If f'(x) = 3x^2 + 2x and f(2) = 3
Then f(1) = ???
integrate
f(x) = x^3 + x^2 + c
put in point to find constant c
3 = 8 + 4 + c
so
c = 3-12 = -9
f(1) = 1^3 + 1^2 - 9
= -7
To find the value of f(1), we need to integrate the given derivative function, f'(x), and then evaluate f(x) at x = 1.
Step 1: Integrate f'(x)
To integrate 3x^2 + 2x, we add 1 to the exponent of each term and divide by the new exponent:
∫ (3x^2 + 2x) dx = x^3 + x^2 + C,
where C is the constant of integration.
Step 2: Evaluate f(x) at x = 1
We are given f(2) = 3, which means f(2) = 2^3 + 2^2 + C = 3.
Substituting x = 2 and f(2) = 3 into the equation, we can solve for the constant of integration:
2^3 + 2^2 + C = 3
8 + 4 + C = 3
C = 3 - 8 - 4
C = -9
Step 3: Find f(1)
Now we can evaluate f(x) at x = 1 using the integrated function:
f(1) = 1^3 + 1^2 + (-9)
f(1) = 1 + 1 - 9
f(1) = -7
Therefore, f(1) = -7.
To find the value of f(1), we need to use the given information about the derivative of the function f(x) and its value at a specific point.
Starting with the given derivative, f'(x) = 3x^2 + 2x, we need to integrate it to find the original function f(x). The indefinite integral of 3x^2 + 2x with respect to x is:
f(x) = x^3 + x^2 + C
where C is the constant of integration.
Next, we are given that f(2) = 3. Substituting x = 2 into the original function, we can solve for C:
f(2) = 2^3 + 2^2 + C = 8 + 4 + C = 12 + C = 3
Solving for C:
12 + C = 3
C = 3 - 12
C = -9
Now that we have the constant of integration, we can rewrite the original function as:
f(x) = x^3 + x^2 - 9
To find the value of f(1), we substitute x = 1 into the function:
f(1) = 1^3 + 1^2 - 9
f(1) = 1 + 1 - 9
f(1) = -7
Therefore, f(1) = -7.