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Project: MTH161
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The limit of nn sides regular convex polygon is a circle when nn approaches to infinity.

@interact def _(n=slider(int(3),int(20),1,default=int(3),label='The number of the corners $=$')): n=int(n) p1=circle((0,0),1,color='black') corner=[[cos(2*i*pi/n),sin(2*i*pi/n)] for i in range(n)] p2=polygon2d(corner,fill=False) show(p1+p2,axes=False)
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Find the deciemal expression of 1729/2018

x=1729/2018.n(digits=1000) print(x)
0.8567888999008919722497522299306243805748265609514370664023785926660059464816650148662041625371655104063429137760158572844400396432111000991080277502477700693756194251734390485629335976214073339940535183349851337958374628344895936570862239841427155599603567888999008919722497522299306243805748265609514370664023785926660059464816650148662041625371655104063429137760158572844400396432111000991080277502477700693756194251734390485629335976214073339940535183349851337958374628344895936570862239841427155599603567888999008919722497522299306243805748265609514370664023785926660059464816650148662041625371655104063429137760158572844400396432111000991080277502477700693756194251734390485629335976214073339940535183349851337958374628344895936570862239841427155599603567888999008919722497522299306243805748265609514370664023785926660059464816650148662041625371655104063429137760158572844400396432111000991080277502477700693756194251734390485629335976214073339940535183349851337958374628344895936570862239841427

Plot the function 2x+6 on the interval (-5,0)

Plot the function x2+5x+6 for x[4,1]x\in[-4,-1]

plot(x^2+5*x+6,(x,-4,-1))
plot(x^3+5*x^2-138*x-304,(x,-20,15))

Find the roots of the polynomial x3+5x2138+304x^3+5x^2-138+304

eq=x^3+5*x^2-138*x+304==0 eq.roots(ring=RR)
[(-15.3103790525455, 1), (2.56286092405128, 1), (7.74751812849423, 1)]
%md Plot the graph of $|x|$ on the interval $(-3,3)$.

Plot the graph of x|x| on the interval (3,3)(-3,3).

Find the roots of x2+7x+61|x^2+7x+6|-1.

g=abs(x^2+7*x+6)-1==0 g.roots(ring=RR)
Error in lines 2-2 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1013, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "sage/symbolic/expression.pyx", line 11256, in sage.symbolic.expression.Expression.roots (build/cythonized/sage/symbolic/expression.cpp:62968) p = self.polynomial(ring) File "sage/symbolic/expression.pyx", line 6698, in sage.symbolic.expression.Expression.polynomial (build/cythonized/sage/symbolic/expression.cpp:41227) return polynomial(self, base_ring=base_ring, ring=ring) File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1290, in polynomial res = converter() File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 220, in __call__ return self.relation(ex, operator) File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1189, in relation return self(ex.lhs()) - self(ex.rhs()) File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 218, in __call__ return self.arithmetic(ex, operator) File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1227, in arithmetic ops = [self(a) for a in ex.operands()] File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 226, in __call__ return self.composition(ex, operator) File "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/symbolic/expression_conversions.py", line 1164, in composition return self.base_ring(ex) File "sage/structure/parent.pyx", line 939, in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9641) return mor._call_(x) File "sage/structure/coerce_maps.pyx", line 282, in sage.structure.coerce_maps.NamedConvertMap._call_ (build/cythonized/sage/structure/coerce_maps.c:6448) cdef Element e = method(C) File "sage/symbolic/expression.pyx", line 1282, in sage.symbolic.expression.Expression._mpfr_ (build/cythonized/sage/symbolic/expression.cpp:10799) return self._eval_self(R) File "sage/symbolic/expression.pyx", line 1209, in sage.symbolic.expression.Expression._eval_self (build/cythonized/sage/symbolic/expression.cpp:10444) raise TypeError("Cannot evaluate symbolic expression to a numeric value.") TypeError: Cannot evaluate symbolic expression to a numeric value.

Plot the graph of x2+7x+61|x^2+7x+6|-1 on the interval (8,0)(-8,0).

g1=plot(abs(x^2+7*x+6)-1,(x,-8,0),thickness=4,legend_label='$|x^2+7x+6|-1$') g2=plot(x^2+7*x+6-1,(x,-8,0),thickness=1.5,color='red',legend_label='$x^2+7x+6-1$') g3=plot(-x^2-7*x-7,(x,-8,0),thickness=1.5,color='green',legend_label='$-x^2-7x-7$') g=g1+g2+g3 g.show(figsize=10)
(-7+((21)^(0.5)))/2.n()
-1.20871215252208