CRL: Calc2 1.1 Riemann MRG 2012.0203

3551 days ago by calcpage123

#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()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{x}
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(3\right)
\newcommand{\Bold}[1]{\mathbf{#1}}1.09861228866811
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{x}
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(x\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\log\left(3\right)
\newcommand{\Bold}[1]{\mathbf{#1}}1.09861228866811
#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()) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3}{2}
\newcommand{\Bold}[1]{\mathbf{#1}}1.50000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{3}{2}
\newcommand{\Bold}[1]{\mathbf{#1}}1.50000000000000
#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)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}1.28333333333333
\newcommand{\Bold}[1]{\mathbf{#1}}0.950000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}1.28333333333333
\newcommand{\Bold}[1]{\mathbf{#1}}0.950000000000000
#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) 
       
# units = ft integrate(v(t),t,0,20) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}880
\newcommand{\Bold}[1]{\mathbf{#1}}880
# 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
#P1.2.1) linear velocity (0,66) to (60,0) units = ft/s v2(t)=-66/60*t+66 plot(v2(t),0,60) 
       
integrate(v2(t),t,0,60) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}1980
\newcommand{\Bold}[1]{\mathbf{#1}}1980
diff(v2(t),t) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{11}{10}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{11}{10}
#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
#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
#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
#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) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}1
\newcommand{\Bold}[1]{\mathbf{#1}}2.50000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}4.50000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}6.00000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}7.00000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}1
\newcommand{\Bold}[1]{\mathbf{#1}}2.50000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}4.50000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}6.00000000000000
\newcommand{\Bold}[1]{\mathbf{#1}}7.00000000000000
#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 
       
show(cop_dist(40)) show(speeder_dist(40)) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2}
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 
       
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]
#p11) var('k') show([3*k+1 for k in range(2,6)]) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left[7, 10, 13, 16\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[7, 10, 13, 16\right]
sum([3*k+1 for k in range(2,6)]) 
       
\newcommand{\Bold}[1]{\mathbf{#1}}46
\newcommand{\Bold}[1]{\mathbf{#1}}46
sum = 0 for k in range(2,6): sum = sum + 3*k+1 print sum 
       
46
46