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merged 2 commits into from
Dec 18, 2024
Merged

Display powers of trig functions in usual way #496

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50949a7
TI3 task 2
siwelwerd Dec 18, 2024
fe59d3d
Refactor task 1
siwelwerd Dec 18, 2024
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50 changes: 34 additions & 16 deletions source/calculus/exercises/outcomes/TI/TI3/generator.sage
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Original file line number Diff line number Diff line change
@@ -1,32 +1,50 @@
#Functions to display powers of trig functions
def print_cosp(self,*args): return f"\\cos ^{{{args[1]}}}({latex(args[0])})"
def deriv_cosp(self,*args,**kwds):
if args[1]==1:
return -1*sinp(args[0],1)*args[0].derivative(args[kwds['diff_param']])
else:
return args[1]*-1*cosp(args[0],args[1]-1)*sinp(args[0],1)*args[0].derivative(args[kwds['diff_param']])

cosp = function("cosp",nargs=2,print_latex_func=print_cosp,derivative_func=deriv_cosp)
def print_sinp(self,*args): return f"\\sin ^{{{args[1]}}}({latex(args[0])})"
def deriv_sinp(self,*args,**kwds):
if args[1]==1:
return cosp(args[0],1)*args[0].derivative(args[kwds['diff_param']])
else:
return args[1]*sinp(args[0],args[1]-1)*cosp(args[0],1)*args[0].derivative(args[kwds['diff_param']])

sinp = function("sinp",nargs=2,print_latex_func=print_sinp, derivative_func=deriv_sinp)

class Generator(BaseGenerator):

def data(self):
# integral with odd power
# Task 1, integral with odd power
x=var("x")

even=2*randint(1,6)

odd=choice([5,7])
n=randint(1,6)
m=randint(2,3)

z = var("z")
if odd==5:
hint = "hint_2"
else:
hint = "hint_3"
k=var('k')
z=var('z')
hint=f"(1-z)^{m}={latex(sum(binomial(m,k)*(-z)^k,k,0,m))}"

trigs=[sin, cos]
shuffle(trigs)

f=randint(1,5)*(trigs[0](x))^even*(trigs[1](x))^odd
trigs=[sinp, cosp]
shuffle(trigs)

F=f.integral(x)
a = randint(1,6)
f = a*trigs[0](x,2*n)*trigs[1](x,2*m+1)
F = sum(a*binomial(m,k)*(-1)^k*1/(2*n+2*k+1)*trigs[0](x,2*n+2*k+1)*trigs[0](x,1).derivative(x)/trigs[1](x,1),k,0,m)


# integral with even powers
# Task 2, integral with even powers

a = randrange(2,6)
m = randrange(2,5)
n = randrange(2,5)
k = 2^randrange(2,5)
g = k*cos(a*x)^(2*m)*sin(a*x)^(2*n)
g = k*cosp(a*x,2*m)*sinp(a*x,2*n)
also_g = k/2^(m+n)*(1+cos(2*a*x))^m*(1-cos(2*a*x))^n
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Suggested change
also_g = k/2^(m+n)*(1+cos(2*a*x))^m*(1-cos(2*a*x))^n
also_g = k/2^(m+n)*(1+cos(2*a*x))^m*(1-cos(2*a*x))^n
assert f.diff() == also_g

Or something like that to ensure no errors were made? Or something that's more clearly the same as g?

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I don't think that will work, the way cosp is created sage really doesn't know that it's a power of cos. We lose a lot of that kind of thing by creating our own function.

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Hmm, it might be quicker to demo than explain what I'm trying to say, one sec

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Maybe there is a way to use assuming to do something. Maybe that's a separate issue t hough to go back and add checks like these wherever we are doing something by hand (across exercises)

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It was not quicker, so I think this is okay to use as-is.



Expand All @@ -35,5 +53,5 @@ class Generator(BaseGenerator):
"F": F,
"g": g,
"also_g": also_g,
hint: True,
"hint": hint,
}
10 changes: 2 additions & 8 deletions source/calculus/exercises/outcomes/TI/TI3/template.xml
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Original file line number Diff line number Diff line change
Expand Up @@ -3,14 +3,8 @@
<knowl>
<content>
<p>
Use trigonometric identities and
{{#hint_2}}
<m>(1-z)^2=1-2z+z^2</m>
{{/hint_2}}
{{#hint_3}}
<m>(1-z)^3=1-3z+3z^2-z^3</m>
{{/hint_3}}
to explain and demonstrate how to find
Using trigonometric identities and the fact that <m>{{hint}}</m>,
explain and demonstrate how to find
<m>\displaystyle \int {{f}} dx</m>.
</p>
</content>
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