WORKSHEET 24 - Fall 19951.a) Find three distinct functionsf(x) such thatf0(x) = 0 for allx.b) Find three distinct functionsg(x) such thatg0(x)=4xfor allx.c) Find three distinct functionsh(x) such thath0(x)=3x−x2for allx.2. Letf1(x) be a function such thatf01(x)=sin(x2). (We do not yet know that there is such a function,but we will see that there is in about a week. We will then describe how to compute it, but we will neverbe able to write it down as a combination of any functions that you already know.) In the followingproblems, your answer will be in terms off1(x).a) Findf2(x) such thatf02(xx2)andf2(pπ/2) = 10.b) Findf3(x) such thatf03(xx2f3(p4) =−18.c) Draw the graph off2(x) as best you can from the information known.(Hint: consider the interval [√kπ,p(k+1)π] for each integerk.)3. Solve the following di±erential equations subject to the prescribed initial conditions.a)dydx=4(x−7)3,x=8,y=10b)dydx=x2+1x2=1c)dydx=x2y2=0d)dydx=4p(1 +y2)3y.(Hint: For c) and d), move anything involving the variable
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