ago
 2 years ago
Find all numbers c that satisfy the mean value theorem, f(x)=e^2x, [0,3]
 2 years ago
 2 years ago

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AntiMatterBest ResponseYou've already chosen the best response.1
mean value theorem states, that if a function is continuous and differentiable on a given interval [w,z], then there exists a value v, where the derivative of the value of the derivative of the function at v is equal to \[f'(v) = (f(z)  f(w))/(zw)\]
 2 years ago

AntiMatterBest ResponseYou've already chosen the best response.1
all right, so if \[f(x)=e^{2x}\] differentiable on [0,3]?
 2 years ago

AntiMatterBest ResponseYou've already chosen the best response.1
i'd say yes, we can graph the function and it is continuous and differentiable
 2 years ago

AntiMatterBest ResponseYou've already chosen the best response.1
so, \[f'(x) = 2 * e^{2x}\]
 2 years ago

AntiMatterBest ResponseYou've already chosen the best response.1
so evaluate f(3) and f(0)
 2 years ago

AntiMatterBest ResponseYou've already chosen the best response.1
solve for the value c, such that \[f'(c) = 2*e^{2x} = (f(3)  f(0))/(3  0)\]
 2 years ago

AntiMatterBest ResponseYou've already chosen the best response.1
\[f(3) = e^{6}, f(0) = 1\]
 2 years ago

AntiMatterBest ResponseYou've already chosen the best response.1
so \[(e^6  1) / 3 = 2*e^{2*c}\] solve for c!
 2 years ago
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