Let's begin by understanding the problem. We know the derivative of some function, which is f'(x) = 4x²+7x−9, and we know that the point (0,1) is on the graph of the original function, which we'll call f(x). We need to use these pieces of information to find the equation for the original function.
The first step in solving this problem is to recognize the relationship between a function and its derivative. If f'(x) = 4x²+7x−9 is the derivative of f(x), then that means f(x) is the antiderivative of f'(x).
The antiderivative is found by performing the opposite operation of the derivative, which is integration. By integrating f'(x), we can get f(x). The integration yields f(x) = 4x³/3 + 7x²/2 - 9x.
We now have our base function, but this is not the final answer. When you take an antiderivative, there is always an unknown constant, which we'll denote as 'c'. So, the more complete equation of our function at this point is f(x) = 4x³/3 + 7x²/2 - 9x + c.
We must find the value of 'c' to get the final function. We do this by substituting the given coordinates of the point on the curve (in this case, (0, 1)) into the equation, and solving for 'c'.
When we substitute x=0, we get f(0) = c, because all other terms drop out. Additionally, we know f(0) should be equal to 1, because the point (0,1) is on the curve. Therefore, c = 1.
Substituting that into our equation for f(x), we get the final equation of the curve:
f(x) = 4x³/3 + 7x²/2 - 9x + 1.
That is the conclude of our calculation process.