Question

find the particular antiderivative of the following derivative that satisfies the given condition.

c(x)=2x² - 5x; c(0)=1,000

c(x)=

Explanation:

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

We know that if \(C^{\prime}(x)=2x^{2}-5x\), then by the power rule of integration \(\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C\) (\(n
eq - 1\)).
\(\int(2x^{2}-5x)dx=2\int x^{2}dx-5\int xdx\)
\(=2\times\frac{x^{3}}{3}-5\times\frac{x^{2}}{2}+K\)
\(=\frac{2}{3}x^{3}-\frac{5}{2}x^{2}+K\)

Step2: Use the initial condition \(C(0) = 1000\)

Substitute \(x = 0\) into \(C(x)=\frac{2}{3}x^{3}-\frac{5}{2}x^{2}+K\).
\(C(0)=\frac{2}{3}(0)^{3}-\frac{5}{2}(0)^{2}+K\)
Since \(C(0) = 1000\), then \(K = 1000\)

Answer:

\(C(x)=\frac{2}{3}x^{3}-\frac{5}{2}x^{2}+1000\)