What is the equation of the line normal to f(x)=x3+4x2 at x=1?

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Ouhamiptkg · Accepted Answer

Explanation:We&#039;ll be using the point gradient formula since we already have a point, x=1, which means y=5; (1,5) no we need a gradient.In order to find the gradient at x=1. we need to differentiate the equation. The differentiated equation is 3x2+8x. We then substitute in x=1, and we have the gradient of 11.However, this gradient is the tangents gradient, not the normal&#039;s. In order to find the normal&#039;s gradient, we know that the gradient of the tangent times the gradient of the normal equals -1. Therefore, the gradient of the normal is −111We the substitute the numbers into the formula to get a equation

llevochalecoiozq · Accepted Answer

Explanation:f(x)=x3+4x2 and x0=1Find the value of the function at the given point: y0=f(1)=5.The 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)=(x3+4x2)′=x(3x+8)Hence, M(x0)=−1f′(x0)=−1x0(3x0+8)Next, find the slope at the given point.m=M(1)=−111Finally, the equation of the normal line is y−y0=m(x−x0)Plugging the found values, we get that y−5=−x−111Or, more simply: y=5611−x11

What is the equation of the line normal

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