Answer:
The equation for the curve is:
![f(x) = 7*e^{2*x}](https://tex.z-dn.net/?f=f%28x%29%20%3D%207%2Ae%5E%7B2%2Ax%7D)
Step-by-step explanation:
We know that for a curve defined as:
y = f(x)
The slope of the curve at the point x is:
y = f'(x)
where f'(x) = df(x)/dx
Here we know that we have a function that passes through the point (0, 7)
We also know that the slope of the curve at every point is twice the value of the y-coordinate. (remember that the y-coordinate is given by f(x))
Then we have two equations:
f(0) = 7
f'(x) = 2*f(x)
From the shape of the equation, we can assume than this is an exponential equation like:
![f(x) = A*e^{k*x}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20A%2Ae%5E%7Bk%2Ax%7D)
Replacing that in the second equation, we get:
![k*A*e^{k*x} = 2*A*e^{k*x}](https://tex.z-dn.net/?f=k%2AA%2Ae%5E%7Bk%2Ax%7D%20%3D%202%2AA%2Ae%5E%7Bk%2Ax%7D)
From that equation, we can conclude that k = 2
Then:
![f(x) = A*e^{2*x}](https://tex.z-dn.net/?f=f%28x%29%20%3D%20A%2Ae%5E%7B2%2Ax%7D)
Now we can use the first equation:
f(0) = 7
With this, we can find the value of A.
![f(0) = 7 = A*e^{2*0}\\7 = A*e^0 = A*1\\7 = A](https://tex.z-dn.net/?f=f%280%29%20%3D%207%20%3D%20%20A%2Ae%5E%7B2%2A0%7D%5C%5C7%20%3D%20A%2Ae%5E0%20%3D%20A%2A1%5C%5C7%20%3D%20A)
Then we can conclude that the equation for the curve is:
![f(x) = 7*e^{2*x}](https://tex.z-dn.net/?f=f%28x%29%20%3D%207%2Ae%5E%7B2%2Ax%7D)