Equation for Tangent Line to Inverse Function at (3,1) of f(x)=x^3+2x^2-x+1

In summary, an equation for the line tangent to the graph of f^-1 at the point (3,1) can be found using the Inverse Function Theorem. The inverse function of f(x) is 1/(3x^2+4x), and when x=3, the slope of the tangent line is 1/21. Thus, the equation of the tangent line is y-1=1/21(x-3), or y=1/21x+6/7. (3,1) is a point on the graph of f^-1 and x in this context refers to the values or y's of f(x). The Inverse Function Theorem was used to find the equation of the
  • #1
kathrynag
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0

Homework Statement


Find an equation for the line tangent to to the graph of f^-1 at the pt (3,1) if f(x)=x^3+2x^2-x+1



Homework Equations





The Attempt at a Solution


I used the Inverse Function Thm
1/(3x^2+4x)
Now do I plug in 3 to this to find slope?
1/21
y-1=1/21(x-3)
y-1=1/21x-1/7
y=1/21x+6/7
 
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  • #2
state the inverse function thm.
what is (3,1) a point of?

also, i think you are mixing up x's. if you look at f^-1 as a function of x, these x's are really values (or y's) of f(x)

i hope you understand this; I'm afraid I've been a bit convoluted.
 

FAQ: Equation for Tangent Line to Inverse Function at (3,1) of f(x)=x^3+2x^2-x+1

What is the slope of the tangent line?

The slope of the tangent line is the coefficient of x in the equation, which is 1/21 in this case.

How do you find the equation of a tangent line?

To find the equation of a tangent line, you need to know the slope of the tangent line and a point on the line. In this case, the slope is 1/21 and the point is (0, 6/7). You can then use the point-slope formula to find the equation: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point. This will give you the equation y = 1/21x + 6/7.

What is the significance of the tangent line in calculus?

The tangent line represents the instantaneous rate of change of a function at a specific point. It is used in calculus to find the derivative of a function, which is a fundamental concept in understanding the behavior and properties of functions.

Can the equation of a tangent line change?

Yes, the equation of the tangent line can change depending on the point on the curve where it is drawn. The slope of the tangent line will also change if the function itself is changing at that point.

How is the equation of a tangent line related to the derivative of a function?

The equation of a tangent line is directly related to the derivative of a function. The derivative of a function at a specific point is equal to the slope of the tangent line at that point. This is why the tangent line is used to find the derivative and vice versa.

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