Exercises 3.4⇒x(t) = 98.1t+ 981e−t/10−981.To find out when the object will hit the ground, we solvex(t) = 1000 fort. Therefore, wehave1000 = 98.1t+ 981e−t/10−981⇒98.1t+ 981e−t/10= 1981.In this equation, if we ignore the term 981e−t/10we will find thatt≈20.2. But this meansthat we have ignored the term similar to 981e−2≈132.8 which we see is to large to ignore.Therefore, we must try to approximatet. We will use Newton’s method on the equationf(t) = 98.1t+ 981e−t/10−1981 = 0.(If we can find a root to this equation, we will have found thetwe want.) Newton’s methodgenerates a sequence of approximations given by the formulatn+1=tn−f(tn)f(tn).Sincef(t) = 98.1−98.1e−t/10= 98.1(1−e−t/10), the recursive equation above becomestn+1=tn−tn+ 10e−tn/10−(1981/98.1)1−e−tn/10.(3.10)To start the process, let
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