Calculus UNIT501A: What is Optimization?

2749 days ago by MATH4R2013

#29 p231 show that f(x)=a*x^3+b*x^2+c*x+d has 2 or 0 extrema! var('a,b,c,d') f(x)=a*x^3+b*x^2+c*x+d g(x)=diff(f(x),x) show(f(x)) show(g(x)) show(solve(g(x)==0,x)) #when b^2-3ac>0 f'(x)--0 has 2 real roots, so f(x) has 2 extrema! #when b^2-3ac<0, f'(x)==0 has 2 complex roots, so f(x) has 0 extrema!! 
       


#23 p231 maximize profit given revenue, r(x)=8*sqrt(x) and cost, c(x)=2*x^2 r(x)=8*sqrt(x) c(x)=2*x^2 p(x)=r(x)-c(x) rp(x)=diff(r(x),x) cp(x)=diff(c(x),x) pp(X)=diff(p(x),x) show(r(x)) show(c(x)) show(p(x)) show(rp(x)) show(cp(x)) show(pp(x)) show(solve(pp(x)==0,x)[2]) show(p(1)) p1=plot(p(x),.1,4,color='red') p2=plot(pp(x),.1,4,color='green') #since p(x)=r(x)-c(x), p'(x)=r'(x)-c'(x) #so p(x) has a min or a max when r'(x)=c'(x)! #max occurs at x=1 (thousand widgets) #and max = p(1) = 6 (mega bux) p1+p2 
       








#20) minimize the time T(x)=t1(x)+t2(x) it takes to row to a village V #20) if you are 2 miles off shore in a boat B and the village is 6 miles from the point closest to the shore S #20) and you row 2mph but walk 5mph! Let L be the point of land fall, LS=x, VS=6, VL=6-x and SB=2. #20) V========================L=========S #20) d1=6-x + x + #20) + + #20) + + #20) + + #20) d2 + + 2 #20) + + #20) + + #20) + + #20) ++ #20) B d1(x)=sqrt(x^2+4) d2(x)=6-x t1(x)=d1(x)/2 t2(x)=d2(x)/5 T(x)=t1(x)+t2(x) Tp(x)=diff(T(x),x) show(T(x)) show(Tp(x)) show(solve(Tp(x)==0,x)) show(solve(25*x^2==4*x^2+16,x)[1].rhs().n())