Question

question 5 · 1 point find the curve whose slope at any point (x,y) is -3x^2 + 4x + 2 if the curve is required to pass through the point (0,6). provide your answer below:

Explanation:

Step1: Recall the antiderivative concept

The slope of a curve is given by its derivative. So, if $y'=-3x^{2}+4x + 2$, then $y=\int(-3x^{2}+4x + 2)dx$.

Step2: Integrate term - by - term

Using the power rule of integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n
eq - 1$), we have:
$y=-3\times\frac{x^{3}}{3}+4\times\frac{x^{2}}{2}+2x + C=-x^{3}+2x^{2}+2x + C$.

Step3: Find the value of C

Since the curve passes through the point $(0,6)$, substitute $x = 0$ and $y=6$ into the equation $y=-x^{3}+2x^{2}+2x + C$.
$6=-0^{3}+2\times0^{2}+2\times0 + C$, so $C = 6$.

Answer:

$y=-x^{3}+2x^{2}+2x + 6$