The concept of a limit is fundamental in calculus and mathematical analysis, representing the value that a function or sequence approaches as the input approaches some point. Limits are essential for defining derivatives and integrals, and they help in understanding the behavior of functions at points of discontinuity or infinity.
The chain rule is a fundamental derivative rule in calculus used to compute the derivative of a composite function. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
Marginal change refers to the small or incremental adjustment to a plan of action, often analyzed in economics to determine the effect of a slight change in a variable on the overall outcome. It is crucial for decision-making processes as it helps in evaluating the benefits and costs associated with minor changes, enabling optimization of resource allocation.
A differentiable function is a function whose derivative exists at each point in its domain, indicating that it has a well-defined tangent line at every point and is smooth without any sharp corners or discontinuities. Differentiability implies continuity, but the converse is not necessarily true, as a function can be continuous without being differentiable at certain points.
Substitution in calculus, also known as u-substitution, is a technique used to simplify the process of integration by transforming a complex integral into a simpler one. This method involves changing the variable of integration to make the integral more manageable, often by reversing the chain rule of differentiation.