MiniProjectsCh6.sagews

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# Mini project: Analysis of a function f(x)=x/(x+1)-cos(40/x) with many characteristic features
#Step 1 (manual)
#Domain= # Complete the line
# lim_(x->infinity)=0 lim_(x->-infinity)=0

#Step 2:Plot the function
f(x)=x/(x+1)-cos(40/x)
p=plot(f(x),-4,10,ymin=-10,ymax=10);p.show(figsize=[5,4])

#Step 3. Find the smallest negative and the largest positive x-intercepts for this function. (Note that the large intercept is REALLY large)
                        #the smallest negative
p=plot(f(x),-0.5,-0.4);p.show(figsize=[5,4])
                       

find_root(f(x),-0.48,-0.47)

-0.4740729100063728

#THE largest positive
                        p=plot(f(x),800.7,800.9);p.show(figsize=[5,4])

find_root(f(x),100,1000)

800.8324463325995

#Step 4 (read Lecture 4, answer the question on Step 4)
# No, we cannot due to the fact that there are numerous solutions.

#Step 5. Find the largest local minimum on the interval (-infinity,-1).
deriv_plot=plot(derivative(f(x),x),-1.35, -1); deriv_plot.show(figsize=[5,4],ymin=-25,ymax=55)
print"Largest Local Minimum"
find_root(derivative(f(x),x),-1.35,-1)

Largest Local Minimum -1.294057347382484