SharedLab 5 / Lab5-turnin.sagewsOpen in CoCalc
Author: Nathaniel Song
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# Lab 5: # Name: Nathaniel Song # I worked on this code with: # Please do all of your work for this week's lab in this worksheet. If # you wish to create other worksheets for scratch work, you can, but # this is the one that will be graded. You do not need to do anything # to turn in your lab. It will be collected by your TA at the beginning # of (or right before) next week’s lab. # Be sure to clearly label which question you are answering as you go and to # use enough comments that you and the grader can understand your code.1 #1 %auto typeset_mode(True, display=False) def weird_function(): pieces = [sin(x), cos(x), arctan(x), ln(x), sqrt(x), exp(x)] f(x)=prod([choice(pieces) for i in range(5)]) + prod([choice(pieces) for i in range(5)]) return f h(x)=weird_function()
#2 (h(5.5)-h(3.5))/(5.5-3.5)
162.643088454926162.643088454926
#3 deltax=[1,.1,.01,.001] vals=[] for y in deltax: p=3.5 a=(h(p+y)-h(p))/y vals.append(a) vals
[63.7603199280383-63.7603199280383, 19.054675259873219.0546752598732, 26.538343839014526.5383438390145, 27.239344615210827.2393446152108]
#4 p=2 vals
[63.7603199280383-63.7603199280383, 19.054675259873219.0546752598732, 26.538343839014526.5383438390145, 27.239344615210827.2393446152108]
#5 def function(p): f(t)=1000*t^2 deltax=[1,.1,.01,.001] vals=[p] for y in deltax: b=((p+y)-p)/y vals.append(b) return vals function(1)
[11, 11, 1.000000000000001.00000000000000, 1.000000000000001.00000000000000, 0.9999999999998900.999999999999890]
#6 def function(p): f(t)=1000000*t^2 deltax=[1,.1,.01,.001] vals=[p] for y in deltax: b=((p+y)-p)/y vals.append(b) return vals function(1)
[11, 11, 1.000000000000001.00000000000000, 1.000000000000001.00000000000000, 0.9999999999998900.999999999999890]
#7 #Limits are necessary when computing derivatives because that is what allows the approximation to be as accurate to zero as possible without actually hitting zero #8 (x^5)/(x^3) x^2 #9 type(x) #10 factor(x^2 + 7*x + 6)
x2x^{2}
x2x^{2}
<type 'sage.symbolic.expression.Expression'>
(x+6)(x+1){\left(x + 6\right)} {\left(x + 1\right)}
#11 var("k") factor(k^2 - 5*k + 6)
kk
(k2)(k3){\left(k - 2\right)} {\left(k - 3\right)}
#12 var("n") f(x)=2^n plot(f(n),(n,-10,10))
nn
#13 #because x is defined as a number and not a variable, so y will then become a result of the funtion #14 i(x)=16*x^2 plot(i(x),(x,0,5)) + point([1,16], color="red",size=40)
#15 plots=[] xmax=[5,3,2,1,.25] for n in xmax: p=plot(i(x),(x,0,5)) + point([1,16], color="red",size=40) plots.append(p) animate(plots)
#16 var("n") @interact def graph(n=(1.1,10)): l=16*x^2 slope=(l(n)-16)/(n-1) m=(slope*(x-n))+l(n) n=plot(16*x^2,(x,0,10))+ plot(m, (x,0,10),ymin=0 ) + point([1,16], color="red", size=40) + point([n,(l(n))], color="red", size=40) show(n) #as the movable point approaches the fixed one, the approximation of the curve at the point (1,16) get more accurate
n
Interact: please open in CoCalc
#17 var("n") @interact def graph(n=(1.1,10)): l=16*x^2 slope=(l(n)-16)/(n-1) m=(slope*(x-n))+l(n) pt=[n,l(n)] n=plot(16*x^2,(x,0,10))+ plot(m, (x,0,10),ymin=0 ) + point([1,16], color="red", size=40) + point([n,(l(n))], color="red", size=40) + text(slope,pt) show(n)
n
Interact: please open in CoCalc
#18 var("n") @interact def graph(n=(1.1,10)): l=16*x^2 slope=(l(n)-16)/(n-1) m=(slope*(x-n))+l(n) pt=[n,l(n)] pts=[1,16] n=plot(16*x^2,(x,0,10))+ plot(m, (x,0,10),ymin=0 ) + point([1,16], color="red", size=40) + point([n,(l(n))], color="red", size=40) + text(slope,pt) +text(32,pts) show(n)
n
Interact: please open in CoCalc
#19 plots=[] points=[-3,-2,-1,1,2,3] equ=3*x^2 for q in points: df=diff(3*x^2,x) line=(df(q)*(x-q)+equ(q)) p=plot(3*x^2,(x,-5,5),ymin=-5,ymax=5)+plot(line,(x,-5,5),ymin=-5,ymax=20) plots.append(p) animate(plots)
plots=[] points=[-3,-2,-1,0,1,2,3] equ=3*x^2#its always a good idea to define the equation separately somewhere so you dont have to type it over and over for q in points: df=diff(3*x^2,x) #you have to put q into the differential equation to get the slope for the line at that point line=(df(q)*(x-q))+equ(q)#this is the equation for the line p=plot(3*x^2,(x,-5,5), ymin=-5, ymax=20)+plot(line, (x,-5,5), ymin=-5, ymax=20)#replace df (which will always be the same) with line. Always specify both plots with the same boundaries so that the picture doesn't look wonky or shift plots.append(p) animate(plots)