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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).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.
This study assessed the ability of university students enrolled in an introductory calculus course to solve related-rates problems set in geometric contexts.
Students completed a problem-solving test and a test of performance on the individual steps involved in solving such problems.
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 ).
The applet shows an image of the snowball with an initial radius of 70 cm.
One approach to finding the rate at which the volume changes is to figure out an equation for the volume, plug into that a formula for how the radius changes with time, thus giving a formula for how the volume changes with time.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.When solving related rates problems, we should follow the steps listed below. This is the most helpful step in related rates problems. 2) Assign variables to each quantitiy in the problem that is a function of time.Each of these values will have some rate of change over time.TUTORIALS HOME GENERAL MATH NUMBER SETS ABSOLUTE VALUE & INEQUALITIES SETS & INTERVALS FRACTIONS POLYNOMIALS LINEAR EQUATIONS QUADRATIC EQUATIONS GEOMETRY FINITE SERIES TRIGONOMETRY EXPONENTS LOGARITHMS INDUCTION CALCULUS LIMITS DERIVATIVES RELATED RATES & OPTIMIZATION CURVE SKETCHING INTEGRALS AREA & VOLUME INVERSE FUNCTIONS Related rates problems require us to find the rate of change of one value, given the rate of change of a related value.We must find an equation that associates the two values and apply the chain rule to differentiate each side of the equation with respect to time.. The derivative, dv/dt would be the rate of change of v.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.We must then substitute these relations into the main equation.General information is information contained in the problem that is true at all times.Particular information is information that is true only at the particular instant that the problem is asking about.