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Feb27-12, 10:51 PM
P: 3
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


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: y-2ln3=2/3 (x-0)

y-2ln3=2/3 x or y=2/3 x+2ln3


f(0.3)≈2/3 (0.3)+2ln3≈2.39722≈2.397


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

Here is where I run into problems. Can anybody help me with the rest? Any help would be greatly appreciated.
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