# Find the equation of the curve passing through the point (3, 2) and having s slope 5x^2 - x + 1 at every point (x, y)

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Subject

**Problem**

Find the equation of the curve passing through the point (3, 2) and having s slope 5*x*^{2} - *x* + 1 at every point (*x*, *y*).

A. $y = \frac{5}{3}x^3 - \frac{1}{2}x^2 + x - \frac{31}{3}$ | C. $y = 5x^3 - 2x^2 + x - 118$ |

B. $y = 5x^3 - 2x^2 + x - 31$ | D. $y = \frac{5}{3}x^3 - \frac{1}{2}x^2 + x - \frac{83}{2}$ |

**Answer Key**

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[ D ]

**Solution**

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$\dfrac{dy}{dx} = 5x^2 - x + 1$

$dy = (5x^2 - x + 1) \, dx$

$\displaystyle \int dy = \int (5x^2 - x + 1) \, dx$

$y = \frac{5}{3}x^3 - \frac{1}{2}x^2 + x + C$

At (3, 2)

$2 = \frac{5}{3}(3^3) - \frac{1}{2}(3^2) + 3 + C$

$C = -\frac{83}{2}$

Hence, the equation of the curve is

$y = \frac{5}{3}x^3 - \frac{1}{2}x^2 + x - \frac{83}{2}$ ← *answer*

You can also solve this problem by trial and error using the choices with (3, 2) on the curve to satisfy the equation.

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