Suppose we have a function defined on a closed interval [c, d].
Suppose we have a function defined on a closed interval [c, d].A local maximum or minimum can not occur at the endpoints of this interval because the definition requires that the point is contained in some open interval (a, b).Tags: Sources Of Literature Review In Research MethodologyReflective Essay On Mental HealthNursing School Essay PromptsAction Research DissertationThesis Statement Of The Story Of An HourHow To Cheat On Mymathlab Homework
Another application of the derivative is in finding how fast something changes.
For example, suppose you have a spherical snowball with a 70cm radius and it is melting such that the radius shrinks at a constant rate of 2 cm per minute. These types of problems are called related rates problems because you know a rate and want to find another rate that is related to it.
The radius of a sphere is increasing at a rate of 2 meters per second.
At what rate is the volume increasing when the radius is equal to 4 meters?
If there are variables for which we are not given the rates of change (except for the rate of change that we are trying to determine), we must find some relation from the nature of the question that allows us to write these variables in terms of variables for which the rates of change are given.
How To Solve Related Rates Problems In Calculus
We must then substitute these relations into the main equation.For example, in this problem, the general information is that the radius is growing at a rate of 2 meters per second.Thus, if we let the radius of the sphere be represented by .Since the function is not defined for some open interval around either c or d, a local maximum or local minimum cannot occur at this point.An absolute maximum or minimum can occur, however, because the definition requires that the point simply be in the domain of the function.The reason why such a problem can be solved is that the variables themselves have a certain relation between them that can be used to find the relation between the known rate of change and the unknown rate of change.Related rate problems can be solved through the following steps: Step one: Separate "general" and "particular" information.3) List all information that is given in the problem and the rate of change that we are trying to find.4) Write an equation that associates the variables with one another.This type of problem is known as a "related rate" problem.In this sort of problem, we know the rate of change of one variable (in this case, the radius) and need to find the rate of change of another variable (in this case, the volume), at a certain point in time (in this case, when ).