Question

Use a linearization to estimate 2.03^5.

Find the differential of f(?) = ?^2 − 3x.

Find the absolute maximum and minimum values of ?(?) = ?^3 − 3?2 on the interval [-1,3].

Answer 1

1. We have to use linearization to estimate (2.03)⁵

Now, consider the function

So we have

Then, a linear approximation of the function f(x) around x=2 is given by

So, now we set

then note that

which is exactly the value we need to estimate. So, our linear approximation for it is given by

Hence a linear estimate for 2.03⁵ is 34.4

2. We have the function

so

Then, for any small change in x, the corresponding change in y, called the differential, is given by

which is our required answer.

3. We have to find the maxima and minima of in the interval [-1,3]

Now, for any extreme point, the first derivative vanishes, so

Thus the function has extremums at x=0 and x=2. Now we use the second derivative tests at each of these points to see if they are a maxima or a minima.

Firstly, the second derivative is , so

Hence the function has a maxima at x=0

and

Hence the function has a minima at x=2

Now, as we have to find the global maximum and minimum in the given interval, we check the values of the function at each of the end points and at the evaluated maxima and minima:-

Thus, the functional values at the evaluated maxima and the right end point are the same, and the left end point and the minima are the same.

Hence the given function has an absolute minimum of -4 attained at x=-1 and x=2 ; and an absolute maximum of 0 attained at x=0 and x=3