Answer:
The equation for the curve is:
[tex]f(x) = 7*e^{2*x}[/tex]
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:
[tex]f(x) = A*e^{k*x}[/tex]
Replacing that in the second equation, we get:
[tex]k*A*e^{k*x} = 2*A*e^{k*x}[/tex]
From that equation, we can conclude that k = 2
Then:
[tex]f(x) = A*e^{2*x}[/tex]
Now we can use the first equation:
f(0) = 7
With this, we can find the value of A.
[tex]f(0) = 7 = A*e^{2*0}\\7 = A*e^0 = A*1\\7 = A[/tex]
Then we can conclude that the equation for the curve is:
[tex]f(x) = 7*e^{2*x}[/tex]
