Equation of Tangent to the Line

In summary, the equation of the tangent line to a given line is typically written in the form y = mx + b, where m represents the slope of the tangent line and b represents the y-intercept of the line. The point of tangency on a line can be found by first taking the derivative of the line's equation, setting it equal to the slope of the tangent line, and solving for the x-value. A line can only have one tangent at any given point, and a line and a curve intersect at a tangent if their slopes are equal at that point. The equation of a tangent line can never be a vertical line because a tangent line must have a defined slope to touch a curve at a specific point.
  • #1
johnq2k7
64
0
3.) Give the equation of the line tangent to the curve at the given point.

a.) (y)(tan^-1 x) = x*y at (sqrt(3),0)

b.) ln y = x^2 +(2)*e^x at (0, e^2)



Please help me with this problem, I am stuck.

Here is some of my work process:

i have no idea how to solve for y in a.)

but for b.) I took the expontential of both sides

then i got y= e^(x^2) + e^(2*e^x)

do I find the derivate of y to get the slope

but how do i plug in the values of the pt.

do i use the y- yl= m(x-x1) formula to find the tangent to the curve?

please help me with these problems, I'm clueless for part a, and i have a bit of an understanding on how to do part b.)
 
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  • #2
johnq2k7 said:
3.) Give the equation of the line tangent to the curve at the given point.

a.) (y)(tan^-1 x) = x*y at (sqrt(3),0)

b.) ln y = x^2 +(2)*e^x at (0, e^2)



Please help me with this problem, I am stuck.

Here is some of my work process:

i have no idea how to solve for y in a.)

but for b.) I took the expontential of both sides

then i got y= e^(x^2) + e^(2*e^x)

do I find the derivate of y to get the slope

but how do i plug in the values of the pt.
Put the x value of the point into your formula for the derivative!

do i use the y- yl= m(x-x1) formula to find the tangent to the curve?
Yes!

please help me with these problems, I'm clueless for part a, and i have a bit of an understanding on how to do part b.)
You may have miscopied part (a). If y is not 0, you can divide through by it and get tan-1(x)= x which means y is not a function of x. I think that there are two points, one positive and the other negative, for which tan-1(x)= x and so the equation is true for all y. Of course, it is true for all x if y= 0 and so its graph is those three lines. If that is really is what (a) says, then since [itex]tan^{-1}(\sqrt{3})\ne \sqrt{3}[/itex], the curve in a neighbor hood of the point [itex](\sqrt{3},0)[/itex] is the line y= 0 and so its tangent is the line y= 0.
 

1. What is the equation of the tangent line to a given line?

The equation of the tangent line to a given line is typically written in the form y = mx + b, where m represents the slope of the tangent line and b represents the y-intercept of the line. The slope of the tangent line can be calculated by taking the derivative of the given line's equation at the point of tangency.

2. How do you find the point of tangency on a line?

The point of tangency on a line can be found by first taking the derivative of the line's equation. Then, set the derivative equal to the slope of the tangent line and solve for the x-value. Plug this x-value into the original equation to find the corresponding y-value. The resulting point (x, y) is the point of tangency on the line.

3. Can a line have multiple tangents?

No, a line can only have one tangent at any given point. This is because a tangent is defined as a line that touches a curve at only one point and has the same slope as the curve at that point.

4. How do you determine if a given line and curve intersect at a tangent?

A line and a curve intersect at a tangent if the line is tangent to the curve at a specific point. This can be determined by checking if the slope of the line at that point is equal to the slope of the curve at the same point.

5. Can the equation of a tangent line ever be a vertical line?

No, the equation of a tangent line can never be a vertical line. This is because a vertical line has an undefined slope, while a tangent line must have a defined slope to touch a curve at a specific point.

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