What is the equation of the line normal to f(x)=7x3+9x2 at x=-2?

Question

titemomo8gjz · Accepted Answer

Gradient of the given equation is it differential
Thus

f^{'} (x) = 21 x^{2} + 18 x

\Rightarrow f^{'} (- 2) = 21 {(- 2)}^{2} + 18 (- 2) = + 48

So the gradient of the line normal to it at this point is:

y = - \frac{1}{48} x + c

...(1)
Let the point on the original curve be

P_{1} \to (x_{1}, y_{1}) = (- 2, y)

f (- 2) = y_{1} = 7 {(- 2)}^{3} + 9 {(- 2)}^{2} = - 20

\Rightarrow P_{1} \to (x_{1}, y_{1}) = (- 2, - 20)

Substitute these values into equation (1) to find c

Emily Green · Accepted Answer

f(x)=7x3+9x2 and x0=2Find the value of the function at the given point: y0=f(2)=92The slope of the normal line at x=x0 is the negative reciprocal of the derivative of the function, evaluated at x=x0:M(x0)=−1f′(x0)Find the derivative: f′(x)=(7x3+9x2)′=3x(7x+6)Hence, M(x0)=−1f′(x0)=−13x0(7x0+6)Next, find the slope at the given point.m=M(2)=−1120Finally, the equation of the normal line is y−y0=m(x−x0)Plugging the found values, we get that y−92=−x−2120Or, more simply: y=552160−x120

What is the equation of the line normal

Answered question

Answer & Explanation

New Questions in Calculus 1