The following diagrams represent mixtures of NO(g) and O2(g). These two substances react as follows:
It has been determined experimentally that the rate is second order in NO and first order in O2. Based on this fact, which of the following mixtures will have the fastest initial rate? [Section 14.3]
MATH 2450 WEEK 6 Chain Rule in Higher Dimensions g(t) = df(x(t) , y(t))= ∂f * dx + ∂f * dy dt ∂x dt ∂y dt R(t) = < x(t), y(t), f( x(t), y(t) ) df(g(t)) = df * dg dt dg dt 2 EX. Use the chain rule to find the df/dt of the function f(x,y) = x + 3xy where x(t) = cos(t) and y(t) = sin(t) Fx= 2x + 3y Fx= 3x dx/dt = -sin(t) dy/dt = cos(t) Now plug in the values you found above into the chain rule equation g(t) = ∂f * dx + ∂f * dy ∂x dt ∂y dt g(t) = (2x + 3y) (-sin(t)) + (3x) (cos(t)) Now make everything is terms of ‘t’. Remember to replace the ‘x’ and ‘y’ with their ORIGINAL values(x(t) = cos(t) and y(t) = sin(t)) not their derivatives g(t) = (2cos(t) + 3sin(t)) (-sin(t)) + (3cos(t)) (cos(t)) g(t) = -2sin(t)cos(t) – 3sin (t) + 3cos (t) Chain Rule in higher dimensions with 3 variables F(x,y,z) R(t) = (t) = df/dt = x * x’(t) +yf * y’(t) z f * z’(t) EX. Use the chain rule to find the df/dt of the function f(x, y, z) = e x*y^(2)*zwhere x(t)= t ,2 y(t)=e, and z(t) = cos(t) 2 x*y^(2)*z F x y *z*e F y 2*x*y*e x*y^(2)*z F z y *x*e x*y^(2)*z x’(t) = 2t y’(t) = e t z’(t) = -sin(t) Plug in the values into the chain rule equation df/dt = f x x’(t) + f *yy’(t) + f * zz(t) = (y *z*e x*y^(2)*)(2t) + (2*x*y*e x*y^(2)*)(e) + (y *x*e x*y^(2)*)(-sin(t))
