Finding the equation of the normal line I have a question to find the equations of the tangent line and the normal line to the curve at the given point. I can find the equation for the tangent line easily but I am not sure what a normal line is and there is no example that I can find.
$y=x^4 + 2e^x$ at (0,2)
From that I do see that if I plug in 0 I get 2 as a result so my guess was that if I plug in another number I can use that to get the slope but it gave an incorrect answer. 
 A: You know that the tangent at the curve is given by
$$y_t = f(a)+f'(a)(x-a)$$
The normal would be a line such that

*

*It also passes through $(a,f(a))$

*It is perpendicular to $y_t$.

Given that a line that passes through $(X,Y)$ and has slope $m$ is given by
$$y-Y = m(x-X)$$
...and that two lines of slopes $n$ and $p$ are perpendicular if and only if $m\cdot n=-1$
Can you find $y_n$?


Given that a line that passes through $(X,Y)$ and has slope $m$ is given by
$$y-Y = m(x-X)$$

Give the equations to

*

*A line with slope $10$ that passes through $(0,1)$

*A line with slope $-5$ that passes through $(-3,3)$

*Given a line with slope $2$ that passes through the origin, find the equation to a line perpendicular to it that passes through $(5,2)$.

*Let $f(x) = x^4+2e^x$. Given the equation of the tangent to $f(x)$ at $(a,f(a))$. Find the normal to $f$ at the same point.

A: Finding the derivative will give you the slope of the tangent line like what Peter T. has shown above.  All you need to do is be able to find what (a, f(a)) and then you will be on your way.
A: Equation of normal is given by,
$$\frac{y-y_1}{x-x_1}=\frac{-dx}{dy}$$
Now $y=x^4 + 2e^x$ 
$\frac{dy}{dx}=4x^3+2e^x =2$ .....at $x=0$
$\frac{-dx}{dy}=\frac{-1}{2}$
and $x_1=0$ and $y_1=2$ , plugging these values in the equation for normal, we get
$$\frac{y-2}{x-0}=\frac{-1}{2}$$
$$x+2y=4$$
were you getting the same answer? or something else..
A: 
Just follow the following steps:


*

*Take the deriviative

*Find the slope of the tangent, and then the slope of the normal by doing a negative reciprcoal.

*Use y=mx+b and sub in values and find equations

*Write a final statement stating the equations.


If you follow the following steps you will never get confused and will understand how to solve the problem in a simple way.
