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# How Calculus Can be Used to Solve Many Problems

Calculus can be used to solve many problems Probably the most important is solving real-life problems or situations An example of this type of problem is a related rates problem A related rates problem is a problem which involves at least two changing quantities and asks you to figure out the rate at which one is changing given sufficient information on all of the others There are several key steps in setting up and solving related rates problems Some important and helpful things to do are as follows 1Draw a picture or make a model of the situation 2Label all quantities which can change as variables 3Identify in terms of the variables and their derivatives what is being asked and what is given 4Use the picture or model to write down a relation among the variables using an equation or something else which relates them 5Using the Chain Rule implicitly differentiate both sides of the equation with respect to time t 6Substitute into the resulting equation all known values for the variables and their rates of change 7Solve for the required rate of change 8Go back over your work and write up a solution Example One vehicle starts driving north and one vehicle starts driving east from an intersection At the time the first vehicle is 5 miles north of the intersection and is traveling at 50 mph At the same time the second vehicle is 12 miles east of the intersection and is traveling at 65 mph At that instant what is the rate that the distance between the two vehicles is increasing Solution The mathematical model of this situation is a right triangle The legs of the triangle are increasing with time If we call the two legs x and y and the hypotenuse s then all three quantities are changing We are asked to find dsdt at a particular time dsdt is the rate

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