Simplify logba×logcb× logac. A continuous function is defined by the following features: d2ydx2>0 for <- 1 and 1<x<3. dydx =0 only when x=-3,1 and 5. y= 0 only when x=1.
Sketch a possible graph of the function. Use Simpson’s rule with three function values to estimate ∫13 log10 xdx . Give your answer correct to three significant figures.
(i) Differentiate logexx
(ii) The curve y =logexx in the domain 1≤x ≤e is rotated about the x-axis. Using the result in (i) Find the volume of the solid formed
logaa×logab×logac =logaalogab×logablogac×logac =logaa =1
f(x)=log10x f(x) 123log101log102log103 ∫13f(x)dx=3-16log101+4log102+log103 =0.564.... =0.56035f i y=logexx u1=lnx v1=x u=1x v=1 dydx=x.1x-lnxx2 =1-lnxx2 ii v=π∫1e(logexx)2dx =π∫1elnxx2dx from(i), lnxx2=-(1-lnxx2-1x2) V=-π∫1e1-lnxx2-x-2dx =-πlnxx+1x1e =-πlnee+1e-ln11+1 =-π2e-1 =π-2πeunits3
10 Questions/Answers
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