#1) exact approach
f(x)=1/x
g(x)=integrate(f(x),x)
show(f(x))
show(g(x))
show(integrate(f(x),x,1,3))
show(integrate(f(x),x,1,3).n())
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#1) approximate approach using rsum with 2 rectangles which overestimates
show(((1)*(1/1)+(1)*(1/2)))
show(((1)*(1/1)+(1)*(1/2)).n())
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#2) use 4 rectangles to sandwich the correct answer
show((1/2)*f(1)+(1/2)*f(1.5)+(1/2)*f(2)+(1/2)*f(2.5))
show((1/2)*f(1.5)+(1/2)*f(2)+(1/2)*f(2.5)+(1/2)*f(3))
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#Ex1.2.2) linear v(t) including (0,0), (20,88) units = ft/s
v(t)=88/20*t+0
plot(v(t),0,20)
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# units = ft
integrate(v(t),t,0,20)
\newcommand{\Bold}[1]{\mathbf{#1}}880
\newcommand{\Bold}[1]{\mathbf{#1}}880
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# units = (ft/s)/s = ft/s^2
diff(v(t),t).n()
\newcommand{\Bold}[1]{\mathbf{#1}}4.40000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}4.40000000000000
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#P1.2.1) linear velocity (0,66) to (60,0) units = ft/s
v2(t)=-66/60*t+66
plot(v2(t),0,60)
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integrate(v2(t),t,0,60)
\newcommand{\Bold}[1]{\mathbf{#1}}1980
\newcommand{\Bold}[1]{\mathbf{#1}}1980
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diff(v2(t),t)
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{11}{10}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{11}{10}
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#Problems 1.2.2)1a) LSUM = LUB
f(x)=4-2^x
(1)*(f(0))+(1)*(f(1))
\newcommand{\Bold}[1]{\mathbf{#1}}5
\newcommand{\Bold}[1]{\mathbf{#1}}5
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#Problems 1.2.2)1b) Exact
n(integrate(f(x),x,0,2))
\newcommand{\Bold}[1]{\mathbf{#1}}3.67191487733311
\newcommand{\Bold}[1]{\mathbf{#1}}3.67191487733311
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#Problems 1.2.2)2a) TRAP = GLUB
n((1)*(f(0)+f(1))/2 + (1)*(f(1)+f(2))/2)
\newcommand{\Bold}[1]{\mathbf{#1}}3.50000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}3.50000000000000
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#Problems 1.2.2)3) Find A(x) by accumulating areas
A1=1
A2=A1+1+0.5
A3=A2+1+1
A4=A3+1+0.5
A5=A4+1
show(A1)
show(A2)
show(A3)
show(A4)
show(A5)
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#Problems 1.2.2)4) cop catches speeder
cop_dist(t)=20*60/2/3600+60*(t-20)/3600
speeder_dist(t)=45*t/3600
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show(cop_dist(40))
show(speeder_dist(40))
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a=plot(cop_dist(x),xmin=20,xmax=50,ymin=0,ymax=1,color='red')
b=plot(speeder_dist(x),xmin=20,xmax=50,ymin=0,ymax=1,color='green')
a+b
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solve(speeder_dist(t)==cop_dist(t),t)
\newcommand{\Bold}[1]{\mathbf{#1}}\left[t = 40\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[t = 40\right]
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#p11)
var('k')
show([3*k+1 for k in range(2,6)])
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sum([3*k+1 for k in range(2,6)])
\newcommand{\Bold}[1]{\mathbf{#1}}46
\newcommand{\Bold}[1]{\mathbf{#1}}46
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sum = 0
for k in range(2,6):
sum = sum + 3*k+1
print sum
46 46 |
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