Find the function with derivative f′(x)=e^(9x) that passes through the point P=(0,4/9).
Well, let's get started with finding the function that has a derivative of f′(x) = e^(9x). The derivative of e^(9x) is e^(9x), so we're looking for a function whose derivative is e^(9x).
To find the original function f(x), we can integrate e^(9x) with respect to x.
∫ e^(9x) dx = (1/9)e^(9x) + C
Where C is the constant of integration.
Now, let's use the point P(0,4/9) to find C. Plugging in x = 0 and y = 4/9 into the equation, we have:
(1/9)e^(9(0)) + C = 4/9
Simplifying this equation, we get:
(1/9)(1) + C = 4/9
1/9 + C = 4/9
C = 4/9 - 1/9
C = 3/9
C = 1/3
So the constant of integration is 1/3.
Finally, the function f(x) that satisfies f′(x) = e^(9x) and passes through the point P(0,4/9) is:
f(x) = (1/9)e^(9x) + (1/3)
And there you have it! A function that fits the bill. Now, I hope you find my explanation integrally amusing!
To find the function with the given derivative that passes through the point P=(0,4/9), we need to integrate the derivative.
We have f'(x) = e^(9x)
To integrate this function, we will use the power rule of integration.
The power rule states that ∫ e^(kx) dx = (1/k) * e^(kx) + C, where C is the constant of integration.
Applying the power rule, we get:
∫ e^(9x) dx = (1/9) * e^(9x) + C
Now, we can find the constant of integration by using the point (0,4/9).
When x = 0, f(x) = 4/9. Plugging these values into the equation, we have:
(1/9) * e^(9*0) + C = 4/9
(1/9) * 1 + C = 4/9
1/9 + C = 4/9
C = 4/9 - 1/9
C = 3/9
C = 1/3
Therefore, the function f(x) is:
f(x) = (1/9) * e^(9x) + 1/3
To find the function that satisfies the given derivative and passes through the point P=(0,4/9), we can integrate the derivative.
Let's start by integrating the derivative f′(x) = e^(9x). The integral of e^(9x) with respect to x is (1/9) * e^(9x) + C, where C is the constant of integration.
Now, we have an expression for the function that satisfies the derivative: f(x) = (1/9) * e^(9x) + C.
To determine the value of the constant C, we can use the fact that the function passes through the point P=(0,4/9). Substituting x=0 and f(x)=4/9 into the function, we have:
4/9 = (1/9) * e^(9 * 0) + C
4/9 = (1/9) * e^0 + C
4/9 = (1/9) * 1 + C
4/9 = 1/9 + C
To isolate C, we can subtract 1/9 from both sides:
C = 4/9 - 1/9
C = 3/9
C = 1/3
Therefore, the function that satisfies the given derivative and passes through the point P=(0,4/9) is:
f(x) = (1/9) * e^(9x) + 1/3
f'(x) = e^(9x)
f(x) = 1/9 e^(9x) + c
Now use point P to find c.