In Problem: (a) Find the slope of the tangent line to the...

In Problem:
(a) Find the slope of the tangent line to the graph of each function $f$ at the indicated point.
(b) Find an equation of the tangent line at the point.
(c) Graph $f$ and the tangent line found in $(b)$ on the same set of axes.
$$
f(x)=x^{3}+3 x-1 \text { at }(0,-1)
$$

Key Concepts

Derivative

The derivative of a function at a point provides the slope of the tangent line at that point. It is a measure of the instantaneous rate of change of the function with respect to one of its variables. In calculus, computing the derivative is fundamental for understanding how functions change and is applied in many problems to determine slopes and rates of change.

Tangent Line

The tangent line to a curve at a given point is a line that 'touches' the curve at that point and has the same slope as the curve does there. It serves as the best linear approximation of the function near that point. Finding the equation of the tangent line involves determining the point of tangency and the slope, often using the derivative.

Point-Slope Form

The point-slope form of a line is a method to construct the equation of a line when the slope and one point on the line are known. It is typically written as y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point on the line. This form is particularly useful when expressing the equation of a tangent line after computing the derivative.

Graphing Functions

Graphing functions involves plotting the set of points that satisfy the function on a coordinate plane. When a function and its tangent line are graphed together, it illustrates how the tangent line approximates the function in the vicinity of the given point. This visual representation can help in understanding the behavior of the function and the concept of linear approximation.

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