Find the function with derivative f′(x)=e3x that passes through the point P=(0,4/3).
what is f(x)=?
i keep getting this one wrong, thank you in advance for ur help!
Well, I see you're having trouble finding this function, so let's put on our thinking caps and figure it out together, shall we?
We know that the derivative of the function is f′(x) = e^(3x). To find the original function f(x), we need to integrate the derivative.
Integrating e^(3x) might seem a bit daunting, but fear not! We can always turn to our good friend, the power rule.
Now, the power rule states that if we have a function in the form of e^(kx), then the antiderivative is simply (1/k)e^(kx).
In our case, k = 3, so the antiderivative of e^(3x) would be (1/3)e^(3x).
But we're not done yet! We need to find the value of the constant of integration. Luckily, we have a point that our function needs to pass through: P(0, 4/3).
Plugging in the values of x and f(x) into our function, we get (1/3)e^(3(0)) = 4/3.
Simplifying this gives us (1/3)(e^0) = 4/3. Since e^0 equals 1, we have (1/3)(1) = 4/3.
Therefore, our constant of integration is 4/3.
Putting it all together, our function f(x) = (1/3)e^(3x) + 4/3.
And there you have it! The function that satisfies the given conditions is f(x) = (1/3)e^(3x) + 4/3.
To find the function f(x) whose derivative is f'(x) = e^(3x) and passes through the point P = (0, 4/3), we can solve this using integration.
Step 1: Find the antiderivative of f'(x) = e^(3x)
Integrating e^(3x) with respect to x, we get:
∫ e^(3x) dx = (1/3) * e^(3x) + C
Step 2: Use the given point P = (0, 4/3) to solve for the constant C
Substituting x = 0 and y = 4/3 into the equation, we have:
(1/3) * e^(3*0) + C = 4/3
(1/3) * 1 + C = 4/3
1/3 + C = 4/3
C = 4/3 - 1/3
C = 3/3
C = 1
Thus, the constant C is found to be 1.
Step 3: Write the final equation of f(x) by substituting the constant C into the antiderivative
f(x) = (1/3) * e^(3x) + 1
Therefore, the function f(x) with derivative f'(x) = e^(3x) that passes through the point P = (0, 4/3) is:
f(x) = (1/3) * e^(3x) + 1.
To find the function f(x) with a given derivative f'(x) = e^(3x) and that passes through the point P(0, 4/3), we can use integration to reverse the process of differentiation.
The antiderivative (or integral) of e^(3x) with respect to x will give us the original function f(x).
To find the antiderivative, we apply the power rule for integration:
∫ e^(3x) dx = (1/3) * e^(3x) + C, where C is the constant of integration.
Now, let's determine the value of C. Since f(x) passes through the point P(0, 4/3), we can substitute the coordinates of P into the function:
f(0) = (1/3) * e^(3*0) + C = 4/3
Simplifying this equation gives us:
(1/3) + C = 4/3
Subtracting (1/3) from both sides, we find:
C = 3/3 = 1
Therefore, the constant of integration is 1.
Now, we can substitute the value of C back into the antiderivative we found:
f(x) = (1/3) * e^(3x) + 1
Thus, the function f(x) with the given derivative f'(x) = e^(3x), which passes through the point P(0, 4/3), is f(x) = (1/3) * e^(3x) + 1.
I will assume you meant:
f′(x)=e^(3x)
then f(x) = (1/3) e^(3x) + c
but (0, 4/3) lies on it, so ...
4/3 = (1/3) e^(3(4/3)) + c
4 = e^4 + c
c = 4 - e^4
continue