NationMaster - Encyclopedia: Related rates

Topics in calculus
Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem For other uses, see Calculus (disambiguation). ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ... In mathematics, the limit of a function is a fundamental concept in analysis. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal (parallel) to the average derivative of the section. ...
Differentiation
Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives This article is about derivatives and differentiation in mathematical calculus. ... In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. ... In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, an implicit function is a generalization for the concept of a function in which the dependent variable may not be given explicitly in terms of the independent variable. ... The exponential function (continuous red line) and the corresponding Taylors polynomial about a = 0 of degree four (dashed green line) In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial... The primary operation in differential calculus is finding a derivative. ...
Integration
Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution This article is about the concept of integrals in calculus. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...

In differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time. Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... Look up time in Wiktionary, the free dictionary. ...

Procedure

The most common way to approach related rates problems is the following:

Identify the known rates of change and the rate of change that is to be found.
Construct an equation relating the quantities whose rates of change are known to the quantity whose rate of change is to be found.
Differentiate both sides of the equation with respect to time (or other rate of change).
Substitute the known rates of change and the known quantities into the equation.
Solve for the wanted rate of change.

Errors in this procedure are often caused by plugging in the known values for the variables before (rather than after) finding the derivative with respect to time. Doing so will yield an incorrect result. An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... This article is about derivatives and differentiation in mathematical calculus. ...

Example

Suppose that there is a 10-meter ladder leaning against the wall of a building, and the base of the ladder is sliding away from the building at a rate of 3 meters per second. How fast is the top of the ladder sliding down the wall when the base of the ladder is 6 meters from the wall?

Calling the distance of the base of the ladder from the wall x and the height of the ladder on the wall y, the ladder, the wall, and the ground represent the sides of a right triangle with side lengths x, y, and 10 (the hypotenuse). The object is to find the rate of change of y with respect to time when x = 6. It is given that when x = 6, the rate of change of x is 3 meters per second. This rate of change is positive because the distance x is increasing. For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...

An equation relating the three sides of a right triangle is the well-known pythagorean theorem, a²+b² = c². In this case, the equation that relates x and y is x²+y² = 10². Differentiating both sides of this equation with respect to time (t) yields In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

$frac{d}{dt}(x^2+y^2)=frac{d}{dt}(100)$

which when solved for the wanted rate of change, dy/dt, gives us

$frac{dy}{dt}=frac{-xfrac{dx}{dt}}{y}.$

It is given that when x = 6, dx/dt = 3. Due to the pythagorean theorem, y = 8. Plugging these values into the equation gives us the answer:

$frac{dy}{dt}=frac{-6times3}{8}=-frac{9}{4}.$

The top of the ladder is sliding down the wall at a rate of ⁹⁄₄ meters per second.

Categories: Calculus | Differential equations

Results from FactBites:

Related Rates (1959 words)
	Related rate problems often involve a situation in which you are asked to calculate the rate at which one quantity changes with respect to time from the rate at which a second quantity changes with respect to time.
	Related rate problems can be recognized because the rate of change of one or more quantities with respect to time is given and the rate of change with respect to time of another quantity is required.
	The rate of change of the volume of the pile is given, the constant multiplier is specified, and there is a request to find the rate of change of the height of the pile when the height is a specified value.

PlanetMath: related rates (599 words)
	The rates of change are no longer scalars, but rather velocity vectors, and therefore the derivative must be regarded as a linear transformation that changes one vector into another.
	The above is the multi-dimensional and coordinate-free generalization of the related rates relation (1).
	This is version 3 of related rates, born on 2002-06-06, modified 2005-11-27.

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