Question #cc09c
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"Suppose that I don't have a formula for #g(x)# but I know that #g(1)
= 3# and #g'(x) = sqrt(x^2+15)# for all x. How do I use a linear approximation to estimate #g(0.9)# and #g(1.1)#?"
The next number is 25.
The formula is
#y=200(1/2)^x, x in NN#
This is an exponential decay or half-life type of question. The number halves each time, so the number after 50 is 25.
The general equation for exponential decay can be expressed as
#y=ab^x#
where #a# is the initial amount, and #b# is the base. In this case we want a base of 1/2, because the number is halving each time.
If the number sequence was 200, 50, 12.5, etc. we would use a base of 1/4 because the number is being quartered each time. The initial amount can also be any number, but in this case #a=200#. Plugging these into the equation, we get
#y=200(1/2)^x, x in NN#
#NN# are the natural numbers (positive integers including zero).
Notice that we had to restrict the values that #x# can take because there are a discrete number of values that we want. For example, #x# can't be 1.5, because that would give a number that is not one of our answers.
The graph looks like this
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The important thing here is that #x# can only take on whole number values, which is why we restricted the domain to natural numbers. The line drawn through the points is only there because I don't know how to turn it off.
