# What is the equation of the normal line of f(x)=x^3-x^2+17x at x=7?

Jan 4, 2017

$y = - \frac{1}{150} x + \frac{61957}{150}$

#### Explanation:

The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point. The normal is perpendicular to the tangent and so the product of their gradients is $- 1$

so If $f \left(x\right) = {x}^{3} - {x}^{2} + 17 x$ then differentiating wrt $x$ gives us:

$\setminus \setminus \setminus \setminus \setminus f ' \left(x\right) = 3 {x}^{2} - 2 x + 17$

When $x = 7 \implies f \left(7\right) = 343 - 49 + 119 = 413$ (so $\left(7 , 413\right)$ lies on the curve) and $f ' \left(7\right) = 3 \cdot 49 - 14 + 17 = 150$

So the tangent passes through $\left(7 , 413\right)$ and has gradient $150$, hence the normal has gradient $- \frac{1}{150}$ so using the point/slope form $y - {y}_{1} = m \left(x - {x}_{1}\right)$ the equation we seek is;

$y - 413 = - \frac{1}{150} \left(x - 7\right)$
$\setminus \setminus \setminus \setminus \setminus \therefore y = - \frac{1}{150} x + \frac{61957}{150}$

We can confirm this solution is correct graphically: