Final answer:
The derivative is the limit of the slope of secant lines as the interval approaches zero, defining the slope of the tangent to the function's graph. Quadratic models may be used for cubic functions locally around extremes. Derivatives measure how a function's output changes with its input and are defined as the limit of the average rate of change of the function as the interval narrows to zero.
Step-by-step explanation:
Relationship Between Limits and Derivatives
The relationship between limits and derivatives is fundamental in calculus. A derivative represents the slope of the tangent line to the graph of a function at a specific point. It is defined as the limit of the ratio of the change in the function's value to the change in the independent variable as the change approaches zero. This means that to find the derivative of a function at a certain point, we compute the limit as Δx approaches zero of the slope of the secant lines near that point.
Using the Quadratic Model with Cubic Functions
While the quadratic model typically describes second-order polynomials, circumstances might require its use for approximation when dealing with a cubic function. This generally occurs if we are interested in the behavior of the cubic function around a local maximum or minimum, as locally it may resemble a quadratic function.
Definition of the Derivative
The derivative of a function is a measure of how a function's output value changes as its input value changes. Formally, if y = f(x), then the derivative of f with respect to x is defined by the limit of the average rate of change of y as Δx approaches zero and can be denoted as f'(x) or dy/dx.