Mean Value Theorem: Estimating f(0.1)

[cheezeitz]

Homework Statement

Let f be a function defined on an interval [a,b]. min f' <= f(b) - f(a) / b -a <= max f' where min f' and max f' refer to the mininmum and maximum values of f' on [a,b]

Then the question is
Using that inequality; estimate f(.1) if f'(x) = 1 / (1 + x^4 cos x) for 0 <= x <= .1 and f(0) = 1

The Attempt at a Solution

I really don't know where to start with this, I have min f'(x) <= f'(c) <= max f'.
The book says to use a calculator too.

 

[Mark44]

For your interval, a = 0 and b = .1. You are given that f'(x) = 1/(1 + x^4 cos x). Clearly f'(0) = 1, and this will be the maximum value of f' on the interval in question. Use your calculator to get an approximate value for f'(.1).

In the following inequality, min f' and max f' refer to the minimum and maximum of f' on the interval [0, .1].
min f' <= (f(.1) - f(0))/.1 <= max f'
==> .1*(min f') <= f(.1) - f(0) <= .1*(max f')

Now add f(0), which is given, to all members of this inequality to get a lower bound and an upper bound on f(.1).

 

[LCKurtz]

Hint: State the mean value theorem if a = 0 and b = .1 and use what you are given.