find the function y(x) satisfying dy/dx = 6x - 7 and y(9)=0. the function y(x) satisfying dy/dx = 6x - 7 and…

Explanation:

Step1: Integrate the derivative

Integrate $\frac{dy}{dx}=6x - 7$ with respect to $x$. Using the power - rule of integration $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $y(x)=\int(6x - 7)dx=6\times\frac{x^{2}}{2}-7x + C=3x^{2}-7x + C$.

Step2: Find the value of C

Substitute $x = 9$ and $y(9)=0$ into $y(x)=3x^{2}-7x + C$. So $0=3\times9^{2}-7\times9 + C$. First, calculate $3\times9^{2}-7\times9$: $3\times81-63=243 - 63=180$. Then $0 = 180+C$, which gives $C=-180$.

Answer:

$3x^{2}-7x - 180$