a.
is a proper joint density function if, over its support, is non-negative and the integral of
is 1. The first condition is easily met as long as
. To meet the second condition, we require
b. Find the marginal joint density of and
by integrating the joint density with respect to
:
Then
c. This probability can be found by simply integrating the joint density:
Answer:
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}
