question find the curve whose slope at any po...
question find the curve whose slope at any point (x,y) is -3x^2 + 4x + 7 if the curve is required to pass through the point (2,11). provide your answer below: y = □
Answer
# Explanation:
## Step1: Integrate the slope function
The slope of the curve is $y'=-3x^{2}+4x + 7$. Integrating with respect to $x$ gives $y=\int(-3x^{2}+4x + 7)dx=-x^{3}+2x^{2}+7x + C$.
## Step2: Find the value of C
Since the curve passes through the point $(2,11)$, substitute $x = 2$ and $y=11$ into $y=-x^{3}+2x^{2}+7x + C$. So $11=-2^{3}+2\times2^{2}+7\times2 + C$. Simplify the right - hand side: $-8 + 8+14 + C=14 + C$. Then $11=14 + C$, and $C=-3$.
# Answer:
$y=-x^{3}+2x^{2}+7x - 3$