What function would have a derivative of ƒ′(x) = 12x2 + 4x − 10? Explain how you arrived at your answer...I believe the answer is this: f(x)=4x^3+2x^2-10x...and I got this using the power rule and multiplying. Thanks.
Your answer is correct. You can add an arbitrary constant to it for the general answer, which is called the indefinite integral. Whatever method you used seems to work.
The integral or "antiderivative" of each term of the form
a x^n is a*x^(n+1)/(n+1)
Just add the different polynomial terms. That apparently is what you did.
To find the function that has a derivative of ƒ′(x) = 12x^2 + 4x − 10, you need to integrate the given derivative.
The power rule states that the integral of x^n is (x^(n+1))/(n+1), except for the case when n = -1, so it does not apply here.
Instead, you need to invert the process of differentiation. Recall that the derivative of a constant is zero, so when you integrate the constant term -10, you get -10x.
For the terms with x, you need to increase the exponent by 1 and divide by the new exponent. Applying the power rule, the integral of 4x is (4/2)x^2 = 2x^2.
Similarly, the integral of 12x^2 is (12/3)x^3 = 4x^3.
Thus, integrating each term individually, the function that has a derivative of 12x^2 + 4x − 10 is:
f(x) = 4x^3 + 2x^2 - 10x.
Therefore, your answer, f(x) = 4x^3 + 2x^2 - 10x, is correct.
You are correct! To determine the function that would have the given derivative of ƒ′(x) = 12x^2 + 4x - 10, you can apply the reverse process of differentiation, which is integration.
To find the integral of ƒ′(x), you can use the power rule of integration. The power rule states that if a function has a derivative of x^n, then the integral of x^n is (x^(n+1))/(n+1), except for the case of n = -1.
Now, let's go through the steps:
1. Start with the given derivative: ƒ′(x) = 12x^2 + 4x - 10.
2. Apply the power rule of integration:
Integration of x^n = (x^(n+1))/(n+1).
Integration of 12x^2 = (12x^(2+1))/(2+1) = 4x^3.
Integration of 4x = (4x^(1+1))/(1+1) = 2x^2.
Integration of -10 = -10x.
3. Combine the integrated terms to obtain the function:
f(x) = 4x^3 + 2x^2 - 10x + C, where C is the constant of integration.
Note that the constant of integration (C) can be any arbitrary constant.
4. Simplify the function: f(x) = 4x^3 + 2x^2 - 10x.
Therefore, the function that would have a derivative of ƒ′(x) = 12x^2 + 4x - 10 is f(x) = 4x^3 + 2x^2 - 10x.