This week we tackled related rates, a category of calculus that still utilizes one of the most important applications of calculus. It is basically moving a formula relating static variables to a formula of rate of change by differentiating the formula implicitly with respect to t (time).
There is a similar strategy for solving related rate problems in comparison to optimization problems...
An example of a problem that uses rates relation is a balloon rising straight up from a level field is tracked form a range fider of lets say 400 feet. At the moment the range finder's elevation angle is pi/4 the angle is increasing at the rate of .14 rad/ minute, How fast is the balloon rising at the moment.
I can see how rates of change and optimization can be applied to the real world. It all makes sense and people's jobs are all about this. It is sweet I'm learning about this and can't wait to learn more.
There is a similar strategy for solving related rate problems in comparison to optimization problems...
- Understand the problem. In particular, identify the variable whose rate of change you seek and the variable(s) whose rate of change you know. A easy way to do this is a simple T-chart.
- Develop a mathematical model of the problem. Draw a picture and label the parts that are important to the problem. Be sure to distinguish constant quantities from variables that change over time. Only constant quantities can be assigned numerical values at the start.
- Write an equation relating the variable whose rate of change you seek with the variable(s) whose rate of change you know. The formula is often geometric, but it could come from a scientific application.
- Differentiate both sides of the equation implicitly with respect to time (t). Be sure to follow all the differentiation rules. Don't forget to chain rule correctly.
- Substitute values for any quantities that depend on time. Notice that it is only safe to do this after the differentiation.
- Lastly Interpret the solution. Translate your mathematical result into the problem setting (with appropriate units) and decide whether the result make sens
An example of a problem that uses rates relation is a balloon rising straight up from a level field is tracked form a range fider of lets say 400 feet. At the moment the range finder's elevation angle is pi/4 the angle is increasing at the rate of .14 rad/ minute, How fast is the balloon rising at the moment.
I can see how rates of change and optimization can be applied to the real world. It all makes sense and people's jobs are all about this. It is sweet I'm learning about this and can't wait to learn more.