Question

find the particular antiderivative of the following derivative that satisfies the given condition.
c(x)=3x² - 2x; c(0)=3,000
c(x)=□

Explanation:

Step1: Integrate \(C^{\prime}(x)\)

Integrate \(3x^{2}-2x\) term - by - term.
Using the power rule \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), we have:
\(\int(3x^{2}-2x)dx=3\int x^{2}dx-2\int xdx\)
\(=3\times\frac{x^{3}}{3}-2\times\frac{x^{2}}{2}+K\)
\(=x^{3}-x^{2}+K\)

Step2: Use the initial condition \(C(0) = 3000\)

Substitute \(x = 0\) and \(C(0)=3000\) into \(C(x)=x^{3}-x^{2}+K\).
When \(x = 0\), \(C(0)=0^{3}-0^{2}+K\).
Since \(C(0) = 3000\), then \(K = 3000\)

Answer:

\(C(x)=x^{3}-x^{2}+3000\)