A tangent line is a line that touches a curve at a single point, without crossing through it. It represents the instantaneous rate of change, or slope, of the curve at that point.
Tangent lines are important because they can help us understand the behavior of a curve at a specific point. They also allow us to find the slope of a curve at a certain point, which is useful in many applications, such as calculating rates of change and optimizing functions.
The slope of a tangent line can be found by taking the derivative of the function at the given point. In this case, we would use the derivative of f(x) to find the slope of the tangent line at x=-1.
To find lines tangent to f(x) at x=-1, we first need to take the derivative of f(x) and evaluate it at x=-1 to find the slope of the tangent line. Then, we can use the point-slope formula (y-y1 = m(x-x1)) to find the equation of the tangent line passing through the point (-1, f(-1)).
Yes, there can be more than one line tangent to a curve at a given point. This can happen when the curve has a sharp point, such as a cusp or corner, at that point. In this case, there may be multiple lines that touch the curve at that point without crossing through it.