View Single Post
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.
Phys.Org News Partner Science news on
NASA team lays plans to observe new worlds
IHEP in China has ambitions for Higgs factory
Spinach could lead to alternative energy more powerful than Popeye