Prepared : Wed Nov 17 16:58:22 CST 1999

MATH 1271

QUIZ VI

1) Use a linear approximation L(x) to an appropriate function f(x), with an appropriate value of a, to estimate

The function will be and a will be 100.(Choose a in such a way that f will be easy to evaluate at that point.) Then the linear approximation formula is

Now let x=102 to give the approximation.

2) Determine the open intervals on the x-axis on which the function

is increasing as well as those on which the function is decreasing.

Increasing/decreasing property of a function is determined by its first derivative.

Hence the zeros of f' are 0, -2 and -1 (Be careful here, actually what we are after are the critical points of the function, which are by definition either the zeros of f' or the points where f' is undefined. But here f' is a polynomial, hence everwhere defined.) So we only need to investigate the sign of f' in the intervals , (-2, 0), (0,1), and .

Pick one point from each interval, plug that into f' and the sign of that will be the sign of f' in the whole interval. Explicitly:

In the interval : Pick point -3. Plug that into f', it gives f'(-3)=-144 and this tells us that in the interval the sign of f' is negative. In terms of the function itself this means that the function is decreasing in that interval.

In the interval (-2, 0): Pick point -1. Plug that into f', it gives f'(-1)=24 and this tells us that in the interval (-2, 0) the sign of f' is positive. In terms of the function itself this means that the function is increasing in that interval.

In the interval (0,1): Pick point 1/2. Plug that into f', it gives f'(1/2)=-15/2 and this tells us that in the interval (0,1) the sign of f' is negative. In terms of the function itself this means that the function is decreasing in that interval.

In the interval : Pick point 2. Plug that into f', it gives f'(2)=96 and this tells us that in the interval (-2, 0) the sign of f' is positive. In terms of the function itself this means that the function is increasing in that interval.

3) Apply the first derivative test to classify each of the critical points of the function

If you have a graphing calculator, you can check your results by plotting the given function.

The first derivative test is given on page 240 of the text, so I will not rewrite the whole theorem, but rather just the basics.

1. If f' is negative before c and positive after c, then f(c) is a local minimum.

2. If f' is positive before c and negative after c, then f(c) is a local maximum.

Since the first derivative test (obviously) depends on the first derivative, let's first find f'.

The only critical point is 1, where f' is zero. But that point does not satisfy the conditions of the either case, hence it is not an extremum point. Done.