Equation of a line that is tangent to f(x)

In summary, the problem is asking to find the equation of a line that passes through a point P(2,7) but is tangent to the graph of the function f(x)=4x-x^2. The given information is that P is not on the graph of f(x) and no other specific function or point is mentioned. The approach taken by the person is to find the derivative of f(x) and then plug it into the equation y=mx+b, but it is unclear if this is the correct method.
  • #1
Squiller
6
0
In order to find the equation of a line that is tangent to f(x) and goes through point P on f, you got to find the derivative of f(x) at P, but how would you go about solving a problem where you have to find the equation of a line tangent to f(x) that goes through point P, but P is NOT on the graph of f.
 
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  • #2


Just to get my facts straight: you have a function [tex] f(x) [/tex], a point [tex] P [/tex] that is not on the graph of [tex] f(x) [/tex], and you want an equation of a line that

  1. passes through the point [tex] P [/tex], and
  2. is tangent to the graph of [tex] f(x) [/tex]

If this is correct, what else is stated in the problem - do you have a specific function [tex] f [/tex], at what point(s) is the line to be tangent, etc. Further, what have you tried?
 
  • #3


f(x) = 4x-x2

Question: Find the equations of the lines that pass through P(2,7) and are tangent to the graph of f(x).

(P is not on f(x).)
Thats all the problem states.

Ive tried finding f'(x) and plugging f' into the Line equation y=mx+b.

y=(4-2x)x+b.

Then plugging in Point P.

7=(4-2x)2+b - I am not really sure if this is heading in the right direction.
 

Related to Equation of a line that is tangent to f(x)

1. What is the equation of a line that is tangent to f(x)?

The equation of a line that is tangent to f(x) is y = mx + b, where m is the slope of the tangent line and b is the y-intercept.

2. How do you find the slope of a tangent line to f(x)?

The slope of a tangent line to f(x) can be found by taking the derivative of the function f(x) at the point where the line is tangent. This derivative represents the slope of the tangent line at that point.

3. Can a line be tangent to a function at multiple points?

Yes, a line can be tangent to a function at multiple points. This occurs when the function has a point of inflection, where the slope of the tangent line changes from positive to negative or vice versa.

4. How do you determine the point of tangency for a tangent line to f(x)?

The point of tangency for a tangent line to f(x) can be determined by finding the x-value where the derivative of the function is equal to the slope of the tangent line. This x-value can then be plugged into the original function to find the corresponding y-value.

5. What is the significance of the tangent line to a function?

The tangent line to a function represents the instantaneous rate of change of the function at a specific point. It can also be used to approximate the behavior of the function near that point, making it a useful tool in applications such as optimization and curve fitting.

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