SOLUTION: Find the tangent and normal lines for {{{f(x)=4-x^2}}}

Question 128195: Find the tangent and normal lines for
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Start with the given function

Derive

to get

Now to find the slope of the tangent line at (1,3), simply plug in into

Plug in

Multiply

So the slope of the tangent line at (1,3) is -2 (note: this is where you made a mistake)

Now let's use the Point-Slope Formula to find the equation of the tangent line

---Point-Slope Formula---
where is the slope, and

is the given point

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)

Distribute

Multiply and to get

Add 3 to both sides to isolate y

Combine like terms and to get

So the equation of the tangent line is

Now simply negate and find the reciprocal of the slope to get to get the perpendicular slope

Now let's use the Point-Slope Formula to find the equation of the normal line note: the normal line has a slope of and goes through (1,3)

---Point-Slope Formula---
where is the slope, and

is the given point

So lets use the Point-Slope Formula to find the equation of the line

Plug in , , and (these values are given)

Distribute

Multiply and to get

Add 3 to both sides to isolate y

Combine like terms and to get (note: if you need help with combining fractions, check out this solver)

--------------------------------

Summary:

So the equation of the tangent line is and the equation of the normal line is

Notice if we graph , the tangent line , and the normal line , we get

Graph of (red), the tangent line (green), and the normal line (blue)

So we can see that is tangent to and that is perpendicular to . So this visually verifies our answer.
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