Show your understanding of the Mean Value Theorem as follows. Check that the con

Question

Show your understanding of the Mean Value Theorem as follows. Check that the conditions (hypotheses) of the Mean Value Theorem hold for the function f(x) = (3x - 1)1/2 on the interval [2,4], and deduce that there must be at least one number c in the interval [2,4] such that Now find a point c in [2, 4] such that Give your answer both in exact form and to four decimal places. (Unrelated to parts (a) and (b)) Use the Fundamental Theorem of Calculus (version 2) to find the derviative with respect to x of the function F(x) defined by

Explanation / Answer

a) f(x) = (3x - 1)^(1/2)

Mean value theorem says, for x in [a, b]

there must be a c such that,

f ' (c) = ( f(b) - f(a) ) / (b - a)

Here, in [2, 4]

f(4) = sqrt(11)

f(2) = sqrt(5)

So, there must be a c in [2, 4] such that,

f ' (c) = ( f(4) - f(2) ) / (4 - 2)

= ( sqrt(11) - sqrt(5) ) / 2

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b) f(x) = (3x - 1)^(1/2)

f ' (x) = 1 / (2 * (3x-1)^(1/2) ) * 3

= (3/2) * 1 / (3x - 1)^(1/2)

f ' (c) = (3/2) * 1 / (3c - 1)^(1/2)

( sqrt(11) - sqrt(5) ) / 2 = (3/2) * 1 / (3c - 1)^(1/2)

(3c - 1)^(1/2) = 3 / ( sqrt(11) - sqrt(5) )

(3c - 1) = 9 / (16 - 2sqrt(55) )

c = (25 - 2sqrt(55) ) / (48 - 6sqrt(55) )

In exact form, c = (25 - 2sqrt(55) ) / (48 - 6sqrt(55) )

To four decimal places, c = 2.9027

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c) dF / dx = sqrt( sin(2x^2) ) * 4x - sqrt( sin(x-5) ) * 1

= 4x sqrt( sin(2x^2) ) - sqrt( sin(x-5) )

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