Find all the roots of the given function. Use preliminary analysis and graphing to find good initial approximations.

f(x) = cos (4x) - 4x^2 + 8x

The function has a root when x≈

Step-by-step explanation:

The quadratic portion of f(x) factors as (-4x)(x -2), so has zeros at x=0 and x=2. The cosine portion of f(x) will have a value of 1 at x=0 and a value of about -0.15 at x=2. Thus, we might expect roots to be slightly negative and near x=2. (It turns out that a not-unreasonable approximation of the cosine function as cos(x) ≈ 1-x²/2 can be usefully used to get better approximations of the roots in each case.)

A graphing calculator makes it easy to find initial approximations of the two roots. The graph shows them to be -0.108 and 1.988.

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Many graphing calculators also include the ability to determine a numerical value of the derivative of a function. This makes it possible to write an iteration function after the fashion of Newton's Method iteration:

  g(x) = x -f(x)/f'(x)

where g(x) gives a new value for old guess x, and f'(x) is the derivative of f(x).

The calculator we used is interactive, so the value of g(x) is found even as you type the argument value x. That makes it possible to achieve full calculator accuracy for the root estimate simply by copying the approximate value into the expression g(x).

  x ∈ {−0.107760269824, 1.98779489573}

45.3592

step-by-step explanation:

1

step-by-step explanation:

it's the only multiple of both and the least