"The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers."
Finding maximum and minimum values, optimization problems, determining concavity and inflection points, etc.
Limits: The foundational concept of calculus, limits are necessary to evaluate derivatives.
Continuity: A function is continuous when it can be drawn without lifting up the pencil. It is an essential requirement to calculate derivatives.
Derivatives: The rate of change of one variable with respect to another.
Calculating Derivatives: Rules for calculating derivatives of basic functions such as polynomial, exponential, logarithmic functions.
Chain Rule: The chain rule is used to calculate the derivative of a composite function.
Implicit Differentiation: A method to find the derivative of a function where the variable is not explicitly given.
Mean Value Theorem: A theorem which states that there is some point within a continuous function's domain where the slope of the tangent line equals the slope of the chord that passes through the same two points.
Second Derivatives: The second derivative represents the rate of change of the first derivative.
Optimization: A process of finding the maximum or minimum value of a function.
Related Rates: The concept deals with finding the rate of change of one variable with respect to another variable in a relationship.
Concavity: A curve is said to be concave up or down based on the concavity criterion.
Inflection Points: A point where a curve changes from concave up to concave down and vice versa.
Curve Sketching: Graphing the curve based on the derivative and the second derivative.
Applications: Applications of derivatives are numerous in fields such as physics, economics, engineering, and statistics.
Optimization: Finding the maximum or minimum value of a function.
Rates of change: Measuring how quickly a variable changes over time.
Tangent line approximation: Approximating a function at a specific point by using its tangent line.
Concavity and points of inflection: Determining whether a function is concave up or down and identifying its points of inflection.
Related rates: Finding the rate of change of one variable with respect to another variable.
Derivatives of inverse trigonometric functions: Finding the derivative of an inverse trigonometric function.
Implicit differentiation: Finding the derivative of an implicit function that is not explicitly stated.
Higher-order derivatives: Finding the second or higher-order derivative of a function.
Average and instantaneous rate of change: Finding the average rate of change of a function over an interval or the instantaneous rate of change at a specific point.
L'Hôpital's rule: Evaluating limits of indeterminate forms by taking derivatives of both the numerator and denominator.
Newton's method: Finding the root of a function using iterative approximations.
Optimization with constraints: Finding the maximum or minimum value of a function subject to constraints.
Riemann sums: Approximating the area under a curve using rectangles.
Rolle's theorem: Finding an interval where the derivative of a function is zero, given the endpoints have the same function value.
Mean value theorem: Finding a point in an interval where the derivative of a function is equal to its average rate of change.
"Functionals are often expressed as definite integrals involving functions and their derivatives."
"Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations."
"A simple example of such a problem is to find the curve of shortest length connecting two points."
"If there are no constraints, the solution is a straight line between the points."
"However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics."
"One corresponding concept in mechanics is the principle of least/stationary action."
"Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium."
"Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle."
"Plateau's problem requires finding a surface of minimal area that spans a given contour in space."
"A solution can often be found by dipping a frame in soapy water."
"There may be more than one locally minimizing surface."
"They may have non-trivial topology."
"Functionals: mappings from a set of functions to the real numbers."
"Variations are small changes in functions and functionals."
"Functionals are often expressed as definite integrals involving functions and their derivatives."
"Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations."
"One corresponding concept in mechanics is the principle of least/stationary action."
"The calculus of variations is a field of mathematical analysis that uses variations to find maxima and minima of functionals."
"Such solutions are known as geodesics."