![]() ![]() Note that both x x and s s are functions of time. The variable s s denotes the distance between the man and the plane. We denote those quantities with the variables s s and x, x, respectively.Īs shown, x x denotes the distance between the man and the position on the ground directly below the airplane. The distance between the person and the airplane and the person and the place on the ground directly below the airplane are changing. The relationship we are studying is between the speed of the plane and the rate at which the distance between the plane and a person on the ground is changing.įigure 4.3 An airplane is flying at a constant height of 4000 ft. The first example involves a plane flying overhead. Let’s now implement the strategy just described to solve several related-rates problems. We examine this potential error in the following example. For example, if the value for a changing quantity is substituted into an equation before both sides of the equation are differentiated, then that quantity will behave as a constant and its derivative will not appear in the new equation found in step 4. Note that when solving a related-rates problem, it is crucial not to substitute known values too soon. Substitute all known values into the equation from step 4, then solve for the unknown rate of change.This new equation will relate the derivatives. Using the chain rule, differentiate both sides of the equation found in step 3 with respect to the independent variable.Find an equation relating the variables introduced in step 1.State, in terms of the variables, the information that is given and the rate to be determined. ![]() Assign symbols to all variables involved in the problem.Problem-Solving Strategy: Solving a Related-Rates Problem Here we study several examples of related quantities that are changing with respect to time and we look at how to calculate one rate of change given another rate of change. In this case, we say that d V d t d V d t and d r d t d r d t are related rates because V is related to r. For example, if we consider the balloon example again, we can say that the rate of change in the volume, V, V, is related to the rate of change in the radius, r. In many real-world applications, related quantities are changing with respect to time. In this section, we consider several problems in which two or more related quantities are changing and we study how to determine the relationship between the rates of change of these quantities. For example, if a balloon is being filled with air, both the radius of the balloon and the volume of the balloon are increasing. If two related quantities are changing over time, the rates at which the quantities change are related. We have seen that for quantities that are changing over time, the rates at which these quantities change are given by derivatives. 4.1.3 Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.4.1.2 Find relationships among the derivatives in a given problem.4.1.1 Express changing quantities in terms of derivatives. ![]()
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