Instantaneous Rate of Change Quiz

Instantaneous Rate of Change in Mathematics

• The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument.

• Derivatives are a fundamental tool of calculus, used to find the instantaneous rate of change of a function.

• The derivative of a function at a chosen input value is the slope of the tangent line to the graph of the function at that point.

• Derivatives can be generalized to functions of several real variables using the Jacobian matrix.

• The process of finding a derivative is called differentiation, and the reverse process is called antidifferentiation.

• The fundamental theorem of calculus relates antidifferentiation with integration.

• A function of a real variable is differentiable at a point a of its domain if its domain contains an open interval I containing a, and the limit exists.

• If f is differentiable at a, then f must also be continuous at a.

• A function that has a derivative is continuous, but there are continuous functions that do not have a derivative.

• The derivative function maps every point x to the value of the derivative of f at x.

• Differentiation becomes a function of functions, where the derivative is an operator whose domain is the set of all functions that have derivatives at every point of their domain.

• Higher derivatives can be found by taking the derivative of the previous derivative, and are written as f′′, f′′′, etc.Understanding Derivatives: Higher-Order Derivatives and Notations

• The derivative of a function f is written as f′ and is called the first derivative of f; the second derivative of f is written f ′′ and is called the second derivative of f; continuing this process, one can define, if it exists, the nth derivative as the derivative of the (n−1)th derivative.

• If x(t) represents the position of an object at time t, then the higher-order derivatives of x have specific interpretations in physics, e.g., the first derivative of x is the object's velocity, the second derivative of x is the acceleration, and the third derivative of x is the jerk.

• A function f need not have a derivative, and even if f does have a derivative, it may not have a second derivative.

• A function that has k successive derivatives is called k times differentiable; if in addition the kth derivative is continuous, then the function is said to be of differentiability class Ck.

• On the real line, every polynomial function is infinitely differentiable; if a polynomial of degree n is differentiated n times, then it becomes a constant function.

• The derivatives of a function f at a point x provide polynomial approximations to that function near x.

• A point where the second derivative of a function changes sign is called an inflection point.

• The symbols dx, dy, and dy/dx were introduced by Gottfried Wilhelm Leibniz in 1675.

• Higher derivatives are expressed using the notation f(n) for the nth derivative of y = f(x).

• One of the most common modern notations for differentiation is due to Joseph-Louis Lagrange and uses the prime mark, so that the derivative of a function f is denoted f'.

• Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative.

• Euler's notation uses a differential operator D, which is applied to a function f to give the first derivative Df; the nth derivative is denoted Dnf.Derivatives: Definition, Rules, and Applications

• Euler's notation is useful for solving linear differential equations.

• Rules for basic functions include derivatives of common functions such as sin(x), cos(x), and ln(x).

• Rules for combined functions include the chain rule and product rule.

• The derivative of a function can be computed using the difference quotient and taking the limit.

• A vector-valued function can be split into its coordinate functions, and the derivative of a vector-valued function is defined as the tangent vector.

• Partial derivatives measure the variation of a function in one direction while holding all other variables constant.

• The gradient of a real-valued function is a vector that maps a point to the vector of partial derivatives.

• Directional derivatives measure the variation of a function in a specific direction.

• The total derivative gives a complete picture of the behavior of a function by considering all directions at once.

• The Jacobian matrix is a generalization of the gradient to functions from Rn to Rm.

• The Hessian matrix is the matrix of second-order partial derivatives of a function.

• Derivatives have numerous applications, including in physics, economics, and optimization problems.Total Derivative: Definition, Formula, and Applications

• The linear approximation formula relates a function f to its total derivative f′(a) at a point a in Rn, with f′(a) being a linear transformation from Rn to Rm.

• The total derivative is the best linear approximation of f at a, and it is unique.

• The total derivative exists if all the partial derivatives and directional derivatives of f exist at a.

• The total derivative of a function is given by the Jacobian matrix, which is the matrix of all the partial derivatives of f at a.

• The existence of the total derivative is strictly stronger than the existence of all the partial derivatives, but if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on a.

• The total derivative can be expressed using the pushforward notation f∗v, which is the total derivative of f in the direction of v.

• The total derivative of a multivariable function does not give another function but instead gives a function from the tangent bundle of the source to the tangent bundle of the target.

• Higher-order total derivatives are not linear transformations and are not built by repeatedly taking the total derivative, but instead, they are jets, which capture higher-order information and take as arguments additional coordinates representing higher-order changes in direction.

• Higher-order total derivatives can be interpreted as maps that take a point x in Rn and assign to it an element of the space of k-linear maps from Rn to Rm.

• The concept of a derivative can be extended to many other settings, and it serves as a linear approximation of a function at a point.

• Calculus is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series, discovered independently by Isaac Newton and Gottfried Leibniz in the mid-17th century.

• The dispute between Newton and Leibniz about who invented calculus first continued until the end of their lives.