1. The problem statement, all variables and given/known data
1. The slope of a function at any point (x,y) is 〖2e〗^x/(e^x+2) . The point (0,2ln3) is on the graph of f.
(A) Write an equation of the tangent line to the graph of f at x=0.
(B) Use the tangent line in part A to approximate f(0.3) to the nearest thousandth.
(C) Solve the differential equation dy/(dx )=(2e^x)/(e^x+2) with the initial condition f(0)=2ln3.
(D) Use the solution in part C to find f(0.3) to the nearest thousandth.
2. Relevant equations
3. The attempt at a solution
(A)
dy/(dx )=(2e^x)/(e^x+2)
At x=0, dy/dx=(2e^0)/(e^0+2)=2/3
Equation of tangent line at x=0: y2ln3=2/3 (x0)
y2ln3=2/3 x or y=2/3 x+2ln3
(B)
f(0.3)≈2/3 (0.3)+2ln3≈2.39722≈2.397
(C)
dy/dx=〖2e〗^x/(e^x+2) → dy=(2e^x)/(e^x+2) → ∫dy= ∫〖(2e^x)/(e^x+2) dx〗
Let u=e^x+2, du=e^x dx
2du=2e^x
Here is where I run into problems. Can anybody help me with the rest? Any help would be greatly appreciated.
