Question

A curve passes through a point P(0, 3) and has the property that slope of curve at every point P is twice the y-coordinate of point P. Use the concept of exponential growth function to find the equation of curve.

Answer 1

The equation of curve is
`y = 3\cdot 7.389^(x)`.

Explanation:

The exponential growth function is represented by the expression described below:

`y = a\cdot b^(x)` (1)

Where:

`x` - Independent variable.

`y` - Dependent variable.

`a` - Initial value of
`y`.

`b` - Base.

By deriving (1), we obtain an expression for the slope of the curve:

`y' = a\cdot b^(x) \cdot \ln b` (2)

If we know that
`x = 0`,
`y = 3` and
`y' = 6`, then we have the following system of equations:

`a = 3` (3)

`a\cdot \ln b = 6` (4)

By (3) in (4):

`3\cdot \ln b = 6`

`\ln b = 2`

`b = e^(2)`

`b = 7.389`

Therefore, the equation of curve is
`y = 3\cdot 7.389^(x)`.