Finding all points where tangent line is perpendicular

In summary, the conversation discusses finding points (x,y) on a given function where there are tangent lines that are perpendicular to a given line. The speaker first finds the derivative of the function and simplifies it. They then set the derivative equal to a fraction and suggest using cross multiplication as the next step, but are unsure of how to proceed.
  • #1
riri
28
0
Hello!

I've encountered a problem of find all points (x,y) on $f(x)=\frac{x-\sqrt{\pi}}{x+1}$ where there are tangent lines perpendicular to $y=-(1+\sqrt{\pi}x+7\pi e^{e^{{\pi}^{110}}})$

So I first found derivative and ended up with $f'(x)=\frac{1(x+1)-(x-\sqrt{\pi})(1)}{x^2+2x+1}$
and then simplified and got to $\frac{1+\sqrt{\pi}}{x^2+2x+1}$
and then I think i equal this to $\frac{1}{1+\sqrt{\pi}}$ because it's perpendicular... and then I don't know what to do. Do they cross out to be $\frac{1}{x^2+2x+1}$?Thank you!
 
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  • #2
Cross multiplying would be the next step...
 

FAQ: Finding all points where tangent line is perpendicular

What is the definition of a tangent line?

A tangent line is a straight line that touches a curve at only one point, without crossing through the curve.

How do you find the slope of a tangent line?

The slope of a tangent line can be found by taking the derivative of the function at the point of tangency. This will give the instantaneous rate of change at that point.

What is the relationship between a tangent line and a perpendicular line?

A tangent line and a perpendicular line are always at right angles to each other. This means that the slopes of the two lines are negative reciprocals of each other.

How do you determine all points where a tangent line is perpendicular to a given line?

To find all points where a tangent line is perpendicular to a given line, you must first find the slope of the given line. Then, take the negative reciprocal of that slope to find the slope of the perpendicular line. Finally, set the derivative of the function equal to the perpendicular slope and solve for the x-values where this occurs. These x-values will be the points of tangency.

Can a tangent line be perpendicular to more than one line?

No, a tangent line can only be perpendicular to one line at a time. This is because a tangent line is unique to a specific point on a curve, and therefore can only be perpendicular to the line that intersects that point.

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