# Exploring Tangent Lines in Cubed Functions

• Mejiera
In summary, the conversation discussed the possibility of having a tangent line in a cubed function. It was mentioned that if a function can be differentiated, it has tangent lines, even if it intersects the curve at other points. The tangent line is defined as a line that touches the curve at a point and has a slope equal to the derivative of the function at that point. It was also mentioned that the tangent line can intersect the curve at the point of tangency.
Mejiera

## Homework Statement

is it possible to have a tangent line in a cubed function

## The Attempt at a Solution

Simply put, if you can differentiate it, it has tangent lines. So you can have tangent lines for things that aren't functions too.

but the tangent line touches a cubed function twice so I am sure if it could really be called a tangent line

Mejiera said:
but the tangent line touches a cubed function twice so I am sure if it could really be called a tangent line
Doesn't matter. The tangent line is just a line that touches a curve at a point (a, f(a)) and whose slope is f'(a). The fact that the tangent line happens to intersect the graph of the function somewhere else is immaterial. Pretty much every odd-degree polynomial will have a tangent line that intersectst the curve somewhere else.

As it turns out, the tangent line to the graph of y = f(x) = 2x + 3 at any point happens to completely coincide with the graph of this function, but that doesn't keep it from being a tangent line.

In addition to what Mark44 said, I will point out that the tangent line can even intersect/cross the curve AT the point of tangency. For example, the tangent to $f(x) = x^3$ at $x = 0$. It's still a tangent line, though.

ok thanks for clearing that up guys.

## 1. What is a tangent line?

A tangent line is a line that touches a curve at one point and has the same slope as the curve at that point.

## 2. How do you find the tangent line of a cubed function?

To find the tangent line of a cubed function, you can use the derivative of the function. The derivative will give you the slope of the tangent line at any given point on the function. You can then use the point-slope formula to find the equation of the tangent line.

## 3. What is the significance of exploring tangent lines in cubed functions?

Exploring tangent lines in cubed functions helps us understand the behavior of the function and its relationship to its derivative. It also allows us to find the maximum and minimum points of the function, which can be useful in applications such as optimization problems.

## 4. Can a cubed function have more than one tangent line?

Yes, a cubed function can have more than one tangent line. This is because the slope of the function can change at different points, resulting in multiple tangent lines with different slopes.

## 5. How can I graph a cubed function and its tangent line?

You can graph a cubed function and its tangent line by plotting the points of the function and then using the slope and point to graph the tangent line. You can also use a graphing calculator or computer software to graph the function and its tangent line automatically.

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